# General Microtuning Thread



## Winspear (Jan 23, 2020)

Seen as a handful of us seem to be discussing various topics of microtuning more often, I figured it would be cool to have a general place to talk about it that isn't a tuning specific thread (thanks for those @ixlramp they are very cool!) or the microtonal music thread (so that theory isn't buried by music/music buried by theory).

I'm still to dive in deeply aside from a 32 tone Just Intonation guitar I once made, but I've been playing around with software synths and various tunings for some years. I'm moving towards having guitars focused around 31 and 22 EDO, as well as trying Kite tuning (41edo with 20.5 edo frets) http://tallkite.com/misc_files/The Kite Tuning.pdf at some point. I'll expand on these and what I like about them when I have more time


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## ixlramp (Jan 23, 2020)

If others are ok with this, i would like to propose that 'microtuning' here is understood to mean anything that is not '12 Tone Equal Temperament' (12TET). So including subtle microtonality like the historical tonal system called 'Well Temperament' ('True Temperament' fretting is an example of this).


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## Winspear (Jan 23, 2020)

Totally! I'll have to dig into those more sometime


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## bostjan (Jan 23, 2020)

Oh wow - you did it!

So where do we even start?

I've probably said it all before, but here goes:
I started getting into microtuning when I studied with a Nashville session guitarist (around 1998 or so). He told me everybody in Nashville was temper-tuning back then. I also studied sitar around 2000 or so... obviously shrutis were different than your average 12edo tuning. I started asking all sorts of questions on AOL (that's what I had back then, yes I'm old), not getting any useful answers, I turned to my own research capabilities and started experimenting, whilst doing broader and broader searches over the internet as search engines started to get better.

The first microtunings I tried out on my own were 7-EDO, 17-EDO, 19-EDO, the harmonic series, and just intonation (JI). After a couple of years, I stumbled on the following micro artists:

Neil Haverstick: 19-edo, 34-edo (session guitarist)
John Starrett: 19-edo (college professor)
Dan Stearns: 19-edo (probably?) (hobbyist?)
Bill Sethares: 19-edo (college professor)
Ralph Jarzombek: 14-edo (brother of Ron and Bobby, incidentally, check him out if you like Jarzombeks)

I swear there was a 17-edo metal band that had something on the internet in the early 2000's. You gotta remember that this was before youtube, and really before google and broadband were widespread; even just before filesharing, so finding audio recordings was a fairly rare thing.

I was enamored by 19-edo from my own experimentation, so I took the plunge and bought a 19-edo strat from Jon Catler, circa 2004-ish. I can't find the emails with the exact purchase date.

I played in a band in the Detroit area, called Ox. 3 piece, synth, guitar and drums. 19-edo. It was a lot of fun, but after a couple years of getting together to practice, we played two gigs in 2006 and then couldn't book any more for a couple of months and everyone moved on. Bummer.

I started recording my own shitty demos of 19-edo stuff and also got my Oni that had some partial 24-edo fretting, so I recorded one demo with that. Pretty much zero interest from anyone in the tuning group, because it was metal, pretty much no interest from here because they were shitty demos, and pretty much no interest from my friends and family because it was microtonal. To be honest, there were maybe only one or two songs from those demos that I, myself, even have any remaining interest in. But, at the time, it was all new for me.

Then I got busy with work, busy with family, and then even busy with other band stuff, and it wasn't until 2015 when I finally hunkered down and got back into it in earnest.

I think I have a pretty open mind about microtonal stuff, but egad! The emails from the tuning group, and now the posts on the facebook groups are just too dense for me to follow along. I have an MS in Physics, and I wrote a dissertation on some incredibly specific stuff at the university, and I really don't have the patience to wade through all of the jargon. I feel like I can bog right down into the mathematics of it, but even that takes a lot of energy, at the level that these guys often play. And then there's just the wording of it all - and it doesn't help that everyone wants to come up with their own terminology and a lot of the ones who understand the maths very well don't know much about music theory.

Ultimately, it seems like it's all about picking up a guitar with alternative fretting and just plugging away. First time I tried 19-edo on a keyboard, it felt like I was trying to play with a hand full of thumbs. On guitar, though, everything is more or less familiar. 19-edo, 24-edo, whatever.

But then there are some things that I've been snagged up on:

1. Non-meantone tunings (i.e. most of them) don't really have equivalencies of notes in different keys. For example, in 22-edo, you can play the major scale. Cool. On the guitar, you can use the same fingering wherever and play the same major scale. Transposition on guitar is great. On keyboard, it ends up a tangled mess, because that note that was C in the key of F is now no longer C, now that you are in the key of Ab. Argh. I don't know how to wrap my brain around it. My solution is to not think too much about notes in the conventional way, but, instead, to think of the tonal center being whatever the root note is, and then think about intervals from there.
2. Bohlen-Pierce tuning/scale/whatever-people-will-end-up-calling-it: 13 equal divisions of the perfect twelfth (octave+fifth). Wow! Such a cool idea. Sounds wild. I have no idea how to communicate about it. How do you even use it in a band? Bass can be an octave down, but that's weird. If the bass is a five-string, then it can be a twelfth down. But then, what do you play  ?!
3. JI - I've got some ideas here I'm working on. But again, I don't know how to communicate these things. JI is wild, because there is so much freedom - yet structure comes from constraints.
4. Thanks @ixlramp for bringing up TT and WT and stuff. It seems kind of contentious sometimes.
5. Tuning standards. I really wish A=440 Hz was translatable into microtuning, but, confusingly, the standard seems to be A4 in 12edo is 440 Hz, then C in whatever tuning is universally C from 12-edo when A=440 Hz. I'm just the past year or so understanding that's the way it is, but it seems frustratingly convoluted.


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## Winspear (Jan 23, 2020)

I enjoyed your 19 recordings - it's a great tuning for sure especially as an entry point to microtonality, and extremely compatible with regular notation/theory. 

1. Totally! It seems hex keyboards are the way to go in general for that stuff. I too think mostly in intervals when playing in such tunings (That tends to be my approach anyway). Somebody did create a notation system for 22 but it's quite different. 

2. Yeah, those are really confusing for me to approach in terms of the bass like you said. I've recently been looking at 35EDT which is essentially a slightly compressed 22edo (octaves around 5 cents flat). I really like how it maps on guitar for 5ths tuning and it improves most things in general (those that it doesn't, come up flat rather than sharp so can be bent). However, even if I'm happy with its non-octave repeating nature (for example, not trying to layer lead guitar parts on top), I'm still stumped as to how to approach bass unless it's going to be copying literally everything an octave down (which is only ok for the most simple parts, in my opinion, and still not how I like to write bass). I guess use of a fretless bass would be a wise choice. I am currently glad I don't have much interest in these non-octave tunings, but I do really like the idea of 35EDT...
Luckily it's compatible with a normal 22EDO guitar, so I don't have to commit to trying it too hard. 

3. Indeed. JI was my entry into actually understanding what was going on with anything microtonal and why. Mapping out the overtone series, learning about consonance, understanding how different EDOs try to hit certain JI interval approximations. But it really is a limitless platform. What are you working on and having trouble communicating?

5. I'm actually not sure about this. Is there a standard? With so much variance in potential tunings, I'm not sure it really matters. Collaboration generally needs to be much more planned it seems, and retuning is always easy enough. If you're just working by yourself you can use whatever. Are you saying you've got the impression people are tuning their C to a regular 12edo tuner set to a440? I.e. C=261.63 ? That's interesting. It's not something I've ever looked into or paid attention to but I do see that's what Scala defaults to. I've often used C=256 when setting up synths just because of clean JI integers that are easier to study. For guitars I'd probably just default to tuning the lowest open string to 12EDO.


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## Winspear (Jan 23, 2020)

2.2 - I suppose one approach and perhaps a correct one to bass in a non-octave scale would be to treat THAT as the fundamental and the guitar parts as intervals off that. Like a piano soloist. Though without that root reference in the guitar part, it's certainly going to sound less obvious and weird. Might be hard work to really make that bass stand out as a fundamental. But such scales as BP have never sounded anything but weird to me yet anyway haha


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## bostjan (Jan 23, 2020)

Winspear said:


> I enjoyed your 19 recordings - it's a great tuning for sure especially as an entry point to microtonality, and extremely compatible with regular notation/theory.
> 
> 1. Totally! It seems hex keyboards are the way to go in general for that stuff. I too think mostly in intervals when playing in such tunings (That tends to be my approach anyway). Somebody did create a notation system for 22 but it's quite different.
> 
> ...



Thanks! I've got a bit more in the works, but life keeps distracting me! Naegleria Fowleri second release just needs vocals, but the bird isn't into it right now...

I got a Trellis board (4 rows of square buttons) a few months ago and rigged it up to an RPi Zero to try to make a cheap microtonal keyboard. I think it could work for 22edo, but I never got the software to the point where I could tolerate the latency. I wanted to mess with it more, but it's no longer high on my to do list. If somebody made a hex keypad for cheap, I'd be all over it. The concept is more intuitive.

I've never messed with 35edt. I was scared of 22edo for a while, because it seemed like too many notes. Once I heard some Brendan Byrnes/Ilevens, though, I was sold. I'm going to have to try 35edt out now on Sevish's site (for anyone who doesn't know, you can test out microtunings with a computer keyboard there). 

I think that the bass part being independent of everything else might be part of the point of BP. For a long time, I approached microtuning with the goal of making things sound better, and, like you said, everything I had heard in BP just sounded bonkers. But the more I revisit 7edo, the more I think I want to go ahead and try BP in earnest. The problem with having guitar as you main instrument is that, unless you nab a really rare removable fret or removable fretboard guitar, you are limited on tunings. And even with a fancy Toglahan or Ron invention, it's still nowhere near as easy to change tuning as a keyboardist has it.

I've put my own constraints onto an approach for JI, and came up with something a lot more limited, but, I think, justifiable. It seems like an approach someone must have done before, but I couldn't find any information. I initially developed it with adaptive softsynth in mind. I'll have to make another post about it.

Well, I always used A=440 Hz, because it's the eadiest reference tone to find if you ever need to tune by ear. I started a collab over FB, and everyone else (4 other people) told me, separately, that they'd prefer C=261.6Hz. I was quite surprised! But, A=440 Hz was, historically, more of an orchestral standard than a band standard.


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## bostjan (Jan 23, 2020)

My approach to JI is to make the following constraints (at least as a starting point):

1. Define seven degrees per octave per scale, following Ptolemy.
2. The scale should start on P1, proceed through whatever tonalities of each consecutive scale degree, in order, and restart on the eighth note at P1 an octave higher. (I had always assumed this was the case, in general for heptatonic scales, but I've had some interesting discussions on other boards, so I realized that's not a fair enough assumption)
3. Define allowable tonalities for each degree. I'm following Gioseffo Zarlino's outline of unisons and octaves always being perfect only, 4ths and 5ths either being perfect, augmented, or diminished, and 2nds, 3rds, 6ths, and 7ths being either major, minor, augmented, or diminished. (hoping to expand this later)
4. Define the steps necessary to go through each possible scale, i.e. quarter, half, whole, grown, extended. Allow a greater and lesser option for each of those.

For example: The major scale is WwhWwWh. W is the greater whole step and w is the lesser one. h is the lesser half step. "H", the greater half step, will be needed when going, for example, from the major sixth to the minor seventh, such that the Mixolydian mode (aka Dominant scale) is spelled WwhWwHw.

5. After running through each scale with all of these constraints, there will be one of each scale degree with one of each of the allowed tonalities:

P1: 1:1, 0 c
d2: 128:125, 41.06 c
m2: 16:15, 11.73 c
M2: 9:8, 203.91 c
A2: 75:64, 274.58 c
d3: 144:125, 244.97 c
m3: 6:5, 315.64 c
M3: 5:4, 386.31 c
A3: 675:512, 478.49 c
d4: 32:25, 427.37 c 
p4: 4:3, 498.05 c
A4: 45:32, 590.22 c 
d5: 64:45, 609.78 c 
p5: 3:2, 701.96 c 
A5: 25:16, 772.63 c 
d6: 192:125, 743.01 c
m6: 8:5, 813.69 c 
M6: 5:3, 884.36 c 
A6: 225:128, 976.54 c
d7: 128:75, 925.42 c
m7: 9:5, 1017.60 c
M7: 15:8, 1088.27 c
A7: 125:64, 1158.94 c

I've spent some time trying to work neutral intervals into this idea by playing with the constraints, but it's a little more difficult.


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## Winspear (Jan 24, 2020)

Those seem like a good set of rules to follow! I checked out the scale briefly and it seems very versatile. A nice amount of usable keys without too many notes to handle. Good work putting that together! I'd love to know how you decided upon some of those higher ratios


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## Winspear (Jan 24, 2020)

Here is my fretboard diagram for the 35EDT based sweetening of 22EDO I've been looking at. Sorry for the unreduced octaves high up - I'm just looking at close fingerings here. I put some frets in behind 0 so that I can also look at 'pinky on the root' type fingerings. This is in 5ths tuning as I've been using that for the most part.
As mentioned, this tuning can be achieved on a normal 22 EDO guitar by increasing the intonated string length by about 1.3mm (if standard scale). This would intonate the 35th fret to a perfect 1902c in theory, but of course it would be sensible to check with lower frets. Yes the frets should technically be placed slightly differently but in practice it doesn't cause more than a cents error or so in various spots.

The main weaknesses of 22EDO to me seem to be the sharp m3,p5,M6,h7,m7 which I aim to improve here. The 5 block charts at the bottom of this screenshot represent intervals within comfortable reach from left to right one,two,three,four,and five strings away from the root. The middle column of each block is the JI interval. To the right, normal 22, to the left, this 35edt tuning, and the error of each.
The strings are tuned 707.3 apart as results from 35EDT, aside from 704 across the bottom two strings to bias toward 6th string rooted chords a little. The charts would read 3c higher for chords rooted on any other string.

All just theory right now but I had to have a play around with it when I heard of 35EDT. It looks to me that just about every shape I play in 5ths tuning would be improved by this guitar setup compared to a perfect 22. The ones that do get noticeably worse such as the octave 3 strings away, the major 3rd etc should be able to be bent correct with ease at least.


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## Winspear (Jan 24, 2020)

Edit time expired - I meant to say "Yes the frets should technically be placed slightly differently but in practice it doesn't cause more than a cents error or so in various spots _which are taken into account in this chart_"

A sweetened guitar tuning of an already niche microtuning is a bit of a specific topic to post about but nevermind 

I'm hoping to manage prototyping a few differently tuned guitars this year before I have to decide which one a luthier is building me toward the end of summer  I am not prepared


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## ElRay (Jan 24, 2020)

ixlramp said:


> If others are ok with this, i would like to propose that 'microtuning' here is understood to mean anything that is not '12 Tone Equal Temperament' (12TET).





ixlramp said:


> So including subtle microtonality like the historical tonal system called 'Well Temperament' ...


I've wanted to go this route for a long time. Bring back "Key Color".


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## bostjan (Jan 24, 2020)

@Winspear that should be very interesting! I had for gotten that I had worked out the maths for something just like this a couple of years ago, but I came across it in my notes. Funny how timing of things coincides sometimes! The two edt's I played with most, though were 19edt and 31edt, since I can get close to those on guitars I already have.



ElRay said:


> I've wanted to go this route for a long time. Bring back "Key Color".


 I want a Werkmeister III fretboard. I think the woobliness would be too much for the TT idea, but Tolgahan uses scotch tape and little tangless bits of fretwire. I have zero excuse not to do this, since I have a bunch of fretwire and scotch tape right in front of me.

-----------

I know I've brought it up before, but, it seems like 50% of the "microtonal" tagged stuff on bandcamp is 24edo or some kind of 12/24edo hybrid. I know one of the users here @nicomortem has some killer 24edo stuff on youtube, and most others who have expressed interest seem to want to go that path. M.A.N., Jute Gyte, and King Gizzard went that route, and Page and Plant (from Led Zeppelin for you young guys and from Greta van Fleet, the Prequel for you really young guys  jk) even did a song in it. It's well established in Arabic and Persian traditions. But, you can't find the saz fretted Ibanez anywhere anymore, that guy who was going to get the microfretted Ibanez sig seems like he never sold any of them, and the rash of saz tuned startups that appeared after the flying banana was released seem to have dried up. 

Did guitarist, in general, lose interest? Or maybe never really had interest?


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## ElRay (Jan 25, 2020)

To follow-on with the 1/4-tone stuff:

Black Flag did a bunch of songs with the "in between strings" tuned 1/4 tone flat (or was it sharp?). That's part of the dissonant sound. It's most notable on "Damaged".
We now have a Bass Clarinet in the house. It came with a note about playing 1/4-tones, so I had to find a 1/4-tone fingering chart. I had to explain 1/4-tones to the 7th Grader. They're curious now.


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## Winspear (Jan 25, 2020)

bostjan said:


> Or maybe never really had interest?



I think so. It's so niche. Even if people think something is cool, likelyhood of buying is pretty slim. Compound that against the chances of a given guitar design fitting someones desires in general, tuning aside. We might be more forgiving of guitar specs if in search of a particular microtuning, but for someone not yet into it, they'd probably need to be tempted into it by a guitar that otherwise fits their general desires too.


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## Winspear (Jan 25, 2020)

ElRay said:


> Bring back "Key Color".



It's a lot of fun. _I think _I've just finalised the tuning system for my first custom micro guitar that's happening later this year. A nice balance between the simplicity of EDO, and varied key colours.

I've gone through a few developments whilst deciding
-31 EDO, hesitant about it being too busy (and the luthier wanting to kill me)

-A 19 tone subset of 31 EDO - no partial frets. To make the board easier to navigate, less cramped, and remove the more obscure notes. This means the subset exists on each string but new notes are introduced here and there, totalling 25 notes of the full 31 available in various places.





The ones it misses completely are Bbb through Cbb. The prioritized 19 subset is Fb through to A#. It could use 12 full frets and 7 pairs of partial frets instead of 19 full frets to keep strictly to the subset, but the two options seem equally useful.

I mapped out the fretboard intervals from every root position. There are only 2 positions without a P5, and it's really interesting to play with different key colours for the first time (some having ~270, 310, 385 3rds for example whilst others have 270, 350, 430 for example).

I developed it further experimenting with taking the subset from higher meantone EDOs. I discovered that 81 rather than 31 gives a generally slightly better tuning overall to the 19 note subset, and the more 'outside' intervals result in slightly lower ratio versions than the 31 tuning in general.

In 31 EDO the 19 subset is created with 12 77c and 7 38c intervals.
In 81 EDO the 19 subset is created with 12 74c and 7 44c intervals. So I also have the benefit of the small steps being _slightly_ wider frets.

Here's an example of the first few fret root maps. Roots begin at the open lowest string and move across one fret with each diagram. The grey are your typical inlay marked frets. The cell widths represent the large/small fret steps.
It uses just the same notation and functions the same as 31. You can see the Gx fret here is one of the ones without a P5.
In total, 40 different intervals happen across the board in total, I believe 26 in each key again (need to check). But my main concern here is just that it maps out and plays logically, and a lot of nice chords can be built in each position. Which seems to be true!






Here's the sheet. 
https://1drv.ms/x/s!Ao7H9HphS60ZgdxKhcfYrLv1X-oZRg?e=kd16II
You can use the 3 tabs at the bottom to change between cents only, notes only, and notes+cents views


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## Winspear (Jan 25, 2020)

A question of my own - Given 25 unique notes with extremely small JI errors, with ~40 total interval combinations, would it be right to refer to this guitar as a 25 tone JI guitar rather than a 19/31/81edo subset monster thing?  It wouldn't be the first time I've seen an entirely straight fret guitar called JI, but they usually have more of a R/5/R/5 style tuning. I'm still in the process of digesting the results of this tuning but it seems in designing for 19 tones per octave, the string to string tuning resulted in a bonus 20th note which is different in each octave - leading to 6 extras across the range of the guitar

Octaves and available notes:





This chart will help me map it out on a synth/test guitar parts via midi until I receive the instrument


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## Winspear (Jan 25, 2020)

Last nerd out on that , in case anyone wants to try it in Scala etc. The full 25 note subset of 81 edo

74 25/24 +4c
118.5 15/14 +1c
148 12/11 -1.5c
192.5 19/17
266.5 7/6 
311 6/5 +2c
340.5 11/9 -7c , 17/14 +4c 
385 5/4 -1.5c
429.5 9/7 -5.5c
459 13/10 +4.5c
503.5 4/3 +5c
578 7/5 -4.5c
622 10/7 +5c
652 16/11 +3.5c , 35/24 -1c
696 3/2 -5c
770.5 14/9 +5.5c
815 8/5 +2c
844.5 13/8 +4c
889 5/3 +5c
963 7/4 -5.5c
1007.4 34/19, 9/5 -10c
1081.5 28/15, 15/8 -7c
1126 48/25 -3c
1155.5 38/18 +4c
1200


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## bostjan (Jan 25, 2020)

I'm intrigued.

So the approach is to start with just intervals, then add small changes to make other keys sweeten up. It's a nutural way of doing it, but so many people start from equal and work the other way. And using 17-limit is bold.

I would call that a well temperament, if tgat's the case. I think the general idea of WT is that it started as JI and evolved into something usable in multiple keys, until people went all the way with the idea of emphasizing equality amongst keys and ended up with ET.

The idea of a WT that points from JI toward 31edo (or any of the series 19edo, 31edo, 50edo, 81edo...) pushes all of my music theory buttons.


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## Winspear (Jan 25, 2020)

Sorry - last interval should be 35/18

Actually, I did come at it from equal but had JI in mind as a goal. I guess given 81edo as a base for the subset it technically isn't anything but ET, though with the very close JI accuracy across the board, 3 different step sizes and more intervals available than steps, it certainly acts much more like something other than ET indeed. I think I've hopefully achieved my goal of finding something that is sizeable but not too large, flexible but with varied colours, and most importantly for this build, a simple guitar to approach and play.

FYI adding 12 partial frets at various points alternating bass side and treble side between the 19 full frets gives you the full 25 frets/notes on every string and fills in those Fb/Cb/x gaps in various octaves. That brings in some 29 cent fret clutter that I'd like to avoid on guitar, but may try a full board on a bass to play along with it where small frets and chord comfort are less of an issue.

By the way, in attempting to adjust the reference pitch up to not result in a low F# any looser than usual, I may have discovered why you found people to be set on A440s C261hz as a reference. The Scala software which I expect is a hub for so many people as a reference and way to port tunings to their synths, is locked to being based around C. You can change the reference, but it might explain why people have a tendency to start there and get accustomed to it. Something you could tweak on some synths but certainly makes life easier to stick with it if you're using Scala often.


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## ixlramp (Jan 25, 2020)

I also propose that this be the place to ask questions about microtonality and historical / alternative / ethnic tonal systems, i think this would help keep this thread more varied, balanced and readable.
(But i have just seen another older thread for questions, but it seems to be quiet and therefore gets forgotten, maybe this thread would be better as it will be bumped by other discussion?)

My advice for those interested in learning more is:
Start with a basic understanding of 'Just Intonation' (JI) intervals.
Because then you will understand what 'harmony' actually is, which is extremely fundamental but something almost never taught in music education or music theory. This basic understanding of interval harmony is essential for understanding tonal systems.

I attempted a basic explanation of JI and 'what harmony is' in this thread from post #24 onwards:
https://www.sevenstring.org/threads/school-me-on-tuning-temperaments.331425/page-2#post-4920114


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## Winspear (Jan 26, 2020)

Definitely good advice! I really struggled to understand what was going on until I took the time to build out the harmonic series and chords from it myself on paper


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## ixlramp (Jan 26, 2020)

A fairly good (but still not very good) place to start with Just Intonation is this webpage by Kyle Gann https://www.kylegann.com/tuning.html
Although i do not like the last section where he tries to demonise Equal Temperaments and modern 'Western' civilisation.
Elsewhere on his site is a fairly good summary of the history of European tonal systems https://www.kylegann.com/histune.html


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## bostjan (Jan 27, 2020)

ixlramp said:


> A fairly good (but still not very good) place to start with Just Intonation is this webpage by Kyle Gann https://www.kylegann.com/tuning.html
> Although i do not like the last section where he tries to demonise Equal Temperaments and modern 'Western' civilisation.
> Elsewhere on his site is a fairly good summary of the history of European tonal systems https://www.kylegann.com/histune.html


Kyle Gann and Hermann Helmholtz were where I started, myself, with JI and WT, repectively.
I love 12edo/ET and think it was a swell invention. He only thing I _don't _like about it is how it's taken as the be-all and end-all tuning, when, in reality, it's just one of a small handful of universally good tuning systems. Even moving into the 21st century, it might not even be one of the best options for long.

One approach I wouldn't mind seeing is something that blends in some ideas from Indian classical music. They essentially have a root+11 note JI system with 10 extra notes to cover some different keys. There are a few wilder scales that do crazy things, but you generally only see the 10 main scales. But, anyway, even with a JI type of system, they manage perfectly well with fretted instruments, keyed instruments, and such. A lot of concepts in Indian classical music are at odds with Western music at the fundamental level, though. Songs are very free form, but the forms underlying the songs are very rigidly dictated...

I think the only way to really get experimenting with this stuff is to develop some kind of guitar that accepts a 3D printed fingerboard and then just keep printing different setups. That way the computer can handle turning maths into measurements and you only need a day or two to prototype something...


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## Winspear (Jan 27, 2020)

Yeah, the Indian system is really nice. I do like that general approach of making a reasonable size JI and then adding some core notes to its core keys.
After seeing the swappable board classical guitars, I have designed what is _hopefully _a rigid, trussrodless neck with magnetic fretboards for electric guitar. I still haven't had time to make it (sigh) but if it works then indeed it would be great for trying a bunch of tunings. Whilst I'd like to make proper fretboards for each one, the ease of 3d printing/milled richlite replaceable boards would certainly outweigh the cost and effort of fretting a permanent board - at least until settling on some that you really want to have done properly. _Especially_ for JI boards. Just drag and drop some little fretlets? Yes please..

That endless race of trying to afford time in the workshop doing things that don't earn me immediate money


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## Winspear (Feb 2, 2020)

Tidied up and completed the fretboard diagram for my 7 string, 19 fret*, 26 note subset of 81 EDO guitar design. *Now 20 frets in the first octave - added a micro fret to give the harmonic 7th adjacent to open strings.
https://onedrive.live.com/view.aspx?resid=19AD4B617AF4C78E!29440&ithint=file,xlsx&authkey=!AIH_Oq3Ty9sEB0w
Green is root, yellow are 3rds options, blue is perfect 5th. You'll need to download the file to utilize the key dropdown menu.

Again, this is extremely similar to the same subset in 31 EDO - I am using 81 to allow compatibility with further expansion on electronic instruments where more notes are practical.
A guitar I stumbled across doing the same thing in 31 (without the harmonic 7th microfret that I put just before the '5th' fret marker)


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## Winspear (Feb 2, 2020)

Sorry, here is the file for Excel compatibility. I also designed a 12 note reduction of the upper octave (I don't want any tiny frets) to result in maximum flexibility - most keys available for scale runs in at least _a_ position, vs all positions but no flat keys, for example.
https://1drv.ms/x/s!Ao7H9HphS60ZgeYoxD0DvgHg4pqCuw?e=VojAaO

Here is the actual fretboard. The lack of decreasing fret sizes in the upper octave feels so wrong  
http://www.ekips.org/tools/guitar/f....5&il[]=28&u=in&sl=single&scale=scala&o=equal


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## ElRay (Feb 2, 2020)

Winspear said:


> ...
> https://onedrive.live.com/view.aspx?resid=19AD4B617AF4C78E!29440&ithint=file,xlsx&authkey=!AIH_Oq3Ty9sEB0w ...


I just tried to look at this file and received the error: "This item might not exist or is no longer available".


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## Winspear (Feb 3, 2020)

Last comment was the replacement  


ElRay said:


> I just tried to look at this file and received the error: "This item might not exist or is no longer available".


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## Winspear (Feb 3, 2020)

bostjan said:


> I would call that a well temperament, if tgat's the case.


Figured out exactly what I've got going on - it's 1/4 comma meantone (approximated by whatever edo it's applied to, 81 in this case). With the added frets the guitar presents both sharp and flat keys of 1/4 comma superimposed, as well as those between B/C E/F. Above the octave, it becomes simple 12 note 1/4 comma but with 622c instead of 578c , as 578 is available on the next string.

At 10:27 people can hear an example of 1/4 comma meantone


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## ElRay (Feb 3, 2020)

Winspear said:


> Last comment was the replacement


Sorry, thought they were two different files.


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## jack_cat (Feb 4, 2020)

I have various thought experiments in the works. 
The nine-string I had built has taken center stage in recent years, and I am starting to think about the possible next iteration. So, the ideas below are my back burner ideas that have arisen peripherally in the process of approximating ET on my 9 string. 

The current stage of the main thought experiment is to have a short scale nylon-string fretless acoustic 5-string built for testing historical fretting patterns. I will use tied frets so as to keep the instrument versatile and simple. The whole idea of swapping out fretted fingerboards --- ehh. The instrument will be very similar to the modern mexican vihuela or to a Baroque guitar.

The string length will be 60 centimeters, this is a convenient length and allows all calculations in clean decimals with a pocket calculator. Given this fretless instrument as a starting place, the experiment branches in multiple directions. Room on the fingerboard for ten tied frets in the standard pattern will do it for most historical fretting schemes. Any more will require a long neck on a short body, a possible design alternative. 

1st Branch: Aristoxenus. Put 60 lineal frets to the octave in .5mm intervals, and in the octave 120:60, you have a smorgasborg of just intervals. Alternatively, mark the fingerboard up with any other lineal division into a highly factorial number, and pick the best ratios to place the frets. 

2nd Branch: The Medieval Arabic Systemist fret pattern: This has only recently come to my attention. It's origin was in pre-Islamic Persia, and it became popular in Arabic music in the 12th and 13th centuries, and then popped up in Italy in the early 15th c. as Prosdocimo's extended Pythagorean keyboard tuning with split keys. It is an extension of Pythagorean tuning which results in a regular logarithmic fret pattern, and is based on calculating two reciprocal Pythagorean circles of fifths, yielding (if taken this far) as much as two complete circles of tones in pairs a Pythagorean comma apart. Prosdocimo took it only to 5 sharps and 5 flats. 

The Systemist system applies this to the fingerboard in the pattern limma-limma-comma. 
Near the nut, the comma is about 7 mm long, I think. (Haven't drawn this yet.)
The first fret is a small Pythagorean half step at 90 cents (limma).
The second fret is at 180 cents (two limmas).
The third fret is really a duplicate second fret at 204 cents, a comma higher. 
The "real" third fret is at 294 cents. 
The fourth fret is at 284 cents, a schisma less than a Just third 5/4 above the open string. 
The duplicate fourth fret a comma higher is at 408 cents. 
The fifth fret is at 498 cents. 
The sixth fret is split into two frets a comma apart just like the 2nd and 4th. 
The seventh fret is at 702 cents. 

You will observe that this resembles the pattern of the 31-division fretboard posted above by Winspeare. 

Now the "trick" of this tuning is: all of the fifths and octaves are perfect, and while the Pythagorean major thirds are at 408 cents, by an accident of number, the enharmonic equivalents of those major thirds come in at exactly 384 cents, and can be substituted for them to make near-perfect triads such as D-Gb-A which will be far sweeter than D-F#-A with the third at 408. 

On Prosdocimo's keyboard, this system had a fatal flaw: he gave it no Fb to give a just third above C. That (making a long story short) is why Zarlino's (or whoever's) Just keyboard tuning was invented - but guitarists can get little from Zarlino's scale, because the different sized whole steps and half steps make crazy patterns across the strings. To make this work on the keyboard, there would be even more split keys required: this is the road to Vicentino's archicembalo. 

On the fingerboard there are still some limitations, but the Systemist pattern reverses itself on the next string tuned a fourth up, so the D# that is missing on the first fret of the D string (where there is a Pythagorean Eb), is found on the 6th fret of the A string. This fills the gap that exists in the keyboard version. 

Anyway, it is clear that on an instrument tuned in fourths there are chords with perfect fifths and near-perfect thirds to be found all over the fingerboard, but the tuning - if I understand it right so far - doesn't circulate. Thus, it performs on the fingerboard the same function that meantone does on the keyboard, which is well-tuned chords in a limited number of common keys, but it will gev better results than meantone, because the fifths are perfect. If I understand it right, one would be limited to the common keys. 

Branch #3: Avicenna or Ibn Sina gave the tuning of the Baroque guitar in Baghdad in the 11th century, and gave it a fretting pattern similar to the systemist pattern, except that the extra 2nd and 4th frets yield an Arabic neutral third instead of the near-just third. This would mean that the common first-position A and D chords could be played with a neutral third instead of a Pythagorean third. 

Now, you are going to tell me that they didn't play chords in medieval Arabic music. Ok, OK, but these fretting patterns have a compelling logic of their own, and at some point I am going to have to make the investigations, and _I_ am surely going to play chords on them, because I want to know what they will sound like. 

These fret patterns were given by Henry George Farmer in Chapter Eleven, The Music of Islam, of Volume One of the New Oxford History of Music edited by Egon Wellesz in the 1950s.


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## Winspear (Feb 5, 2020)

That's really interesting! I see what you mean about the happy accident with the major third. It sounds like an interesting and versatile fretting pattern. Personally I think the comma frets would be too small for me to enjoy (I considered 7mm a limit for the upper end of the fretboard (which is why I dropped to 12 tones there rather than 19), let alone near the nut!). But the great thing with tied frets is you aren't committed to the one tuning  Go for it!


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## ixlramp (Feb 5, 2020)

jack_cat said:


> two reciprocal Pythagorean circles of fifths, yielding (if taken this far) as much as two complete circles of tones in pairs a Pythagorean comma apart


I have spent a long time trying to learn about the origins of, and the Just Intonation basis for, Indian Classical music, and it seems that this may have been the ancient basis. From the resulting 24 tones they then removed the comma offset versions of the tonic and fifth, resultng in the 22 shrutis.

It has been suggested that they then realised that the 384c interval is 2c away from an exact JI major third 5/4 at 386c, so then adjusted the system by shifting half of the tones up by 2c. Essentially it becomes Pythagorean based on the tonic plus Pythagorean based on the JI major third.


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## ixlramp (Feb 7, 2020)

Hmm stated that somewhat wrong.

As jack_cat detailed earlier, if you continue a Pythagorean (3-limit) chain of JI fifths far enough (in both directions from the tonic), you end up with complex-ratio intervals only 2 cents away from far more consonant 5-limit intervals, like 5/4 or 8/5.
The ear would obviously hear these intervals as slightly mistuned 5/4 or 8/5, so it seems to make sense to replace the more distant Pythagorean intervals with the 5-limit intervals.

This is detailed here http://www.tonalsoft.com/monzo/indian/indian.aspx
Visualised as a 3-5 tonal lattice, there is a central long line which is the Pythagorean chain of fifths, plus a 'box' around the tonic formed by the replacement 5-limit intervals.

This webpage https://en.xen.wiki/w/A_shruti_list seems a good investigation of the JI basis for Indian Classical music. You can see that the JI intervals are a mix of extended Pythagorean and 5-limit, which supports the above.
That page then later, unfortunately, degrades into 'Tuning-List-temperament-gibberish', best to ignore all that.


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## c7spheres (Feb 7, 2020)

I wonder why there aren't more removeable fretboard guitar's out there. Even in the custom market. You'd think even just for players wanting to go to fretless or certain strings fretless and a couple alternate temperments i would be at least a little more popular. I'm certainly more into it than fanned frets.


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## Winspear (Feb 10, 2020)

c7spheres said:


> I wonder why there aren't more removeable fretboard guitar's out there. Even in the custom market. You'd think even just for players wanting to go to fretless or certain strings fretless and a couple alternate temperments i would be at least a little more popular. I'm certainly more into it than fanned frets.




I reckon it's just such a hurdle it's the kind of thing you only really see from someone who's really into it and has a lot of coin, or where the luthier themselves wants the product. With electric I'm guessing relief can become a real issue so you've also got the challenge of needing to make a rigid trussless neck (that's at least my assumption and plan).

What's everyone using here if messing with software? I've just finished a Reaper configuration using a fantastic little Reaper plugin I'd like to share with any other Reaper users; https://1drv.ms/u/s!Ao7H9HphS60ZgepH4o5novr7HawjVA?e=BvWff6
I've also just finished configuring a setup to use it with an after-synth audio pitch shifter plugin as opposed to the usual synth tuning file/script/pitch wheel control, for any instruments that lack those features, or for global care-free changing between them. I can expand on this if anyone wants a guide.


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## Winspear (Feb 22, 2020)

Some further tweaks to my fretting.
One challenge is selecting which frets to use to reduce the set to 12 frets for the upper octave of the fretboard. The weighting for what I'm looking at is shifted to the treble side of the guitar here. I feel like the current selection is a pretty good balance between flats and sharps in varied positions, allowing fairly free usage of the keys 2 or 3 steps in either direction around the circle of fifths. It seems restrictive when looking at it alongside the full 19, but I must remember it's still as good as historic meantone, that bending is a thing, and that differences in an accidental or two could be potentially inaudible in a scale run. What do you think?


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## Winspear (Feb 22, 2020)

Oh, and a bass  Note that these fret patterns are not identical - it has been shifted to again be weighted towards the cleanest scale in the bass register, and treble register above the octave.





I'm very excited about the potential of these meantone high edo subset frettings in general. Just as easy to produce as an EDO in terms of no partial frets, and the cons of meantone (the few odd coloured notes in the treble due to straight frets, an unclosed circle of fifths/wolf fifth in one key/forced subminor/supermajor 3rds in a couple of keys) do not seem particularly significant to me. Perhaps so in 12 where you lose a significantly higher percentage of keys, but meantone19 seems very flexible.


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## jack_cat (Feb 24, 2020)

Since I am coming into tuning theory from antiquity instead of starting now and working back, 
will somebody please explain what EDO means? thanks, (signed) clueless in Podunk...

On the subject of the "systemist" limma-limma-comma fretting (which I mentioned in my post above, and which I got from H G Farmer's article on the music of Islam in the older "New" Oxford History of Music from the 1950s), I upload here 



a drawing of this fret pattern applied to the tuning A2-D3-G3-B3-E4 on a 60-centimeter fingerboard, of which the first 20 c. is shown, to the seventh fret. The drawing is to scale and the proportions were made with a pocket calculator. The only thing not to scale is that the fingerboard does not taper. With regard to the apparent problem of the closely spaced frets, I can't judge this until I have such an instrument in my hands. 

There are fifteen-note circle-of-fifths chains in the areas between the comma-separated frets. The first runs from the open B string to the Bbb on the inferior second fret. This Bbb makes a quasi-just third for the F diagonally below it. Crossing the comma line starts a new chain. The second begins on C# on the superior 2nd fret of the B string and runs to Cb, which is the just third for a G chord. The third begins on D# and runs to Db, which substitutes for a just C# as the third for an A chord. Another partial chain begins on E#, but I only drew as far as the 7th fret, so it runs off the drawing.

I note again the similarity to Winspeare's design posted and discussed above, except that his has a more frequent, although irregular, iteration of the doubled fret; this could solve some problem - if I were clear what problem that might be! Adding more comma-separated frets to the plan I have drawn could be an option, but it is clear that this makes for a very confusing fingerboard. 

I assume that the same issues were encountered with split-key keyboards - too much trouble for practical music in the long run. 

Perfect fifths and octaves are available for all notes except those on the extreme ends of the circle-of-fifths chains. There seems to be a pattern by which once should know that perfect intervals are chosen by one set of relationships, and imperfect 5-limit consonances are found by a different pattern, depending on how the related notes are positioned in relation to the comma-separated pairs of frets. 

Moustafa Gadalla claims that this pattern was used in ancient Egypt; Curt Sachs and Henry George Farmer both trace it to pre-Islamic Persia. So, it is far older than equal temperament. Galilei, the inventor of the ET based on all semitones of 18:17 plus a little fudge to adjust the saddle, in his "Fronimo" sneered at lute players who added "little trastinis" to improve their intonation. 

That's enough for now, I guess!
jack


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## ixlramp (Feb 24, 2020)

EDO is 'Equally Divided Octave', or maybe 'Equal Divisions (of the) Octave'. A tonal system created by dividing an octave into X equal steps of pitch.


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## jack_cat (Feb 28, 2020)

Thanks. In other words EDO x = ET x, with a less ambiguous term. Having read Barbour's Tuning and Temperament, it appears that there are a relatively finite number of these that work in various ways, depending on what you want to do, and Barbour's book seems to suggests that the limma-limma-comma system is a subset of a 53-division, although it appears to have evolved in 17th century Arabic music into a nominal 24-division without frets, and although Barbour also interprets this Systemist fretting - wrongly, I believe - as a 17 EDO - which it might become easily if the mathematical basis were forgotten and the intervals fudged. So I see that EDOs are fun for some musical minds, and that whatever harmonies you can make will circulate completely. While this is of course interesting, it is not my primary interest at the moment.

What I have been focused on in recent months are medieval and renaissance monochords and fretting patterns, which are very closely related concepts and use the same kinds of rational divisions. It appears that 12-tone ET came on the scene only with Galilei in the 16th century, using his 18:17 semitones. I think that Galilei's (ah, I THINK it was Galilei's) comment that lutes had "always" been tuned in ET since the instrument was invented, must be a delusion, because I find only ratio-based fretting systems reported in historical sources before that time, which are Arabic because there are no European fretting systems before 1500 that I know of - even the first tuning mentioned is from only 1482 in Ramos de Pareja who gives G2-C3-E3-A3-D4 as the tuning of a "lyra", and he does not give the fret scheme.

The really major historical division in tuning systems (since remote antiquity and up to the 18th c.) appears to me to have been the different systems for (A) the harp family including the kanun and the keyboard, and (B) the lute family in which fretting patterns are essential. When Zarlino invented Just Intonation is when the keyboard tunings diverged from the medieval Pythagorean systems which the keyboard appears to have shared with the lute up to that time.

There are quite a number of other fretting systems from the sixteenth century, and some interesting ones are proposed by Juan Bermudo (1555), such as his proposal for a seven-string tuned to the roots of the seven recta hexachords G-C-F-G-C-F-G, in which the six syllables Ut Re Mi Fa Sol La will have the same fretting in each hexachord on each string, for a unity of intonation not possible on a tuning in fourths-and-a-third. One might have thought that Bermudo might take advantage of this opportunity to perhaps try a fret pattern using Zarlino's just scale of 9:8 - 10:9 - 16:15, etc - which is an inviting experiment - but no, Bermudo was a die-hard Pythagorean and went with two 9:8 whole steps and a 90 cent limma, with semitones in two sizes, 90 and 114 cents.

Before the 16th century, there are no European fretting systems detailed that I know of, but there are lots of monochords, which are the same core concept. On the other hand, there are a number of Arabic fretting systems from about 850 CE up to the 13th c. at least; sometime after that the oud players stopped using frets, which probably must represent some fairly major style change. It appears by the comparison of fretting patterns that Arabic and European music may have been more similar in the middle ages and later diverged.

The monochord which Guido of Arezzo gave in the Epistola de Ignotu Cantu is dog-simple and purely diatonic, and compares badly with the sophisticated fretting patterns current in Arabian music for two centuries before him. Ramos de Pareja said Guido was "better monk than musician", and in reading the Epistola, I find that I agree: Guido's system is really dumbed down, and it seems amazingly so compared to his Arabic contemporaries. I am even inclined to guess that Guido's works were written later to falsely explain the European origins of staff notation and solfege. Juan Bermudo states matter-of-factly that Guido wrote in the year 1320, 300 years after what the modern histories say. There is an odd postscript to Guido's diatonic monochord: he says that "some musicians" want to add Bb and F# to it, but that these are unnecessary. If he just invented it, how does it happen that other musicians were already pointing out its deficiencies?

Juan Bermudo's 1555 book devotes a hundred pages or so to diatribes against "barbaric" musicians who use chromaticism and mix their modes, and I believe that these are probably directed against the Muslim musicians still active in Spain, and that European and Muslim music came very close together in Spain and perhaps used the same fretting system and overlapping modal systems in which the European modes were a subset of the Hispano-Arabic modes. Bermudo had high praise for a vihuela player named Guzman; Julian Ribera wrote of a Cordoban family named Ibn Kuzman who had been musicians and poets for several generations, back to the caliphate. A list of vernacular Spanish chord names found in Peru from the 17th c. includes the "Guzmanillo", but I can't determine what chord this might be. (Some others are easy: "Patilla", the buckle, is the first pos. A-Major chord, and the "Cruzado" is E-minor after the common alfabeto symbol which is a Maltese cross.) 

Anyway, some fine day in the next few years, I am going to have to have an instrument built with which I can test all of these ancient fretting systems and monochords for myself. I am working up to it, and will think about it until I have the cash in hand.

Postscript: an ancient concept which also bears examination is of equally spaced frets in which the fingerboard is divided lineally by some very factorial number. Of these the simplest is the lineal 12-division which gives the well known ratios 12:9:8:6 for the fourth, fifth and octave, and has a 12:11 large half step, a 12:10 = 6:5 minor third, and a 12:7 fret which splits the difference between the fifth and the octave. A 40-division is mentioned from Persia which could only work on an instrument with strings tuned a fifth apart, as it has no fifth. Aristoxenus's 60-parts appear to correspond exactly to a 60-part lineal division of a string. Because 60 is not divisible by 9, there is no 9:8 Pythagorean whole step, but there are lots of other 3- and 5-limit intervals.

jack


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## ixlramp (Feb 28, 2020)

Well ... EDO and 'Equal Temperament' have the same pitches but are not really the same thing.
EDO is a purely mathmatical division. Equal Temperament is a 'temperament', so a system generated by a Just Intonation interval which is then 'tempered' (altered) to the point that the result is equally spaced pitches.
So the standard 12ET is an ET where the generating fifth has been tempered, and not really an EDO.
I am sure you know all this though =)


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## Winspear (Mar 1, 2020)

As I'm looking at having my subset fretting done on a 28.7" scale in F, I figured I'd defret my Agile 30" VI, put it in E with the same strings, and try cable tie frets to quickly test it and make sure I wasn't going to regret anything. Happy to confirm it's all working as intended and extremely playable (aside from the relief and action required to make cable tie frets even slightly functional  Good enough for test purposes however).
Whilst I had put a limitation on myself to use frets no smaller that roughly a regular guitars 24th, I'm happy to say that it's significantly nicer to play when that size is further down the neck (given string flexibility, comfort, and slightly lower action). Much less choked and easy to play fluently (whereas I usually don't enjoy the upper range of a guitar at all). I still can't say I'd want to go much smaller, and would certainly consider the comma size proposed by @jack_cat an issue for chordal playing, but after trying this, I can definitely consider some of the very small fretted microtonal guitars I've seen (slightly) more reasonable than I previously thought. Still keeping my intervals no less than 74c above the octave fret though, as much as I'd like to have some 44c frets up there.

I found a video of an extremely similar setup however where partial frets were used where I have all entire frets resulting in more flexibility and some 'less useful' notes at the cost of stricter finger positioning for chords


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## Winspear (Mar 6, 2020)

@bostjan I stumbled across this in my adventures yesterday and wanted to reply - I was pleased to deduce it was you who posted https://en.xen.wiki/w/Talk:Golden_Meantone
I don't know what Talks on xenwiki are or how to partake in them(???), so I figured I'd reply here.

In short I think you're thinking too much into it - I don't personally believe the pattern breaks down after 81EDO. All the patterns are still there as expected, with the nice diatonic MODMOS(?) containing 696 just the same. I'd look at it more as an observation that simply, once getting into such large EDOs, there may be better approximations of Just ratios contained on scale steps other than those the scale was designed to use. Even Pythagorean could be considered a very small example of that, with 384 appearing as an alternative to 408 after however many fifths.
I wish it was possible to take a nice clean subset from 131 using that better fifth! (and the same in the case of its harmonic 7th). Do correct me if it is, I'd be very happy!

I wish tunings with a better 5th than meantone were suitable for guitar subsets such as that posted above, in general. Unfortunately if going above 12 for key flexibility and the good 3rds, 19 note Pyth types tunings (such as a subset from 53) ends up with too small of a fret size for the small intervals. That small interval gets drastically smaller with every slight sharpening from meantone 5th! I'm glad it's flat rather than sharp however - generally easy to bend slightly. I'd love a Pyth 19 guitar if it was playable - the 294 317 384 407 3rd options in addition to a perfect 5th is so awesome. Though the 266, neutral, and 430 of meantone are cool for being more xen 

As a fan of 19EDO, would you be interested in the 19 of 31/50/81 type subsets such as video above and my spreadsheet?


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## Winspear (Mar 14, 2020)

Sharing with you extended 35-tone Meantone, 17 5ths up and down

```
42.85714
 75.00000
 117.85714
 150.00000
 160.71429
 192.85714
 235.71429
 267.85714
 310.71429
 342.85714
 353.57143
 385.71429
 428.57143
 460.71429
 503.57143
 546.42857
 578.57143
 621.42857
 653.57143
 696.42857
 739.28571
 771.42857
 814.28571
 846.42857
 857.14286
 889.28571
 932.14286
 964.28571
 1007.14286
 1039.28571
 1050.00000
 1082.14286
 1125.00000
 1157.14286
 2/1
```

It contains an interval structure of diesis (43c), septimal diesis (32c) and semicomma (11c) - a reduction of the 43+74 of Meantone 19 or the 43+32 of Meantone 31. So really it's a Meantone 50 (32+11) with its extremities dropped i.e a Meantone 31 with Cbb Fbb Ex Bx tagged on.
Essentially 3 historic meantone 12s stacked on 0, 150, and 236c. Can be looked at similarly to 24EDO as 19 is to 12, in that the outer intervals are all in quartertone territory - so you can use it for quartertone with perfect thirds.
Not useful for guitar with those 11cs, but having fun with it on the DAW!


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## bostjan (Mar 15, 2020)

Winspear said:


> @bostjan I stumbled across this in my adventures yesterday and wanted to reply - I was pleased to deduce it was you who posted https://en.xen.wiki/w/Talk:Golden_Meantone
> I don't know what Talks on xenwiki are or how to partake in them(???), so I figured I'd reply here.
> 
> In short I think you're thinking too much into it - I don't personally believe the pattern breaks down after 81EDO. All the patterns are still there as expected, with the nice diatonic MODMOS(?) containing 696 just the same. I'd look at it more as an observation that simply, once getting into such large EDOs, there may be better approximations of Just ratios contained on scale steps other than those the scale was designed to use. Even Pythagorean could be considered a very small example of that, with 384 appearing as an alternative to 408 after however many fifths.
> ...



I'm super interested in those, which have been all at the apex of my interest for 15 years or so.

I think that the term "meantone" might not be so clearly defined, at least for me.

I've been trying to get a discussion going there for a while. People seem to just drop in and add to their own pages and duck out. Every once in a while, somebody drops some info on there that just really strikes me as out-of-place. I really don't like the idea of deleting anything unless if it was purposeful vandalism, but it's tough getting people there engaged. There's the facebook group that has a lot of discussion, but things get lost easily. Then there's here, with only a handful of people posting about xen/alt tuning, but the infrastructure is better and the discussions run deep.

As much as I strongly gravitate toward 19edo, 31edo, and JI (the easily structured ones), I think I really need to dive into something outside of that box- maybe 14edo, that doesn't follow those rules, just to get enough hands-on knowledge.


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## Winspear (Mar 15, 2020)

bostjan said:


> I'm super interested in those, which have been all at the apex of my interest for 15 years or so.



Well, you'll be able to hear some metal on a 19 fret subset of 31 EDO (actually of 112 but I'll get to that later - I'd taken it from 81 previously in the design process but 112 is my final choice) before too long, as a guitar I'm having made for it has just begun construction! I think it may be my new 'standard'.



bostjan said:


> I think that the term "meantone" might not be so clearly defined, at least for me.



What is it you're looking to understand?

Personally I am yet to explore anything but JI, Pyth, and Meantone, and struggle to hear more 'Xen' music as out of tune versions of those

Something like 14 or 7 can be nice though with the right timbre, for basic diatonic melody based music like African , Chinese etc


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## Winspear (Mar 21, 2020)

I'm pleased to share with you my digital tab playback method, and also an idea for a (personally) more readable microtonal tab notation.







Firstly - In 12edo I have long preferred to compose via digital tab with playback. So I am very pleased to have achieved fully flexible microtonal guitar playback with tablature.

The notation staff here is just for harmonic reference/composition. It is muted. It shows the chromatic 19 tone scale.

The first tab staff shows full 19 tone tablature. This is the staff used for composition and playback. This staff has accurate live playback achieved by tuning the strings 19 semitones apart in software and routing out on virtual MIDI cable to your usual microtonal synth control setup. This is possible for EDOs in ANY Tab software program - keeping in mind a 128 note limit.
(Note: This example for my guitar is actually not 19 EDO, but rather an unequal Meantone 19 subset of 31 EDO. Accurate playback for any and all Non-equal tunings like this can be achieved by using multiple single string tabs running on individually tuned channels. The first staff you see here is this way - However I can only confirm Sibelius is able to drag the strings close together to look normal like this.)
I am happy to walk anyone through these methods in detail if needed.

The second tab staff shows the collapsing of enharmonics and how this scale would be played on a 12edo guitar.

The final staff shows my Hybrid tab. Not suitable for playback, but this is personally what I would prefer to read in musical context. After years as a 12edo guitarist, it is hard to respond to numbers so far apart as being positions quite close together. I would prefer to retain the usual numeric fingering relationships. Traditional marker positions are often used on microtonal guitars too - and it is hard to shake the idea that they represent 3 5 7 9 and 12.
Thus, I have written regular 12edo tab with the enharmonic frets given a fractional distinction. It is then readable on both instruments with similar muscle memory. Note functions remain clear to a guitarist born in 12. Areas of microtonal interest are immediately visible. A small arrow like shown can be used to indicate which way the note should be rounded depending on harmonic function if limited to 12 notes. Multiple methods could be used, such as v4 instead of 3.5 and so on - the idea remains the same. Would you prefer your tablature like this, or are you comfortable adapting to high numbers long term?


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## shellsj (Apr 12, 2020)

bostjan said:


> Kyle Gann and Hermann Helmholtz were where I started, myself, with JI and WT, repectively.
> I love 12edo/ET and think it was a swell invention. He only thing I _don't _like about it is how it's taken as the be-all and end-all tuning, when, in reality, it's just one of a small handful of universally good tuning systems. Even moving into the 21st century, it might not even be one of the best options for long.
> 
> One approach I wouldn't mind seeing is something that blends in some ideas from Indian classical music. They essentially have a root+11 note JI system with 10 extra notes to cover some different keys. There are a few wilder scales that do crazy things, but you generally only see the 10 main scales. But, anyway, even with a JI type of system, they manage perfectly well with fretted instruments, keyed instruments, and such. A lot of concepts in Indian classical music are at odds with Western music at the fundamental level, though. Songs are very free form, but the forms underlying the songs are very rigidly dictated...
> ...



printed fretboard! Yes, I was thinking the same thing.


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## shellsj (Apr 12, 2020)

I don’t think I can provide a link yet to my website, harmonicsofnature dot whathaveyou - but a few points I can share:

1. I discovered a strange effect with a tone generator at 5.4, 7.2, and 10.8 Hz. A swirling sound heard over Bluetooth headphones stopped at these frequencies, indicating some kind of infrasonic interference pattern. When I extrapolated these frequencies into a harmonic series I discovered several interesting things:
- they are the notes F, Bb and F.
- the A from this series works out to 432 Hz - which Zarlino also documented, in fact, Zarlino matched this same set of frequencies - I have section on Zarlino with a screen-shot from his book
- the 11th harmonic of 7.2 Hz corresponds to 316.8 Hz, which if you’ve read John Michel’s book, The Dimensions Of Paradise, is the Earth number that is found at Stonehenge, Glastonbury, the great pyramid and Revelations and speaks to the squaring of the circle.
- the lemma, which is the reason that temperaments were invented is not an awkward gap but a sub-frequency of the note you started your harmonic scale with. E.g. 7.2 Hz (my Bb) x 3 is its 5th (F), and if you keep going up in 5ths you come to 237.6 Hz as Bb whereas the octave for 7.2 Hz should be 230.4 Hz. So the lemma is 237.6 Hz minus 230.4 Hz = 7.2 Hz, the frequency we started with. So, the lemma is a fractal of the starting frequency, and any attempt at covering it up by averaging, eg EDO or equal temperament, destroys the fractal nature of harmonic propagation

- to which people will say, “but I want to play in any key” to which I’ll say, do you want the Earth to spin at different speeds to match your idea of creative freedom? because it turns out that the ancient Hebrew measure of time, the Helek, was 3.3333 rec seconds, or 1/72nd the time it takes the Earth to rotate one degree. And one vibration per Helek is 1/3.333 secs, corresponds to 2.4 Hz which is a fifth below my Bb of 7.2 Hz, which would indicate that the world is a big old clock (obviously, that’s what our clocks are based on) and it produces frequencies which indicate the harmonic scale of natural music. So, I believe an instrument tuned to the harmonic series of 7.2 Hz is the only set of notes worth having - you can play different modes of that, but the goal is to hit those frequencies. And I think a bent-fret instrument with a fundamental Bb low note would be the closest thing to the sacred mono-chord of yore. 

- also, I’ve found that the ancient template for temple design, such as Gobekli Tepeh, used dimensions which exactly match the wavelength of Bb, F and C frequencies from this harmonic series, all 5ths.


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## shellsj (Apr 12, 2020)




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## shellsj (Apr 12, 2020)

Zarlino’s frequencies - which match mine derived from Bb as 7.2 Hz and F derived from 5.4 Hz - even though he has transposed the labels next to these frequencies for some reason.


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## ixlramp (Apr 15, 2020)

shellsj said:


> I don’t think I can provide a link yet to my website, harmonicsofnature dot


https://harmonicsofnature.com/


shellsj said:


> the A from this series works out to 432 Hz


There is nothing significant about 432Hz, there is just an ignorant delusional pseudoscientific movement related to that. The same applies to their rival '528 Hz LOVE' and 'The ancient solfeggio frequencies'.
432Hz was also occasionally historically used for the pitch of A, amongst all the many other arbitrary frequencies that have been used for A.
However, i will not discuss the 432Hz movement any further here. Please do not debate the 432Hz movement in this thread, as such threads tend to be locked in this forum because they become so ridiculous, endless and tiresome, and this is a good rational thread for microtonality.


shellsj said:


> the 11th harmonic of 7.2 Hz corresponds to 316.8 Hz, which if you’ve read John Michel’s book, The Dimensions Of Paradise, is the Earth number that is found at Stonehenge, Glastonbury, the great pyramid and Revelations


I do not doubt any possible significance of this number, but stating it as 'Hz' makes it arbitrary, because 'Hz' is based on the 'second', which is a completely arbitrary human-created period of time with no universal significance.
Firstly, the rotation speed of the Earth is arbitrary, all planets rotate at different speeds and the Earth has no significance over any other planet in the universe.
Secondly, the second is derived by dividing the Earth day by 24, then 60, then 60 again, 86400 is a rather arbitrary number.
Attaching significance to numbers stated in units of 'Hz' is a mistake pseudoscientific movements repeatedly make.


shellsj said:


> the lemma, which is the reason that temperaments were invented is not an awkward gap [...] and if you keep going up in 5ths you come to 237.6 Hz as Bb whereas the octave for 7.2 Hz should be 230.4 Hz. So the lemma is 237.6 Hz minus 230.4 Hz = 7.2 Hz


There is no such thng as a 'Lemma' in tuning theory. You are referring to the interval between unison and 12 stacked Just Intonation fifths, this is called the 'Pythagorean Comma' https://en.wikipedia.org/wiki/Pythagorean_comma
There is a Just Intonation interval called a 'Limma' or 'Pythagorean Limma', one of the Pythagorean semitones, size 256 / 243 or 90.2 cents https://en.wikipedia.org/wiki/Semitone#Pythagorean_tuning


shellsj said:


> the ancient Hebrew measure of time, the Helek, was 3.3333 rec seconds, or 1/72nd the time it takes the Earth to rotate one degree. And one vibration per Helek is 1/3.333 secs, corresponds to 2.4 Hz which is a fifth below my Bb of 7.2 Hz


Why would the ancient Hebrew measure of time have any significance over any other measure of time used by many other cultures (human or alien) throughout time and space?
What about the hundreds or thousands of other units of time that do not form a simple interval with the 7.2Hz you claim is special? It seems you have ignored those and are picking out the occasional inevitable coincidences.


shellsj said:


> So, I believe an instrument tuned to the harmonic series of 7.2 Hz is the only set of notes worth having


'Worth having' for you personally i hope.
This is sad, by doing so you have discarded the vast majority of the world's beautiful instruments, tonal systems, scales, and music. It should be obvious to you that beautiful music exists outside your very narrow definition of what is personally acceptable. You should trust what you find beautiful, not ignorant and delusional pseudoscientific theories.


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## ixlramp (Apr 15, 2020)

shellsj said:


> E.g. 7.2 Hz (my Bb) x 3 is its 5th (F), and if you keep going up in 5ths you come to 237.6 Hz as Bb whereas the octave for 7.2 Hz should be 230.4 Hz. So the lemma is 237.6 Hz minus 230.4 Hz = 7.2 Hz, the frequency we started with.


Up a JI fifth is multiplying frequency by 3 / 2 or 1.5, not 3, although i understand it does not make a difference if you are shifting results by octaves.
230.4 / 7.2 = 32 which is 5 octaves instead of the correct 7, so there is an error here.

The Pythagorean comma (what you call the Lemma) is the difference between 12 JI fifths and 7 octaves.
Up 12 fifths is multiplying frequency by 1.5 ^ 12 = 129.746337891.
Up 7 octaves is multiplying frequency by 2 ^ 7 = 128.

Start with 7.2 Hz:
Up 12 JI fifths:
7.2 * 129.746337891 = 934.173632812 Hz
Up 7 octaves:
7.2 * 128 = 921.6 Hz
934.173632812 - 921.6 = 12.573632812 Hz which is not 7.2 Hz.

The example from your site:

"We start with our “still point” frequency of *7.2* Hz. But that’s too low so we octave it up to 460.8 Hz to give us a B-flat we can hear – and we start the cycle-of-fifths with that frequency. As above, 460.8 x 3/2 gives us a fifth harmonic (an F of 691.2 Hz). If we multiply that by 11, it should take us around the cycle of fifths 11 more times to bring us back to B-flat. What frequency do we actually get (once we octave it down a bit)? 475.2 Hz. So, what’s the _size_ of the Lemma? 475.2 Hz minus 460.8 Hz = *14.4 Hz. *Do you recall that our “still point” vibration for B-flat is 7.2 Hz that’s half of 14.4 Hz – an octave below."

Start with 460.8 Hz:
Up 12 JI fifths:
460.8 * 129.746337891 = 59787.1125 Hz
That result down 7 octaves:
59787.1125 / 128 = 467.086816406 Hz
467.086816406 - 460.8 = 6.286816406 Hz which is not 7.2 Hz or an octave of 7.2 Hz.

The error may be this:

"As above, 460.8 x 3/2 gives us a fifth harmonic (an F of 691.2 Hz). If we multiply that by 11, it should take us around the cycle of fifths 11 more times", specifically "If we multiply that by 11".

To go up 12 fifths you must multiply by (3 / 2) ^ 12, '^' being 'to the power of' or 'exponent'.
You do not multiply by (3 / 2) * 11, you multiply by (3 / 2) 12 times, which is different.


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## shellsj (Apr 16, 2020)

ixlramp said:


> Up a JI fifth is multiplying frequency by 3 / 2 or 1.5, not 3, although i understand it does not make a difference if you are shifting results by octaves.
> 230.4 / 7.2 = 32 which is 5 octaves instead of the correct 7, so there is an error here.



I've taken the simple approach of a guitar string: If you play a harmonic half way along its distance, the frequency of the harmonic is double what you started with. If you play a harmonic a third of the way along the string, the frequency of the harmonic is 3 times what you started with. So, although the generally accepted fraction is 3/2 to keep the resulting frequency within the same octave, I like to keep it simple: harmonics produce frequencies that are x times the original frequency, according to the whole number fraction of the string-length they are played at.

So, my whole approach is simply based on harmonic propagation - which is a natural phenomenon.

Playing harmonics at simple, whole number divisions of the string length will give you the Octave (half way), 5th (touch a 1/3 of the distance), 3rd (touch a 1/5 of the distance), 7th (touch a 1/7 of the distance), and 9th. Now, this doesn't give us a complete musical scale, but you can layer the harmonics on top of harmonics: Being as the 5th is the loudest harmonic after the octave, it makes sense to determine its 3rd, 5th, 7th and 9th, and so on. I know that the generally accepted approach is just to focus on the 5ths - the cycle of fifths, but I'm interested in the way the overall harmonics propagate.

The trouble comes when you go too far with this approach - e.g. if I start with Bb at 230.4 Hz, and generate its 5th (F = 345.6 Hz), and its 5th (C = 518.4), and then generate the 7th harmonic of C to get me back to Bb, I get 518.4 x 7/8 = 453.6. So, now I have two frequencies for Bb, the one I started with, 230.4 Hz (or 460.8 Hz, as an octave) and 453.6 Hz. But what's the difference - the gap - between these? 460.8 Hz - 453.6 Hz = 7.2 Hz. And 7.2 Hz is a sub-octave (6 octaves below) of our starting frequency of 460.8 Hz. (460.8Hz / 64 = 7.2 Hz) This is an interesting, physical phenomenon. The gap is always a sub-octave of either the frequency you start with, or a harmonic of that frequency. You can do it with any starting frequency, 440 Hz, what have you. And there's a page on my site where I've explored this. I hadn't read about this anywhere else - I just did the math one day and found it. The page is called, "The Cycle of Fifths - the Lemma is fractal".

You say, "the Pythagorean comma (what you call the Lemma) is the difference between 12 JI fifths and 7 octaves"

Lemma just means "gap" - there are gaps along the whole harmonic series, as I've explored in the page I mention. Sometimes, there's a gap on top of a gap. And that's what makes harmonic propagation fractal: the bits left over are reflections of the bits you started with.

You say this: "To go up 12 fifths you must multiply by (3 / 2) ^ 12, '^' being 'to the power of' or 'exponent'.". You're right, I over simplified it. If you take a step-by-step approach with each harmonic generated, as I have in the attached table, and as I describe above, you can see this phenomena occurring with each harmonic that contradicts one of the earlier harmonics we've generated - such as the Bb to Bb example above. It's not just about the 5ths, because the way that vibration propagates is in terms of all whole number multiples of the original frequency.

I didn't set out to prove 432 Hz, or the Solfeggio, or any of that. I simply found a phenomena with a tone generator (there's a video on my home page) and generated the harmonic series from those frequencies. What I discovered is that there does seem to be a lot of coincidence of these frequencies in exhumed ancient flutes, and Chinese cast bells, even in the world of cymatics where the first frequency that takes shape is 345 Hz (I found 3 frequencies on my tone generator that day (5.4, 7.2, and 10.8 hz) - 5.4 and 10.8 are both sub-octaves of 345.6 Hz. So, of course, I get your point that all this could be completely arbitrary, and with the internet I could postulate that the meaning of life is apple sauce and custard and fine corroborating evidence. But there does indeed seem to be a relationship between these numbers as frequencies in Hz, and these numbers as measures of distance. Which sounds very odd until you consider that a rotating item (like the Earth) is moving at a frequency, and also expressing distance as it rotates. So, I'm not going to be able to explain it here - I've started to investigate it on my blog pages. But nature isn't arbitrary - just take the fact that the moon is positioned and sized precisely so that it blocks out the sun exactly at an eclipse. Higgs Boson also indicates that if anything was just the slightest bit different in the balance of matter and anti-matter, the whole thing would collapse. 

Either way, we agree that JI is a more harmonious kind of music. And until the invention of the harpsichord and fretted instruments like the guitar, fretless stringed instruments and horns were all about letting the musician hit the note that felt right. So, the question for me then, being as the guitar is such a tricky instrument, if I'm going to go to the trouble to put frets in strange places to make it JI, is there a specific key and base frequency I should be starting from. I believe my experience with 7.2, 5.4 and 10.8 Hz provides that base frequency, and strangely enough, the 7th of 7.2 is A at 432 Hz. Like I said, I'm not here to defend 432 Hz. But Zarlino documented it, as I've posted above, so if I'm going to try to have a JI guitar, I'm going to tune it to Bb based of 7.2 Hz. The result is only a few Hz different than 440 Hz, so it's not doing anyone any harm, after all.

I appreciate you taking the time to consider what I had posted. We're all just interested in the same thing here.


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## ixlramp (Apr 18, 2020)

Sorry, i made the mistake of challenging a 'pseudoscientific cosmic frequency theory' in this thread. It is difficult for me not to, having a physics degree and a good understanding of the science of music.
I think it is best we do not discuss this type of topic in this thread, whether that is 432Hz, 528Hz, 7.2Hz or whatever, to avoid swamping the thread with intense and unresolvable arguing.
Such topics have little to do with guitar microtuning anyway.


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## shellsj (Apr 19, 2020)

@ixlramp I said,
"E.g. 7.2 Hz (my Bb) x 3 is its 5th (F), and if you keep going up in 5ths you come to 237.6 Hz as Bb whereas the octave for 7.2 Hz should be 230.4 Hz. So the lemma is 237.6 Hz minus 230.4 Hz = 7.2 Hz, the frequency we started with."

And you responded with,
"230.4 / 7.2 = 32 which is 5 octaves instead of the correct 7, so there is an error here."

I didn't say anything about "7 octaves". What I was explaining was that 7.2 x 64 = 230.4 (some number of octaves, it doesn't matter how many, above). And *the difference between 237.6 and 230.4 Hz is 7.2 Hz*, which is the number I started with - indicating that the actual *frequency* of the gap is harmonically related to the starting frequency. We started with 7.2 Hz and ended with a gap of 7.2 Hz.

That has to be something of interest to someone with a curiosity for harmonics and physics, as you have.

Now, I do my analysis of the cycle of fifths, fifth by fifth in the gaily coloured chart, above. But we can take the exponential route as you have:

You said, "Up 12 fifths is multiplying frequency by 1.5 ^ 12 = 129.746337891".

Now, it's really important not to round this stuff, in order to reveal the gap-on-top-of-harmonic-gap phenomenon: 1.5 ^ 12 = exactly 129.746337890625

(I use the calculator on my phone, and if I turn it sideways it gives me 16 digits to work with)

If we use 0.9 Hz as our starting frequency for the cycle of fifths (a low sub-octave of 7.2 Hz to keep the numbers small), then the 12 cycles-of-fifths would be:
129.746337890625 x 0.9 Hz
*= 116.7717041015625 Hz*

Now, a higher octave of our starting frequency (0.9 or 7.2 hz) is *115.2 Hz*. So, that's where we would have liked the cycle-of-fifths to have ended up.

*First gap:*
What's the gap: 116.7717041015625 Hz - 115.2 Hz
= 1.5717041015625

Tiny decimal fractions of a Hz, I know! Fun times - but stick with me, because this is science 
Let's octave that up a few times to see it more clearly: 1.5717041015625 x 64 = 100.5890625​*Second gap:*
The 7th of Bb (7.2 Hz x 7 = 50.4 hz (Ab) x 2 to bring it into this octave) = 100.8

So, what's the difference between the "gap" and its nearest harmonic of Bb:
100.8 Hz (Ab) - 100.5890625 Hz = *0.2109375* Hz

Another tiny gap, OMG, give me a break, I hear you say.​
*Third gap:*
Let's again octave it up to look at it: 0.2109375 x 128 (7 octaves, doesn't matter how many octaves really) = 27 Hz. Not yet recognisable to most people: so 27 Hz x 16 (4 more octaves) = 432 Hz.

Now, 432 Hz is the major third (x 5) of the perfect fifth (x 3) of our starting note, i.e.
Bb at 7.2 Hz x 3 (fifth, an F) = 21.6 Hz
21.6 Hz x 5 (third of F is an A) = 108 Hz (x 4 = 432 Hz)

So, it ended up that the gap at the end of 12 cycles around the cycle of fifths, starting with a frequency of 0.9 Hz (Bb), ended up being the major 7th (harmonically) of that starting note (an A).​
I go through all this for you not to proudly proclaim 432 Hz at the end. I didn't know where this particular calculation would end up, but I have found that that if you analyse the gap at the end of your cycle of fifths against the closest harmonic of your starting frequency - (and sometimes its gap on top of gap). *It does come to an exact harmonic of your starting frequency, every time. *

*It doesn't matter what frequency you start with. * So, this needn't be a conversation about flooding this chat with unresolvable, circular conversations about particular starting frequencies as being cosmic. You can do this with *440* Hz as your starting frequency. Or even 1 Hz, or whatever you like.

Give it a try. It only works though if you compare the resulting gap with a whole-number harmonic of your starting frequency.

*A random example:*
Let's try with a starting frequency of 1 Hz

1 Hz x 1.5 ^ 12 = *129.746337890625 Hz*, the frequency we end up with after 12 cycles of the cycle-of-fifths.

Now, 7 octaves (as it happens) above 1 Hz is *128* Hz. So, the gap between our starting frequency (at whatever octave) and ending frequency) is:
129.746337890625 Hz - 128 Hz = 1.746337890625 Hz
Let's octave up this number 1.746337890625 Hz so we can work with it: x 64
= 111.765625 Hz

Now, I haven't done the harmonic sequence for 1 Hz before, but let's see what's a close harmonic to that:
7 x 1 Hz would be the 7th harmonic of 1 Hz. Octave 7 Hz up 4 octaves (x 16) and you get 112 Hz. Pretty close to 111.765625 Hz.

So, what's the gap between the closest harmonic and our number at the end of the cycle of fifths: 112 Hz - 111.765625 Hz = 0.234375 Hz

Tiny number, let's octave it up so we can work with it: 0.234375 Hz x 64 = 15 Hz.

So, what is this gap of 15 Hz in relation to our starting frequency of 1 Hz?
1 Hz x 3 (its fifth) = 3 Hz.
And the third harmonic of 3 hz (x 5) = 15 Hz.
So, the gap we end up with is a third of a fifth, AKA the "major 7th" of the frequency we started with (which, by the way is what an A is to Bb in the first example).

So, a lot of math. Probably most people on this thread don't care. But being as this is a thread about JI, and the alternative to JI is equal temperament, and the reason that equal temperament came about is that people didn't like the ugly lemma at the end of the cycle of fifths and devised ways to apportion it across the other notes of the scale to "hide" it; I would say this this is fascinating stuff because what it reveals is that the "gap" at the end of the cycle of fifths is actually, *always*, a sub-harmonic of whatever frequency you started with.

This is why sometimes in science it's good to be simple-stupid, and look at the data - the frequencies in this case - as that's when you discover intricate insights into the nature of things. This isn't "pseudo science" - it's just math applied to harmonic propagation to discover something that has just been sitting there in plain sight.

I'd be happy if you'd acknowledge the efficacy of this approach, but if not, that's OK, I've spent a lot of time demonstrating it, and that's all I need to say.


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## Winspear (Apr 19, 2020)

I may be wrong because I haven't dabbled much in JI or done any deep analysis of interval relationships - but, wouldn't one assume that JI being a ratio based system means that _of course_ any interval, including said lemma, found in any combination of JI notes, is also a ratio of some other relationship between the notes?


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## shellsj (Apr 19, 2020)

I agree, in retrospect, but that doesn’t seem to be common knowledge, though evidently true. To, me it shows that harmonics are fractal: the gaps are themselves harmonics


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## ixlramp (May 1, 2020)

shellsj said:


> Lemma just means "gap"


I have searched the web for the definition of 'lemma', i cannot find any meaning being a 'gap' of any sort, in music theory or not.
I also searched the xenharmonic wiki, which is where all the technically-minded microtonal and JI tuning geeks store their knowledge, the only mention of 'lemma' is as a 'mathematical proposition'.
The 'gaps' found in JI theory you are explaining, for example between 12 fifths and 7 octaves, are called 'commas':
https://en.xen.wiki/w/Comma
Can you link me to any definition of 'lemma' as a 'gap'?


shellsj said:


> if you analyse the gap at the end of your cycle of fifths against the closest harmonic of your starting frequency - (and sometimes its gap on top of gap). *It does come to an exact harmonic of your starting frequency, every time. *


Ok, that is possibly true, i only found one particular mistake in one of your examples where a power should have been used.
I may have misundersood one of your examples, i find the wording of your explanations a little unclear.


shellsj said:


> This is why sometimes in science it's good to be simple-stupid, and look at the data - the frequencies in this case - as that's when you discover intricate insights into the nature of things. This isn't "pseudo science"


Most of the content on your site is not 'science'. On your site you describe yourself as being 'scientific', you are not. These are things i cannot restrain myself from objecting to.


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## Winspear (May 2, 2020)

It is usually spelled Limma I've found  It seems to be more comparable in size to a chroma as far as I can see - bigger than a comma, and diesis too
http://www.tonalsoft.com/enc/l/limma.aspx
From Xen gallery of just intervals:
20/19 88.800698 19uy1, nuyo unison A1519 small undevicesimal limma, small undevicesimal semitone
256/243 90.224996 sw2, small wa 2nd m2 Pythagorean limma, Pythagorean minor second, |8, -5>
135/128 92.178716 Ly1, layo unison A15 major limma, |-7, 3, 1>


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## shellsj (May 2, 2020)

ixlramp said:


> I have searched the web for the definition of 'lemma', i cannot find any meaning being a 'gap' of any sort, in music theory or not.
> I also searched the xenharmonic wiki, which is where all the technically-minded microtonal and JI tuning geeks store their knowledge, the only mention of 'lemma' is as a 'mathematical proposition'.
> The 'gaps' found in JI theory you are explaining, for example between 12 fifths and 7 octaves, are called 'commas':
> https://en.xen.wiki/w/Comma
> ...


@ixlramp fascinating - lemma used to be there on the web, I’m sure. Must be linguistic revisionism . and I’m pretty sure that Jacob Bronowski uses the term in the Ascent Of Man where he talked about Pythagoras. We can call it a “[email protected] if you like. And, as I mentioned, we can call It a “gap”. The point is, not the word, but the concept which I bothered to explore in my “unscientific” way. The point is not what it’s called, but that the “gap” between harmonics and their closest living relative is itself a harmonic, or a combination of harmonics. Looking at the world through a prism of “cents” does not reveal that truth. A truth which makes it clear that any kind of equal temperament destroys harmonic content. Anyway, it’s possible that other people might find it interesting. You’re an interesting guy - sometimes so willing to help. I suppose all belief systems are protected fiercely by the people who have invested in them - but slowly they incorporate new insights.


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## shellsj (May 2, 2020)

Here you go - looks like the preferred translation from the Greek is “limma” or “leimma” https://en.m.wikipedia.org/wiki/Limma


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## bostjan (May 4, 2020)

Not sure where to ask this, but does anyone have any idea what a 23 tone/octave JI neck might run, ballpark?


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## Winspear (May 5, 2020)

bostjan said:


> Not sure where to ask this, but does anyone have any idea what a 23 tone/octave JI neck might run, ballpark?


Total guess but I'd imagine like $550-650?
Tuning details?


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## Necris (May 5, 2020)

bostjan said:


> Not sure where to ask this, but does anyone have any idea what a 23 tone/octave JI neck might run, ballpark?


JI fretting is basically a luthier's nightmare job, you're more or less asking the moon and the price is going to reflect that -- as I'm sure you're aware. If you find someone who will do it I wouldn't be surprised if you were scraping up against or even surpassing $1000 for a neck.

It may not be the most direct comparison, but for context maybe 6 or 7 years ago now I got a quote for a 33-edo bass conversion from a well-regarded luthier who had done some work for people in a facebook group I was part of, and that was around ~$1000 before shipping both ways.


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## bostjan (May 5, 2020)

Winspear said:


> Total guess but I'd imagine like $550-650?
> Tuning details?



Open strings AEADGBE (7) or AEADF#B (6)

A = ref
Bbb 41.06 ¢ (128:125)
Bb 111.73 ¢ (16:15)
B 203.91 ¢ (9:8)
Cb 244.97 ¢ (144:125)
B# 274.58 ¢ (75:64)
C 315.64 ¢ (6:5)
C# 386.31 ¢ (5:4)
Db 427.37 ¢ (32:25)
Cx 478.49 ¢ (675:512)
D 498.05 ¢ (4:3)
D# 590.22 ¢ (45:32)
Eb 609.78 ¢ (64:45)
E 701.96 ¢ (3:2)
Fb 743.01 ¢ (192:125)
E# 772.63 ¢ (25:16)
F 813.69 ¢ (8:5)
F# 884.36 ¢ (5:3)
Gb 925.42 ¢ (128:75)
Fx 976.54 ¢ (225:128)
G 1017.60 ¢ (9:5)
G# 1088.27 ¢ (15:8)
Gx 1158.94 ¢ (125:64)

This would work fine in the key of A major, A minor, A diminished, A augmented, or any hybrid of those four, but isn't going to work so well for any other key. I think I have come to grips with that.



Necris said:


> JI fretting is basically a luthier's nightmare job, you're more or less asking the moon and the price is going to reflect that -- as I'm sure you're aware. If you find someone who will do it I wouldn't be surprised if you were scraping up against or even surpassing $1000 for a neck.
> 
> It may not be the most direct comparison, but for context maybe 6 or 7 years ago now I got a quote for a 33-edo bass conversion from a well-regarded luthier who had done some work for people in a facebook group I was part of, and that was around ~$1000 before shipping both ways.



Yeah, I knew it's a big ask. I'd not be too surprised by anything at this point, but would like to know if it'd be something I could budget. Whether $100 or $1200 would be acceptable would depend, I guess on the builder's experience and confidence, and how likely I gauge that the thing will actually get done.

My 19-edo seven string took five luthiers and five work order (with five deposits) before one project completion (or any sign that any work at all had even been done). A big reason for the low success rate was that I was trying to save a little money.


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## Winspear (May 6, 2020)

Nice!
I guess I would ask Sword, given his EDO prices are pretty low and he definitely has experience beyond EDO fretting.
Very much the kind of work I hope to offer myself some day - it would be very satisfying !


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## Winspear (May 12, 2020)

I know this is a guitar forum, but is anyone here into keyboard? Or any other microtonally capable instrument?
I've been very excited for a long time about the release of the Lumatone, and it's finally happening this summer.
It will retail at $4000 but you can secure a $1000 discount on the release if you subscribe to their email list at https://www.lumatone.io/
The price point on it whilst high is quite incredible considering the previous similar options that cost as much or more for a frankly significantly inferior instrument.

Here is the 31EDO layout I will use on the Lumatone. You can play this on a phone or tablet, though you'll need a Windows touchscreen to control your own synths ( see below )

Although a keen guitarist, I've long felt that is mainly due to comfort and familiarity and that I need to branch out. After making a touchscreen isomorphic MIDI controller with https://savethehuman5.com/chameleon/ , I'm fairly convinced the Lumatone could become my main instrument. Chameleon costs $5 (highly recommend for anyone with a touchscreen on Windows - I now consider this software & a ~$300 touchscreen perhaps the ultimate entry point microtonal instrument). I can't wait to play such a layout with a physical interface, velocity sensitivity and much more. 



I also have a cheap trombone arriving (today!) that I am keen to explore microtones on.


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## bostjan (May 12, 2020)

I build a prototype isomorphic keyboardkeypad out of a Trellis and a Raspberry Pi about two years ago. It cost me something like $35-40 for the parts. I meant to make a case for it, but never got around to it. If I ever find the time, I do plan on making something a little more useful, but, honestly, it's about the 101st thing on my to-do list.

Professional isomorphic keyboards are just too expensive, IMO. I know a lot of time goes into them, but $3000-4000 buy a heck of a lot of instrument in any other context. And most of the ones I see in the microtonal group are at the kickstarter/gofundme stage- some of those have been stuck at that point well over a year now.

Maybe a bit of a rant, but I'm curious about other people's experience getting exposure or collaboration going...

Ron says his shop is too busy cranking out all sorts of microtonal guitars. Where are all of these going?! There are seriously about a dozen-and-a-half microtonal guitarists active on bancamp, making any music. There are maybe just as many of which I am aware that are not really active. The Cryptic Ruse guy is one of them, but he must own a dozen micro-converted guitars. We are either about to see a big boon or else people just aren't releasing stuff anywhere I can find it.

I've tried getting a collab put together in 19-edo a few times, and, even though I had three people commit, I only had one person ever actually record anything, and as awesome as that was, he only ever sent me an unfinished dub for part of one song. I had also requested similar in 24-edo, thinking it would be more likely to garner activity as a more popular tuning, but again, only one person expressed any serious interest, but then put his involvement on indefinite hold. For reference, I've done tons of standard-tuning collabs, and, although there seems to be about a 1:1 ratio of aborted projects versus completed projects, I don't think I've even ever had a non-successful project die as miserably as my micro-tonal collabs.

I've also tried to get some local guys to pick up my instruments at home and just jam, but that only leads to awkward facial expressions, followed by "do you have a real guitar/bass I can play?" Even when I have had paying gigs and tried to get bass players to help back me up, the story always goes the same, where I over-explain the fact that they have to use a special bass, they see too many words and don't read, commit anyway, then insist on using their own bass with 12-edo fretting and just ignore the verbose chord spellings on the charts. I hate to say it, but, in a live context, it's not a deal-breaker anyway, as lame as it seems to be playing half-microtonal-half-not music, when part of the gimmick is that it's microtonal. 

Last holiday season, I asked around about whether anyone would want to jump in on a holiday microtonal music compilation release, but, since CD's are going the way of the payphone/dinosaur, and I have an potential audience of single digits myself, I couldn't get any significant interest. I will likely try again next season and record enough to release something along those lines by myself if necessary.

I worked with Michael from Biptunia a couple years ago, when he was putting together a charity compilation, and I had a lot of fun contributing. The contract he sent out asked for an exclusive song, and it seemed like only one of the "bigger" microtonal artists did anything one-off for the album. I was really proud of the piece I submitted, and went all in with the subtle references and whatnot. I don't even think anyone bothered to notice any of it. Oh well.

The overall interest in microtonal music out in the wild is maybe negligible, at least as of January 2020, when live music was even still a thing. Back in 2015/2016, when I made my last big push to get it going, there at least was a fair amount of curiosity about it. But if Ron is cranking out 19-edo fretboards like they might be the next toiletpaper or N95 masks, then maybe something is coming somewhere in the pipe.


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## Winspear (May 13, 2020)

Feel you on the controller prices! The Lumatone seems like the first to have earned its price tag, to me. It is still in launch stage, but with some units out in the wild already along with its previous version the Terpstra, I'm fully confident in it. 

It's definitely an incredibly small niche in the big picture. And I think you are right about people just not releasing. In this online guitar circle, especially as it extends to Facebook, I have a tendency to forget that not everyone is posting about every guitar they buy, posting clips, publicly working on music. I assumed there weren't more than the couple of 31EDO guitars I could find on Youtube (because if you have a 31EDO guitar, you're _definitely_ going to show it off, right? ), but then found at least two other people with them. Similarly regarding the 19 of 31 subset tuning - I found a couple of guitars just like the one I'm having made posted in the Facebook group but with owners who are seemingly inactive online. Which is fine, and makes sense in the bigger picture I guess, but indeed I've come to expect everyone to be doing something visible (even though I myself haven't shared anything I'm working on either  ).

Thanks to the likes of Jacob Collier and Adam Neely there has definitely been a huge increase in interest in the last couple of years. I think things will pick up quite a bit. Though the physical financial nature of microtonal guitar definitely is a set back there. ~90% of my microtonal contacts are synth guys willing/able to invest very little money


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## bostjan (May 14, 2020)

I just did some brute force algorithms to see which tunings match the JI I posted earlier with the fewest notes. I input the JI intervals and calculated the minimum difference betweeneach JI interval and all of the EDO intervals, and then scored them in buckets. I then took the score divided by the number of notes, added some subjective scoring based on how uniquely the EDO intervals fit the JI set, and penalized a small amount for each unused option in the EDO set.

I guess I shouldn't have been surprised, but 12-EDO came out with the highest score, by a margin of more than 20% 

Next up, though was 19-EDO, which makes sense, since it's where I gravitated, and, even though I was trying to be as objective as possible with the algorithm, everything from the choice of JI interval definitions to the scoring buckets is subjective.

So I rewired my algorithm for Ptolemy's JI system and surveyed the most popular tonalities and weighted the scoring according to that, and , that time, 12-EDO came out ahead by an even larger margin of 35%. But again, second-up was 19-EDO.

After that, whether I used my scoring with my own intervals or the less subjective scoring with Ptolemy's intervals, there was a difference. My scoring recommends 6-EDO, 22-EDO, 53-EDO, and 34-EDO. The other scoring came up with 5-EDO, 17-EDO, 29-EDO, and 24-EDO, for 3rd to 6th place, respectively. I only went up to 65-EDO, since, aside from 53-EDO, which is really neat, the score divided by number of notes becomes too penalizing toward higher-numbered systems.

I find it interesting that the result, based on my inputs, nailed my personal taste so well. I love 19-EDO, and I am quite fond of 22-EDO. I am a fan of Brendan Byrnes (22-EDO), Claude Debussy (6-EDO), and Neil Haverstick (34-EDO). I also really like the Mercury Tree (17-EDO), though, but I'm not as fond of 24-EDO stuff as most people are.

Also, I typically don't give a toss about numerology, but I had noted long ago that the best EDO's, in my opinion, followed a pattern 5, 7, 12, 19, 31, ... following a series Xn = Xn-1 + Xn-2

Some EDO tunings are "meantone," meaning that they temper the fifth flat in order to improve thirds. 12-EDO is one. 19-EDO and 31-EDO also do this. 7-EDO really isn't (from the perspective of intent), but it ticks all of the technical boxes of having a fifth flattened by less than a comma. So these numbers fit also into a pattern of all being meantone tunings (at least for the next few in the series, then it starts to break down since there are too many different fifths available).

I just looked at the Riemann Zeta function, and these numbers start popping up there, too: 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, ... Take the difference and sums of those numbers, and you get more series, all three series share common numbers, like 5, 7, 12, 19, 31, 53, 72, ...

Of the EDO tunings used in Gamelan music and other traditions, they are 5-edo, 7-edo, 12-edo (of course), 19-edo, 24-edo, and 31-edo. 53-edo is also really cool, and I just checked out 72, and, besides having a slew of notes, it's got some good stuff going on. The more popular xenharmonic tunings with equal divisions seem to be 10-edo, 15-edo, 22-edo, and 41-edo, which are in the zeta funtion set, but missing from either the differential or the integral...

All of this means probably nothing, but it's just blowing my mind how all of these seem to keep popping up over and over. 13-edo and 18-edo tunings seems to be completely disconnected from everything useful (and notorious for it), and never pop up in these series. But it's probably all confirmation bias in a roundabout way.

It seems like a lot of people aren't incredibly excited about 19-edo, either because it's got wide ninths or because it sounds too plain. Every time I try to examine tunings with the least amount of objectivity, though, it appears to be the next best low-note-count option to 12-edo.


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## Winspear (May 15, 2020)

Interesting post! Indeed, I don't quite understand the math behind it but I frequently see discussion on how adding together tunings with particular properties results in tunings with those properties - hence the 7 12 19 31 pattern of meantone.
People saying things such as "41 is good at x because it = 19+22". Very interesting how that works out.
I actually did something very similar to you measuring the error of tunings to help decide which meantone to fret my guitar for.
Interesting how our results differed - probably because I was already biased towards a 19 note subset like that which exists in 31 or is approximated by 19 - and was using that as my JI measure and not penalising for extra notes (as I wouldn't have them on my guitar). The EDOs closest to 1/4 comma meantone like 112, 81 and 31 came up best. Pretty much a sliding scale of error from 31 to 19 EDO (in order: 31 112 81 131 50 119 69 88 107 126 19 - The ratio of L:S collapsing through this scale)
What you said about people not being excited about 19 edo - I already mentioned to you how I dislike its enharmonic diesis and chroma for melodic motion, but really I think the main downfall for me is that tunings this close to 1/3 comma meantone are so far flat
on the harmonic 7th and subminor 3rd (aug 6 and aug 3) - 19edo equating them with the diminished 7th and 3rd - these two intervals being the most exciting draw about extending meantone out to 19 for me.
50 EDO, approx 3/11 comma meantone, is a reasonably sized nice middle ground between 1/4 and 1/3 (31 and 19) - hits those two intervals well especially the subminor, and has both m and M thirds within a few cents. The whole cent numbers are pleasing also haha. It's diesis is 48 cents which is quite an improvement on 31edo for a guitar - I'm surprised I didn't swing more that way (I went with 112 EDOs 43 cents small fret).


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## Winspear (May 15, 2020)

I don't know what tuning I will explore next outside of meantone. 17edos superpyth seems like something I could enjoy for very specific sounds on guitar, and is a very good size to have the full EDO on the guitar.
Beyond that, probably 41 EDO because I am extremely interested in the thirds tuned Kite guitar and how effectively it handles it. At that kind of size Pyth pretty much covers all grounds , giving all the benefits of meantone just with the inverted spellings.
It's a shame the S interval is (imo) too small to make a 19 subset of Pyth usable on a regularly tuned guitar.

The tuning I have settled on for computer based work for now is a full 124 EDO (as 4x31 (actually using it as 5x31 with the outers as enharmonics)) - as it contains 31/my upcoming guitar+intended Lumatone layout, is large enough for free-JI with practically realistic accuracy, and I have an extremely efficient workflow and notation system for it - far more approachable than full-cycle large meantones such as 112 or 131.

Outside of this I do utilize smaller JI scales when synthlike perfection and attention to beating etc. is a requirement and I'm using something more accurate than acoustic instrument samples. I've been invested what seems to be a new approach to JI used by my friend and former teacher - I've refrained from posting it until she finishes her paper on it, but you can hear and see an example here:

I will elaborate but can say I'm finding this system much more enjoyable personally than the typical JI approach of assembling a gamut of various low limit intervals


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## Winspear (Jun 17, 2020)

Just dropping in with another release from my friend 

And a progress update on my meantone Carillion build!


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## ElRay (Jun 17, 2020)

Winspear said:


> ... And a progress update on my meantone Carillion build!


 I've for a looooong time wanted a "well tempered" guitar to get actual "key color" instead of the minor timbre difference you (might) get playing in one key vs another.


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## Winspear (Oct 3, 2021)

https://ventifacts.bandcamp.com/album/ventifacts
Extremely strong new music! very cool.

Anyone up to anything in the microtonal world recently? I've been exploring harmonics 24;48 with intention to apply them to a guitar, and also having a lot of fun with this 53 tone harmonic series tuning (all harmonics up to 47, and then filler to flesh out the roughly 53edo framing) on keys that you can try here on touchscreen (you can also use your mouse and spacebar to make chords). Been having fun in general applying harmonic series segments to EDO frameworks like this as far as they can support.


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## gnoll (Oct 3, 2021)

Microtuning scares me. I have enough trouble with just 12 notes.


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## bostjan (Oct 3, 2021)

gnoll said:


> Microtuning scares me. I have enough trouble with just 12 notes.


Bohlen Pierce has fewer notes.


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## gnoll (Oct 3, 2021)

bostjan said:


> Bohlen Pierce has fewer notes.



It looks cool and it is interesting stuff.

Sometimes I get frustrated by my dumb 12tet music that feels lame and samey but I still think I have some exploring left to do in that realm before delving into microtones. When I listen to good music I get reminded of what really is possible and how little I have done.


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## bostjan (Oct 3, 2021)

gnoll said:


> It looks cool and it is interesting stuff.
> 
> Sometimes I get frustrated by my dumb 12tet music that feels lame and samey but I still think I have some exploring left to do in that realm before delving into microtones. When I listen to good music I get reminded of what really is possible and how little I have done.


12tet has tons of great ideas contained within it. It just seems to me like most of the best ideas have already been done by someone else. I IV vi V only has a few ways it can be made into a song and early late 90's pop punk pretty much covered them all already.

Something like 19tet gets you into a subspace that's open to almost all of the same old ideas, but with a bunch of new real estate. 24tet has all of the same tones, but with some pretty spicy alternate options. Having tried 14 and 7 and 22- they are all a little more restrictive, but have fresh openings as well. But, ultimately, it's like painting with a different pallete. You can make a painting that looks almost exactly the same out of mixtures of different paints. Even something crazy like Bohlen Pierce can be used to play a well known melody and tell what it is.


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## Winspear (Feb 4, 2022)

Hey @bostjan here is the tab system I said I'd recommend and how it pans out in 19 for you. It's very good for retaining muscle memory and musical function over from the regular 12 framework which music theory is compatible with _but not based on_.







We're going to label the frets adhering to their musical interval naming patterns, retaining the same information that is contained on a regular guitar.
We can start with the normal marked frets 0 3 5 7 9 12. These form a minor third from 0-3 and 9-12, and major seconds between 3579, also hosting the octave/powerchord shape in these pairs.
Work similarly from these frets with the octave/powerchord/major second shape to discover 2 from 0, 4 from 2. 10 from 12, 8 from 10. In the middle we hit a line of symmetry where we have an upper 6 and a lower 6. We can't label fret 6 on this guitar in a non ambiguous way. We can mark it with arrows or colours, here I use blue for flat/down and orange for sharp/up.
The remaining frets have enharmonic names, just as 19edos enharmonics (B#/Cb E#/Fb) occur on them. We can label them both as downs or ups, and the one it makes most sense to use will depend on context at a given time. 
For example using the octave shapes again, we have an octave from 10 to 8, and would want to use ^6 to notate an octave from 6 to 8, because unlike v7 it retains the 2 numerical step. If we continued to work down we'd pair ^4 with ^6, and so on, really just musical interval logic applied to tab numbers. We might want to use the v7 alternative in the context of a diminished fifth from the open string. Contextual choices, just like musical accidental notation. It's not like it will change the result of what's played at least, but at least makes it slightly easier to perceive and read what's going on. 

I've been working with this system for a while and love it, given I inlaid the usual frets and an additional side dot marker for 2 and 10, navigation is pretty well fleshed out and I'm getting ok at reading the additional colours. Dorico can colour tab numbers which is nice. Other software might have to rely on overlaying some arrow text or similar.


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## Winspear (Feb 4, 2022)

Also, Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# , if we take this chain of fifths and consider that it is symmetrical if tuning the center (D) in unison with 12edo D, and the notes closest to the middle will be most in tune with 12edo and the ones closest to the edges most out of tune, the above tab system should generally make for the most appropriate choices if playing along with music reading regular tab. You'd use fret 2 for F#, rather than Gb, which would have been twice as much out of tune with the track your playing along with. It'll have you avoiding more distant keys as a composer would avoid the alternatives with the most accidentals.

It's probably more likely that you're tuning C or A in unison with 12edo which will push this off to the side a bit (so am I), but I just wanted to point this out as a bit of a justification, as the fret numbering system has some symmetry going on that is in line with the symmetry of D dorian being the middle of meantone, the circle of fifths, etc.


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## Winspear (Feb 4, 2022)

I missed a sentence to point out something obvious, after "We can't label fret 6 on this guitar in a non ambiguous way." - Just see the normal grey numbers along a single string, it's the 12 tone regular interval names - the only ambiguous one being 6, between the Perfects, the aug4/dim5


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## Winspear (Mar 12, 2022)

@bostjan Hmm, looks like my diagram link above expired, but anyway - I just came here to say that my octave=12 tab numbering method above whilst correct for 19edo and usable for my 19 subset of 31edo in isolation, was arrived at by an unnecessarily complicated method and I realise had some oversights if I was trying to play my tab on a full 31edo guitar. The system below provides cross compatibility between 31, 19, and 12edo.
31edo frets:

*0*​*^0*​*v1*​*1*​*v2*​*2*​*^2*​*v3*​*3*​*^3*​*4*​*^4*​*v5*​*5*​*^5*​*6*​*6*​*v7*​*7*​*^7*​*v8*​*8*​*v9*​*9*​*^9*​*v10*​*10*​*^10*​*11*​*^11*​*v12*​*12*​
19edo frets:

*0*​*v1*​*1*​*2*​*v3*​*3*​*4*​*^4*​*5*​*6*​*6*​*7*​*v8*​*8*​*9*​*^9*​*10*​*11*​*^11*​*12*​

The central 6s must be labelled with ^ and v also as there are two of them, though the meaning is not the same for them around that central mirror point.

With these numbers (I use coloured numbers, bluev and orange^), one can read a 31edo tab on a 19edo or 12edo guitar, and a 19edo tab on a 12edo or 31edo guitar by simply ignoring arrows they don't have a fret for, and have the notational functions retained. The exception being if one is writing with _doubleflats and doublesharps in 19edo. _Functional doubleflats and doublesharps from 31 or 12 edo are cross compatible to all 3 tunings.


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## bostjan (Mar 12, 2022)

I've been spending a lot more time with 14edo. It's a different approach being so disconnected from meantone and pythag, but still sounding vaguely like 12. I think your notation could work for some of the weirder tunings, but it might be messy compared to just numbering the frets. OTOH, anything multiple or divisible into 12 should be way easier, and for anything meantone, it keeps you in the proper mindset. Anything with partial frets, I think, has to take an approach like what you're showing us here. And I feel like the Saz necks and flying banana necks are where most people are starting, so it'd make sense to grow the notation along the typical learning path.


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## Winspear (Mar 12, 2022)

I agree, that makes sense! I think it would be a struggle to apply to something so different than meantone or pythagorean, and especially when the number isn't so far from 12 the muscle memory position benefits aren't much of an issue - I'm sure reading tab for simple numerical 14 is not bad at all


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## sven karma (Jun 11, 2022)

Hi there, was googling microtonal stuff and ended up here. Basically a keyboardist. Went down the rabbit-hole during lockdown and soon concluded 12-edo is like the McDonald's burger of tunings.

At the same time the whole key/modulation stuff it regularizes do work. I've done stuff in 17 and 19, and while 17 does lovely timbres and 19 can be a bit quirky, 17 kinda has a gamelan thing whereas with 19 you can feel the circle of fifths thing kick in, if you know what I mean.


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## Winspear (Jun 12, 2022)

sven karma said:


> Hi there, was googling microtonal stuff and ended up here. Basically a keyboardist. Went down the rabbit-hole during lockdown and soon concluded 12-edo is like the McDonald's burger of tunings.
> 
> At the same time the whole key/modulation stuff it regularizes do work. I've done stuff in 17 and 19, and while 17 does lovely timbres and 19 can be a bit quirky, 17 kinda has a gamelan thing whereas with 19 you can feel the circle of fifths thing kick in, if you know what I mean.


17 is very cool, and 29 as an extension of it. Working on a 17-out-of-29 fretboard with a 17 treble end right now!


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## bostjan (Jun 12, 2022)

Winspear said:


> 17 is very cool, and 29 as an extension of it. Working on a 17-out-of-29 fretboard with a 17 treble end right now!


Yeah, it's weird how EDO's sum the way they do, for example 19 is good, 12+19=31, which is an improved 19. 17 is good, 12+17=29, which is an improved 17. 15 is a good blackwood tuning, 12+15=27, which is like a better version of 15. 10 is popular, 22 is better, but shares some of the weirdness of 10.

I'd love a full 29edo guitar, but there are just so many notes! Heck, to me, 24edo is too tightly packed for guitar, but a lot of folks make it work.


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## sven karma (Jun 12, 2022)

bostjan said:


> Yeah, it's weird how EDO's sum the way they do, for example 19 is good, 12+19=31, which is an improved 19. 17 is good, 12+17=29, which is an improved 17. 15 is a good blackwood tuning, 12+15=27, which is like a better version of 15. 10 is popular, 22 is better, but shares some of the weirdness of 10.
> 
> I'd love a full 29edo guitar, but there are just so many notes! Heck, to me, 24edo is too tightly packed for guitar, but a lot of folks make it work.


Yeah the logistics of microtuning is a pain. 19 works ok on a conventional keyboard layout, the 12+7 thing makes the octaves go up c-g-d-a.

And then I did some 26, 'cos 19 + 7 are both prime numbers.


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## bostjan (Jun 12, 2022)

sven karma said:


> Yeah the logistics of microtuning is a pain. 19 works ok on a conventional keyboard layout, the 12+7 thing makes the octaves go up c-g-d-a.
> 
> And then I did some 26, 'cos 19 + 7 are both prime numbers.


There was a musician here who specialized in 26edo, but I haven't seen him post in 3 years. He did some cool stuff with microrhythms, too. Here's his bandcamp.


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## Winspear (Jun 13, 2022)

Yeah it's an interesting pattern!

I've settled into some traditions for myself on guitar I think, utilizing subsets and lower temperaments.
My meantone guitar I took the 19 note MOS of 31edo on the first half of the fretboard (basically the naturals/flats/sharps, leaving out the most uncommon notes). Then at the top of the fretboard I drop to a 12 MOS because I just wanted to avoid any tiny frets whatsoever. So yes soloing in some positions means I might only have a G# and no Ab, etc. Haven't found it a particular issue so far.

However I am digging into (what I mentioned above with combining 17 and 29) tempering the upper half of the fretboard to the lower EDO as a more flexible compromise.
So with 29 again, I'm taking the 17 note MOS (a combination of 41c and 82c frets) on the first half of the fretboard, favouring it over 17EDO (much like I considered the 19 subset of 31 preferable to 19edo). Then in the upper octave I will switch to actual 17edo to avoid having any 41c narrow frets at all. You'd think it would throw stuff very much out of tune in comparison, but most of the highly affected notes between the two EDOs are the ones I already dropped in using a subset. In other words, the smaller the fifths chain, the less the error stacks. The most distant notes in comparison between 17edo and 17/29 when restricted to the 17 subset are something like 17c out of tune, but most are in the 5-10c range, with only the most distant notes in the chain being tempered to an inbetween unison.
It still sounds like a lot of error considering many of us use microtonality to get things 5-10 more accurate haha, but as a compromise between not having those notes available at all or having a cramped guitar which is no fun to play, I'll take it. I'm mostly using bends and vibrato and such up there anyway! So far I only have experience combining the two EDOs via MIDI, but it sounds absolutely fine.
Should have the 29+17 board finished in a month and I intend to swing that way for my next 31 guitar too rather than limit myself to 12 tones up top.

In other news I recently finished up a richlite+EVO Gold 16EDO replacement classical board for a dude in Canada! Fun project


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## Winspear (Jun 13, 2022)

sven karma said:


> Yeah the logistics of microtuning is a pain. 19 works ok on a conventional keyboard layout, the 12+7 thing makes the octaves go up c-g-d-a.
> 
> And then I did some 26, 'cos 19 + 7 are both prime numbers.


A fun one to try is using the traditional 'split black keys' approach that some old meantone organs/pianos did.
You can try all white naturals as normal, and 5 sharps on the blacks. And then use a keyswitch or pedal to change to another midi channel that has 5 flats on the blacks. 17 tones just missing out on the EF/BC bridges this way (alternately a full 17edo)


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## sven karma (Jun 13, 2022)

Winspear said:


> A fun one to try is using the traditional 'split black keys' approach that some old meantone organs/pianos did.
> You can try all white naturals as normal, and 5 sharps on the blacks. And then use a keyswitch or pedal to change to another midi channel that has 5 flats on the blacks. 17 tones just missing out on the EF/BC bridges this way (alternately a full 17edo)


Sounds like a good plan, if my music theory were strong enough to figure where to go when! I actually quite like looking at the keyboard and not having a clue which note is what and just having to work stuff out by ear!


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## Winspear (Jun 14, 2022)

Second song from the upcoming Scarcity record is up!


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## bostjan (Jun 14, 2022)

Winspear said:


> Second song from the upcoming Scarcity record is up!



What tuning?


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## sven karma (Jun 14, 2022)

Winspear said:


> Yeah it's an interesting pattern!
> 
> I've settled into some traditions for myself on guitar I think, utilizing subsets and lower temperaments.
> My meantone guitar I took the 19 note MOS of 31edo on the first half of the fretboard (basically the naturals/flats/sharps, leaving out the most uncommon notes). Then at the top of the fretboard I drop to a 12 MOS because I just wanted to avoid any tiny frets whatsoever. So yes soloing in some positions means I might only have a G# and no Ab, etc. Haven't found it a particular issue so far.
> ...


Due to user error/synth quirk I did a piece with one in 13 and one in 17 and after the initial wtf like you I was impressed with the effects you can get.

On account of a dream (only the 2nd music related one I've ever had) I'm doing stuff in 25 at the moment. At times it's like an overpadded sofa being so close to 12/24, at other times (vegetarians etc look away now) like one of those Argentinian cuts of steak no one else does. I went a bit Hans Zimmer ha ha


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## Winspear (Jun 15, 2022)

bostjan said:


> What tuning?


Haven't seen it said direct from them but someone said 72 so that must have come from somewhere.
As far as I understand there is a fairly established 'school' using 72 going back quite some years as a higher resolution off the quartertone thing.
A lot of pitches of course, so I think in most cases it is used as just a structural+navigational system from which to select pitches for a piece. I think it's very likely that the guitars being used here are regular 12 tone or at most 24, with select 72 EDOstep string offsets for the piece.
Definitely a very nice system that keeps the familiar 12edo as a basis and gives you some simple easy to remember relationships such as
+1 EDOstep for the 6/5 m3
-1 for the 5/4 M3
-2 for the 7/4 H7
-3 for the 11/4 H11
Tuning of core JI intervals works out exceptionally well in general in 72.

Seems this record is going to be a flow-through single-track movement structured thing where each movement prettymuch focuses on hammering home one harmonic series structure. Super effective.

@sven karma that's really damn cool!


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## Winspear (Jul 15, 2022)

That record is out and very, very good  Confirmed 72 edo recorded across 6 layered guitars tuned a comma apart


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## sven karma (Aug 19, 2022)

Came across this on Reddit:


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## Winspear (Aug 27, 2022)

sven karma said:


> Came across this on Reddit:



Forgot Oni had built that. Super cool. 22 has such a distinct sound, Brendan Byrnes is an artist who has mastered it pretty well - check him out if you haven't https://brendanbyrnes.bandcamp.com/


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## marmoset (Sep 26, 2022)

Great thread! Does anyone have any recommendations for ear training for just intonation intervals? I'm trying to learn the sounds of them but struggling at the moment.


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## Winspear (Sep 27, 2022)

marmoset said:


> Great thread! Does anyone have any recommendations for ear training for just intonation intervals? I'm trying to learn the sounds of them but struggling at the moment.


This website is great for playing with microtonal things and checking out sounds. A few different syntaxes to learn so I've prepared a file for you, that stacks the harmonic series up to 16 in various octaves. Add ---- for sustain. You can click the > at each line to start playback there.
Note the neat format for a chord lists all the ratios at once, like 4:5:6:7:8 is 5/4 and 6/4 and 7/4 and 8/4 all on the root, 4. With the intervals between being 5/4, 6/5, 7/6, 8/7.
Definitely focus on the base harmonic series for internalizing JI. It goes up in order of complexity.
You can also look at 'modes' of it, started on different roots, which is where some different chords come from (such as 6:7:9, a cool subminor triad).
Practice singing over drones, the 5/4 should be easy enough, 7/4, the 'barbershop 7th', (the 4th line in this file) is really first 'new' interval in the harmonic series and should be somewhat of an "aha!" moment to feel and sing. Try feeling and singing each new note of this over the drone before it comes in on the next line. If you want to try it over a single low E distorted guitar drone, they should be much more prominent to hear, as the distortion amplifies them. Speaking of which, try a distorted low E + open G string. Then slowly downtune the G string, you should hear JI at -16 cents (10/4, octave of 5/4). Also try E+D, downtune the D, you should hear JI at -33 cents (7/4). These three should sound great together.

You can also mess with opening two tabs of this. Set it nice and low like 200Hz and move the other one slowly up, see if you can find some nice resting points and hear how approaching them sounds and how it settles in peacefully at clean numeric ratios.


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## marmoset (Sep 27, 2022)

Winspear said:


> This website is great for playing with microtonal things and checking out sounds. A few different syntaxes to learn so I've prepared a file for you, that stacks the harmonic series up to 16 in various octaves. Add ---- for sustain. You can click the > at each line to start playback there.
> Note the neat format for a chord lists all the ratios at once, like 4:5:6:7:8 is 5/4 and 6/4 and 7/4 and 8/4 all on the root, 4. With the intervals between being 5/4, 6/5, 7/6, 8/7.
> Definitely focus on the base harmonic series for internalizing JI. It goes up in order of complexity.
> You can also look at 'modes' of it, started on different roots, which is where some different chords come from (such as 6:7:9, a cool subminor triad).
> ...


Thanks so much! This is all super helpful.

That xenpaper website is great, I'll have an experiment. Thanks for setting up those examples too - some of those chords sound really nice to me so will be interesting to see which ratios they have in.

I'll have a go with those guitar experiments. I wouldn't have thought to use distortion but makes sense when you point it out. 

The two tone generators thing is a nice trick! I think it's my headphones but I've noticed that when I play intervals with sine waves I get some low frequency noise too. E.g. with 200Hz and 250Hz in the two tabs I get a major third but also a noisy buzzing.


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## CanserDYI (Sep 27, 2022)

Man I really want to like Microtuning stuff so badly, I really do, but man, my western ear is just screaming at me whenever I listen to anything microtonal.


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## Winspear (Sep 27, 2022)

marmoset said:


> Thanks so much! This is all super helpful.
> 
> That xenpaper website is great, I'll have an experiment. Thanks for setting up those examples too - some of those chords sound really nice to me so will be interesting to see which ratios they have in.
> 
> ...



50hz difference tone  You should find however (as is the case here, it is a low octave fundamental reinforcement) that the difference tones in JI are in tune with the chord.
252hz (12edo major third) whilst only sounding 16 cents out of tune with the 200hz, will produce a 52hz difference tone, a huge 70 cents away from 50hz! A difference tone that is now almost a semitone out of tune with our bass. This is another thing you can easily observe with solo range distorted guitar, have a go with some close dyads on the upper frets of the B and E strings , bend them around slightly and see how much it affects the difference tone.


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## Winspear (Sep 27, 2022)

CanserDYI said:


> Man I really want to like Microtuning stuff so badly, I really do, but man, my western ear is just screaming at me whenever I listen to anything microtonal.



There really is such a huge range of deviation, application, weirdness, _careful effective use _of weirdness making it more approachable, etc. Some application can be very subtle.
Let's remember classical music was originally tuned differently - it doesn't stand out too much when the harmonic gestures used are the basic tonal diatonic stuff that we are used to in classical, but the difference in vibe and resonance should be apparent:



A few links of the highest quality and most approachable micro music I know:





https://www.youtube.com/watch?v=uIYg8b2p8JY&ab_channel=MikeBattaglia
https://www.youtube.com/watch?v=ZIn6uis5duw&ab_channel=ZheannaErose


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## CanserDYI (Sep 27, 2022)

Winspear said:


> There really is such a huge range of deviation, application, weirdness, _careful effective use _of weirdness making it more approachable, etc. Some application can be very subtle.
> Let's remember classical music was originally tuned differently - it doesn't stand out too much when the harmonic gestures used are the basic tonal diatonic stuff that we are used to in classical, but the difference in vibe and resonance should be apparent:
> 
> 
> ...



Okay, that Brendan Byrnes stuff, at least the opening track to the 2227 album there, that's really cool, I'm definitely going to be listening to that album this week, seriously thanks!


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## sven karma (Oct 11, 2022)

Guitar output to CV input converters exist? The Hydrasynth has 2 CV inputs and a mod matrix that routes to the oscillator pitches so conceptually y'all can pile in on my 'variable multiple just intonation'.

The HS has preset and imports microtonal tunings. It also has separate key track for each of its 3 oscillators. So at 100%, 25 ji scale does 25. At 200% you get a 12 1/2 ji, at 25% you get a 100 ji. 

Rabbit hole, dive in. First it was 24% on one oscillator ...


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## ElRay (Oct 12, 2022)

sven karma said:


> Guitar output to CV input converters exist? The Hydrasynth has 2 CV inputs and a mod matrix that routes to the oscillator pitches so conceptually y'all can pile in on my 'variable multiple just intonation'.


I'm not sure how useful for this specific project is, but the MOD Dwarf (along with the MODEP/piStomp OSS Projects) has Analog-to-CV and MIDI-to-CV plugins. Works pretty well for "Guitar Synth" purposes:
Guitar Synths - Introduction and Overview - YouTubehttps://www.youtube.com › watch​


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## sven karma (Nov 9, 2022)

Winspear said:


> There really is such a huge range of deviation, application, weirdness, _careful effective use _of weirdness making it more approachable, etc. Some application can be very subtle.
> Let's remember classical music was originally tuned differently - ...



All the way back in the 16th century:


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## Winspear (Nov 11, 2022)

Real sweet new BB album!


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## sven karma (Nov 28, 2022)

Still working on the sweetness here lol. Bass synth in 16ji, top synth in 15ji. Which inspired the title.



Going back to the topic of things you can do with the Hydrasynth CV inputs, one easy thing is mini mono jack cable between CV pitch out and mod in, instant pitch tracking as a mod source.

And if you have some more cables lying about, a passive splitter and a pedal like an MF ring mod with an lfo out, you can have an analog lfo (going up to double the speed of the internal HS ones) working its charm on the digitalness.


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