# School me on... tuning temperaments...



## odibrom (Aug 17, 2018)

Hi there folks.

Yeah, lately this subject has come to surface quite often and made me think about it. My knowledge on this is VERY limited, so I'm recruiting yours to be more aware about what is out there to experiment and specially, to hear.

What I've gathered so far is that the "temperament" is how the octave is divided, but there are more than one way to do so.

So, lets make this thread about this and see where it goes to. I know that there are already some threads on this, but I believe those are yet a bit sophisticated. I was into "start from scratch".

Equal Temperament, Just Temperament, others, Micro tonal... what else is out there, what are the differences between those...?

A piano or a guitar is a tempered instrument, but a violin is...? Yah, I know, BASICS, sorry...

What say you?


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## bostjan (Aug 17, 2018)

I think "microtonal" is kind of a catch-all term for whatever isn't the most popular...

Piano and (fretted) guitar, obviously, have a finite set of notes they can play. A violin or cello or trombone has an infinite number of notes it can play, or, more precisely, an uncountably large number of notes. Pick any two notes on a violin, and there exists a note in between those two, no matter how close together they are. Other instruments make their notes based off of natural harmonics, for example, a bugle, and cannot be tuned to any temperament at all.

I'd say that even on guitar, if you look outside your standard blues/rock/metal schools of thought, there are different ways to tune the strings in order to get different temperaments. A lot of country players deliberately tune their 2nd string flat to get a sweeter sound out of common open chords.

Just about everyone talking about the "True temperament" system is describing the thing using false hyperbole and flat out myths to describe it. I think it's cool as hell, but I also know that it's not "more in tune" than any other fretboard or anything close to that. It's just a different sort of tuning. It's like saying that chocolate ice cream is definitively better than strawberry. While I would choose chocolate 4 times out of 5, it's only a matter of personal taste and nothing else, and they both have their advantages and disadvantages.

My approach is to categorize tuning systems into these buckets:
1. Player-intonated: Instruments that provide a continuum of notes, that is, where the player determines how intonation is affected. For example: violin, double bass, slide guitar, cello, slide trombone, etc.
2. Harmonically/just-intonated: bugle, shofar, whirlie, etc.
3. Well-tempered: western instruments during the baroque period
4. Equal-tempered: fretted and keyboard instruments
5. Hybrid: steel guitar is a mixture of equal-temperament (positions on the fretboard), well-temperament (strings and pedals are tuned to match the chord, rather than the equal temperament notes), and player-intonated (it's played with a metal slide); flute is intonated so that keys pressed down give notes true to equal temperament, but many upper-register notes are based off of the harmonic series, achieved by overblowing.

When you put multiple instruments together in concern with each other, the differences in timbre usually cover up most of the discrepancies, but, at the same time, if you are arranging for orchestra, you can't be totally stupid about things, otherwise it'll sound like a mess. I ran into this ages ago, when I worked on a movie score, and the producer wanted some really oddball instrument combinations, like sitar, banjo, and bagpipe, and, man, what a hot mess of intonation problems!


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## synrgy (Aug 17, 2018)

Was just talking with my Dad about this again last night. He understands it _way_ better than I do, but I'll try to contribute.

The depths to which one might explore the topic are limitless. The loose idea is that the math renders 'perfect' tuning impossible; only 11 of the 12 notes can ever be perfectly in-tune. "Temperament" takes the remaining 23.46 cents (about a quarter semitone; aka 'The Pythagorean Comma') that make the 12th note 'off', and divides them among all 12 notes.

Just play a basic chord on any 'in-tune' piano, and - if you're listening for them - you'll hear the faint wobbles of (ever so slight) dissonance: Fifths are flat, fourths are sharp, etc.

Prior to the 20th century, instruments were most-often tuned in temperaments that minimized the dissonance of the key they were playing, which had the side-effect of maximizing the dissonance for the chords that weren't complimentary to that key.

The entire 'character' of a chord can change from one temperament to the next; in one it might be 'bright', and in the next it might be 'dark', even though it's still technically the same chord.

What really messes with me is that what we've known as 'the sound of a piano' our whole lives (unless there are any centenarians here?) isn't what they sounded like when, say, Bach was composing.

I recently acquired a clip-on strobe tuner that comes pre-programmed with different temperaments. I've only tried a couple on my guitar(s), but I've definitely noticed a subtle difference between,'standard guitar tuning' and 'James Taylor tuning'.

Here's a fun little video that does a decent job of breaking some of this down:



My Dad handed me a copy of this six pound tome last night and joked that there would be a quiz in the morning.  Anyway, if you really wanna dig deep, you should try and find a copy of it, physical or digital.


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## bostjan (Aug 17, 2018)

This video about dulcimers is actually pretty telling, in relation to how equal temperament, just intonation, and well tempered tuning compare and contrast:


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## TedEH (Aug 17, 2018)

bostjan said:


> A lot of country players deliberately tune their 2nd string flat to get a sweeter sound out of common open chords.


Only tangentially related, but I also usually tend to tune very slightly flat on whatever is the largest string on an electric guitar. It's a tradeoff between compensating for attack, particularly when playing open notes, the very slight amount that notes tend to get pulled sharp from fretting, the slight difference that the first fret or two tend to be off because of the cut and placement of the nut, etc. It's only a small difference, but enough that I hear it. Like the other cases discussed already, it's a matter of tuning in a way that supports what and how I play that probably wouldn't work for someone else.

Similarly, I've heard cases of bassists who intentionally tune just sliiiiiiightly flat compared with the guitarists they play with, for reasons I can't remember. Something about certain playing styles that cause most notes to fall sharp probably.


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## bostjan (Aug 17, 2018)

One point, along these lines, that needs to be pointed out, maybe, is the "Sweetened tuning" that a lot of older players talk about.

If you listen to James Taylor, you should be tuning as follows:

1-E - minus 3 cents
2-B - minus 6 cents
3-G - minus 4 cents
4-D - minus 8 cents
5-A - minus 10 cents
6-E - minus 12 cents

This combines three things:

1. Sweetened temperament (which I sort of touched on before, but I might need to get into)
2. What @TedEH brought up about slacking the lower strings a bit more
3. Tuning down a small amount for various reasons.

So...

1. Sweetened tuning, like how I mentioned slacking off the second string (B string) a little bit. This is done a lot, and another method is to play the last chord of the song you are about to play, and then adjust the tuning of each string that isn't on a root note of that chord until the chord sounds "okay." Then, as you play your song, some chords will sound good and some chords might not sound as good, but the idea is that the last chord you play will sound fucking amazing, and leave a good taste in the audience's proverbial mouth. There are other methods of doing this. In the late 80's and early 90's when I had just started playing, these charts sometimes appeared in magazines, and my guitar teacher at the time walked me through a few of them, since, he said, tuning was the most important step prior to playing the guitar. I thought I would quickly pull a few of these off of google or some sites I frequent, but it looks like interest in this topic is basically zero nowadays. Maybe it was a relic of a bygone generation of guitarists even then and now it's knowledge that's simply lost to attrition. I know for a fact, though, that my guitar teacher had a special tuning chart he wanted me to use for my lessons, and that it really did make certain open chords sound noticeably better. Oh well. Maybe I'll come across that old chart someday and preserve it for the masses of modern guitarists who inevitably won't give a shit how some guy they never heard of used to tune, even if he did tons of session work in the 80's.

2. Slacking off the lower strings. Yeah, the thicker, spongier-feeling wound strings do tend to go sharp when you fret them, so it makes sense to tune the open string a few cents flat to compensate.

3. Tuning down a small amount, if you are trying to get up to some high notes, gives you a tiny shade of relief, which might make the vocals sound better. Also, in general, lower notes sound more relaxing to the audience, so by tuning down a few cents, you allow them to connect with the music a tiny fraction of a percent more.  You pretty much never see any of those tuning charts I mentioned having you tune sharper, on average. They might take one string up a tiny bit, but only if other strings are taken down at least as much.

Overall, I'd say that the Taylor chart is a little extreme on the low end. But who am I to argue with James Taylor?

Let's look at Buzz Feiten's system.

First off, you use a shelved nut, so you can't simply tune your regular old guitar with your regular old nut this way, but you can install a shelved nut and then do this.

So the nut is such that the distance from the first fret to the nut's pinch point is less. IIRC, it's most pronounced on the third string, but maybe BFTS is the same all around. There were a few similar systems, like Earvana and MTS, so I might not be 100% correct on that.

Then, when you tune the open strings, you slack the wound strings all by 2 cents, slack the G string also 2 cents, and the high B one cent. Leave the high E string as is. Then, set intonation so that all of the strings are in tune at the twelfth fret except that the D and G strings are 1 cent sharp.

This leaves you with a very mildly tempered tuning that makes all of the common chords sound a little sweeter. But...note that this system never really caught on much outside of Washburn guitars that, for a short time, came with this factory installed. IMO, the problem was that it was much ado about something very very subtle. Yeah, chords sound sweeter, but, well, +1 to -2 cents isn't really going to get noticed, especially once it's spread out so evenly (what I mean by that is that, if the third is 2 cents more in-tune, but the root note of the same chord is also 1.5 cent down, the third is effectively sweetened by only half a cent, not two, as alluded). So it makes a lot of effort to give you some pretty unimpressive gains.

Then, True Temperament, which, as I have said, is just an alternate tuning, not a solution to all of the guitar's tuning troubles.
(adjustments all in cents)
E -2
A 0
D +2
G +4
B -1
E -1

Then, the frets are wiggled around and wavey such that the chromatic scale is
E -2
F 0
F# -4
G +4
G# -4
A 0
Bb -4
B -1
C +2
C# -4
D +2
Eb -4

except, evidently, on the first string (?)

This helps almost all of the common guitar keys sound a little better. Note that it's more aggressive than the Buzz Feiten tuning system, with +4 or -4 cent changes from equal temperament, and they are not localized to one area of the fretboard only. So, it's going to be a much more profound change overall.

The biggest problem, to me, is that the minute you switch to drop D or DADGAD or whatever, none of it works anymore. Those wavey frets only work for *one tuning*.

So, what can you do?


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## odibrom (Aug 17, 2018)

I now have a reason to use this "smile" thing...

@bostjan @TedEH @synrgy



You guys ROCK  big time. Thank you.

So, the True temperament is probably derivative of the Just temperament...? Calling on the video about the equal temperament, it divides the octave in equal frequency parts. Tempering a guitar is then a process of calculating the distances of frets according to the desired temperaments and that's why frets differ from one temperament to the other. It has to do with the intonation of notes along the fretboard. For the equal temperament there is a known formula which I've learned here:

Scale Length = A
Fret Number = B
Number of frets by Octave = C
Formula = A-(A/(2^(B/C)))
... or one can also use the other method of using the constant *17.817* (for a 12 notes per octave system only) as a divider for calculating the frets position:

Scale Length = A
Constant = *17.817*
Formula > A/*17.817 =* 1st fret distance to nut, for the 2nd fret we have (A-(A/17.817))/17.817, and so on...
So, the first part is clear by now, which is how one can intonate a given octave note number division (most common is 12), but then we can divide the octave with 17, 19, 13.5 notes... (using the 1st formula and a spread sheet software is quite easy to get crazy results and experiments). What do we call these exotic / alternative octave divisions? are these EDOs?... what does it means? This to say that, for example, one could do a just or true temperament of a 17 notes per octave kind of thing, not only the equal one... right?

... I'm quite sure the fretless instruments or those that allow for a continuous frequency range (versus de stepped that a temperament instrument gives) have a proper name (not just the negative of the previous)... or am I daydreaming...?

Nevertheless, I must say again, THANK YOU very much on all explanations (meaning time and effort taken to write these walls of text). You ROCK.


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## All_¥our_Bass (Aug 18, 2018)

odibrom said:


> are these EDOs?... what does it means?


Yes, those are EDOs
EDO = *E*qual *D*ivision of the *O*ctave
Regular western tuning for guitar/piano/keys is 12edo.

If you want to stay within the western 'meantone' way of thinking 19edo and 31edo are good choices (nice tunings for the major and minor chords, keys work mostly the same way).

https://en.wikipedia.org/wiki/19_equal_temperament
https://en.wikipedia.org/wiki/31_equal_temperament

There's tunings like 17edo that give you three flavors or third/second/seventh (major, minor and neutral)
22edo is a neat type of quarter-tone system.

If you are on facebook, hit up these folks:
https://www.facebook.com/groups/xenharmonic2/


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## odibrom (Aug 18, 2018)

... and suddenly I've seem the light.

Yah, one doesn't know what doesn't see/hears.

@All_¥our_Bass much appreciated for your super direct reply.


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## All_¥our_Bass (Aug 18, 2018)

http://split-notes.com/brendan-byrnes-neutral-paradise/


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## bostjan (Aug 18, 2018)

I love 22 EDO. It does major and augmented stuff so well. It sounds sad, but kind of optimistic.

True temperament is really a sort of well temperament, a tuning that compromises different keys almost equally.

The equal divisions of the octave are plentiful, but only a few get any notice on guitar:

12 EDO is standard guitar/piano tuning. 12 notes.
24 EDO is double that. Quarter tones...used in Turkish, Arabic, and Persian music, traditionally. Lots of guitarists use it. Ibanez even made some production guitars in 24 EDO.
19 EDO is very similar sounding to standard. I use it, and so do John Starrett, Neil Haverstick, Dan Stearns, and others.
22 EDO is very emotive, used by Brendan Byrne's and Paul Erlich.
17 EDO is also out there in the wild. The Mercury Tree uses it.
16 EDO is used by Last Sacrament.
26 EDO was used by our own @The Omega Cluster 
31 EDO is another that sounds like standard tuning, but you're starting to get a lot of notes.
34 EDO is also sweet sounding. Used by Neil Haverstick, more recently.
I'm sure I forgot a lot of examples. If anyone wants to expound on the list, I'd appreciate it. Of course, we have a whole thread of examples.


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## odibrom (Aug 18, 2018)

Saying you guys rock isn't enough, but words are escaping me, so sorry for my lack of vocabulary...

Personally, I'll keep myself at 12 EDO and maybe 24 on the fretless, randomly. One can also pass by on the 24 EDO with fretted guitars doing quarter note bendings or using botleneck slides and similar apparatus.

Well, I'm satisfied with all your answers, I feel like going back to music school again, but with greater curiosity on my behalf (30+ years ago it was painfull)...


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## ElRay (Sep 1, 2018)

Not sure if this belongs here or should be in its own thread ...

I don’t get “True Temperament”. Yes it’s a trademark, but they make it sound like every note in every key plays “correctly”. That can’t be, so what key(s) is it optimized for? If you have pure 5ths and 3rds in C, then they’ll be off the further you get away from C-Major.

They talk about the physical harmonics of the strings lining up with the fretted notes. That would just mean that the frets line up with the harmonics. If that was true, then the frets would still be straight, just not EDO. Also, if you’re not in “The Key”, then those physical modes are not going to line up with the correct overtones of the notes in the scale. 

Is it an attempt to have a guitar on one of the historical “well” termperaments? Is it just a different “well” temperament that is well suited for typical guitar music?

There’s definitely a lot of hype and salesmanship that conflicts with my understanding of temperaments.


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## ElRay (Sep 2, 2018)

ElRay said:


> Not sure if this belongs here or should be in its own thread ...
> 
> I don’t get “True Temperament”. Yes it’s a trademark, but they make it sound like every note in every key plays “correctly”. That can’t be, so what key(s) is it optimized for? If you have pure 5ths and 3rds in C, then they’ll be off the further you get away from C-Major.
> 
> ...


Ok. I think I’ve made sense of this.

The True-Temperament necks implement 12-EDO, but have offsets to deal with the non-ideal-ness of real strings. That’s the added “magic”. Their non-True Temperament necks actually implement alternate temperaments with their “magic” added in.

The down side is that the necks will only work with specific tunings and specific string gauges. Good luck if you like alternate tunings, progressive tensions, wound 3rds, etc.


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## bostjan (Sep 2, 2018)

We had some discussion about this in the microtonal thread. I'm pretty certain that True Temperament is a well tempered tuning. I think that's pretty darn cool, but it is contrary to some of tge promotional materials out there.


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## ElRay (Sep 14, 2018)

bostjan said:


> We had some discussion about this in the microtonal thread. I'm pretty certain that True Temperament is a well tempered tuning. I think that's pretty darn cool, but it is contrary to some of tge promotional materials out there.


It took a while to wade through the website and distill reality from marketing. 

There’s really two product lines. The first is plain-old 12EDO, but the wiggly frets compensate for the realities of real strings, non-zero nut height, etc. esentially, they figured-out/trial-and-errored/etc where the string should be fretted to hit that 12EDO frequency as correctly as possible. 

The other products implement other temperaments with the same fine tuning. 

For me, the biggest problem is that it only supports specific tunings, with a narrow range of strings, on necks with specific geometries. So, if you like heavy strings, wound 3rds, wide necks, tunings other than E-standard, scales other than 25-1/2”, etc. you’re SOL


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## Winspear (Sep 16, 2018)

ElRay said:


> TT stuff


http://www.sevenstring.org/attachments/tt-comp-png.60373/
Here is their tuning analysed in each key - I look at the minor and major thirds and perfect 5ths. Green is an improvement, orange is worse. I didn't bother to colour the 5ths. I noted what I heard playing it on a synth.
I know they have done multiple/more extreme temperaments, but no, the standard one is not just correcting the strings to 12 EDO - it is improving the most common chords in 12 EDO - it is what I've analysed above. They seem to say the 'string fixing' adjustments are implemented _on top _of this tuning system rather than using simply the theoretical perfect fret placements for this microtonal tuning. I've said before I'll withhold judgement before trying, but I've never found any such compensation to be necessary (that is to say, a normal 12EDO guitar intonates 12EDO perfectly for me when set up well). The temperament does seem to be a pretty nice one as far as improving common chords go. I'm still not convinced it's noticeable enough to warrant the cost and fast wearing frets though


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## ElRay (Sep 16, 2018)

So it is a bit of woo. The hype makes it sound like everything is better in tune throughout the neck. That’s why I thought it was just compensation for non-ideal strings, etc.


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## The Omega Cluster (Sep 16, 2018)

It's only improving them as far as you think of "just intonation" as being better than equal temperament, considering its flaws (sounding worse in some keys) as well. That sounds quite backwards to me, especially since they are very secretive about it; i.e. not telling customers what they really are buying. Even then, the improvement is so slight because it's not a true JI fretting that there's really no noticeable difference. At worst, it might sound very bad when accompanied by other instruments or keyboards that are tuned in regular 12EDO because while you don't hear much of a difference when the instrument is played alone (under 5¢ is not really noticeable), it will sound significantly worse when played with another instrument, because a 5¢ difference in two notes being played together is easily noticeable and will "howl" as hell.


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## ixlramp (Sep 17, 2018)

Yes TT is what is known as a 'Well Temperament', they have created their own type of Well Temperament which is fairly subtle compared to historically used Well Temperaments (such as used by J.S.Bach).
It also seems they have also added per-string per-gauge per-fret finer adjustments as compensations for the general imperfections of guitar.

Reading the official site and watching video interviews with the TT guy doesn't explain the details much, and these official sources don't make the ridiculous claims that almost everyone else do. I have found that almost everyone else (the non-official sources) who talks about it, including many great musicians like Steve Vai and the current highest profile users of TT, don't understand it and come out with hype and incorrect claims.
The few of us here who have studied microtonality and tuning systems do understand it.


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## The Omega Cluster (Sep 17, 2018)

Yeah it's such a shame. Whenever I see a musician I like and respect with one of these instruments, there's that one part of respect that flies off, chipped away because of their credulousness...


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## ixlramp (Sep 17, 2018)

odibrom,
A 'temperament' is a tonal system created from 'tempered intervals', it's just one type of tonal system.
A 'tempered interval' ('tempered' as in 'changed') is a 'just intonation' ('just' meaning 'correct') interval that has been altered in size.

For example, many tonal systems have been created by stacking equally-sized fifths on top of each other but altering the size of the fifths away from the 'Just Intonation' fifth (which is 702 cents or 7.02 12TET semitones). These systems are called 'regular temperaments' and 'Meantone' is one example.
"Let's start with Europe's most successful tuning, if endurance can be equated with success. Meantone tuning appeared sometime around the late 15th century, and was used widely through the early 18th century. In fact, it survived in pockets of resistance, especially in the tuning of English organs, all the way through the 19th century. No other tuning has survived in the west for 400 years."
From https://www.kylegann.com/histune.html

The modern and popular 12TET is a 'temperament' because it is created by stacking 700 cent fifths (slightly flat of 'just intonation'). The result is 12 equally-spaced pitches in an octave, so it is called '12 tone equal temperament'.

The most important subject to study is 'just intonation' (also known as 'natural intonation', 'pure' or 'perfect' intervals), as all other tonal systems either arise from this, are derived from this, or are compared against this. 'Just intonation' also explains what 'harmony' actually is.


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## Winspear (Sep 18, 2018)

^ What a brilliant webpage, hadn't stumbled across that before! Useful charts


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## ixlramp (Sep 18, 2018)

The page for an introduction to Just Intonation is https://www.kylegann.com/tuning.html
A list of Just Intonation intervals showing their frequency ratio and interval expressed as cents https://www.kylegann.com/Octave.html

I feel Just Intonation can be explained more clearly though.

////////////////////

What is 'harmony'? What is it about 2 pitches that makes the interval sound 'harmonius' and 'in tune'?

The 'octave' is a clue. This is the most harmonius sounding interval (after the 'unison') and most know that an octave is an exact doubling of frequency, so the ratio of the frequencies is exactly 2:1.
Therefore, in the time that the lower pitch completes 1 cycle of vibration, the higher pitch completes exactly 2 cycles of vibration, after which they have both come back to their starting points again and this behaviour repeats.
See the top 2 waves in the image, they form an octave interval:




If you have 2 pitches, keep the lower one constant and smoothly sweep the higher one, the audible harmonicity (or consonance) of the interval goes through many peaks and valleys. This is usually shown as a dissonance curve:




The most consonant intervals correspond exactly to simple frequency ratios, that is, 'x:y' where 'x' and 'y' are fairly small numbers. These are 'Just Intonation' intervals.

On the dissonance curve the next most consonant interval is a frequency ratio of 3:2. This is an interval of 7.02 semitones (702 cents) which is of course extremely close to the very consonant 12TET interval called the 'fifth' at exactly 7 semitones (700 cents).

3:2 or 702 cents is the 'Just Intonation fifth', and 3:2 is the next most complex ratio after 2:1.
In the time that the lower pitch completes 2 cycles of vibration, the higher pitch completes exactly 3 cycles of vibration, after which they have both come back to their starting points again and this behaviour repeats.
See the lower 2 waves in the first image, they form a 'Just Intonation fifth' interval.

Consonance is determined by how simple the frequency ratio is.


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## ixlramp (Sep 18, 2018)

On the dissonance curve, one of the next most consonant intervals is a frequency ratio of 4:3. This is an interval of 4.98 semitones (498 cents) which is of course extremely close to the very consonant 12TET interval called the 'fourth' at exactly 5 semitones (500 cents).

4:3 or 498 cents is the 'Just Intonation fourth'.
In the time that the lower pitch completes 3 cycles of vibration, the higher pitch completes exactly 4 cycles of vibration, after which they have both come back to their starting points again and this behaviour repeats.

///////////////////////

So far these intervals have been extremely close to 12TET intervals, however when we get to 5:4 which is the Just Intonation major third at 386 semitones, we find that the 12TET major third at 400 cents is now 14 cents sharp, about 1/6th of a semitone sharp.
Guitarists especially notice this because they often use distortion which amplifies dissonance, and why they have tuning issues when playing the very common major triad chord.
It's in tune with 12TET but sounds out of tune. If they retune a string to tune the third by ear they will of course tune it to 386 cents which is out of tune with 12TET, then that out of tune string causes issues for other chords they play.

6:5 is the Just Intonation minor third at 316 cents. The 12TET minor third at 300 cents is 16 cents flat, about 1/6th of a semitone flat.
Likewise the 12TET sixths are out of tune by similar amounts.

9:8 (not labelled on the dissonance curve but is one of the downward pointing spikes to the left of 6:5) is the Just Intonation major second at 204 cents, this is very close to the 12TET major second at 200 cents.

In 12TET the major second, fourth and fifth are close to Just Intonation, all other intervals are up to 1/6th of a semitone away.


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## ixlramp (Sep 18, 2018)

The history of tuning is a shift from Just Intonation, through Meantone and Well temperament, to 12TET, in order to gain the ability to freely modulate to a larger number of keys, at the expense of perfect harmony.
This has happened mostly due to the limitations of keyboard instruments with their limited number of fixed pitches per octave.

Indian classical music took the other route, it sacrificed modulation to preserve Just Intonation. Each Raga is in one key with no modulation. The harmony can sound exotic but it also sounds extremely 'in tune':


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## The Omega Cluster (Sep 18, 2018)

That's interesting, but how did you (or the one who made the graph) calculate dissonance? It's not specified and I'm curious.


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## ixlramp (Sep 18, 2018)

The curve was taken from https://music.stackexchange.com/que...ss-curve-have-a-dip-for-complex-intervals-lik But it's the curve for 'all audible harmonics' that is linked to in the 'answer'. The same author answered here too https://music.stackexchange.com/que...asure-the-consonance-or-dissonance-of-a-chord
So it's actually computed instead of a human experimental response, however it seems reasonable.
The curve depends on how many harmonics you consider to be in the tones, and if it's just odd harmonics (for example a clarinet).
Nice that it's somewhat 'fractal'.


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## The Omega Cluster (Sep 18, 2018)

This is very interesting because prior to that I had to rely on a subjective dissonance curve. I'm intrigued to see that the octave is not perfectly consonant too. The BP one is really cool, but I'd like to make such curves for some other temperaments. It's a shame that the researcher doesn't include their equations...


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## ixlramp (Oct 26, 2018)

To complete my Just Intonation explanation ...
Polyrhythms are a very good analogy for harmony, they are essentially 'slow harmony'.

If you have two drummers where one plays two beats in the exact same time period that the other plays one beat, it's a simple 2:1 polyrhythm and is an analogy for the octave interval. The simplicity of the polyrhythm is an analogy for how consonant an octave interval sounds.

If you have two drummers where one plays five beats in the exact same time period that the other plays four beats, it's a 5:4 polyrhythm and is an analogy for the Just Intonation major third interval. The polyrhythm is more complex, which is an analogy for how a J.I. major third interval sounds consonant, but less consonant than an octave interval.

If they have the timing precise you would describe them as drumming 'in perfect harmony'. Similarly, tuning a major third interval so that the frequencies are in the ratio 5:4 is a perfectly tuned (Justly Intonated) major third.
If one drummer is slightly slow they will go out of phase, they are no longer in harmony, this is the 12ET major third we are used to.

If you record a 5:4 polyrhythm and speed it up until it turns into musical pitches, you will actually hear the Just Intoantion major third musical interval. A note is a fast rhythm, harmony is fast polyrhythms.


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## ixlramp (Nov 13, 2018)

Here's a big list of Just Intonation intervals and their size in cents (1/100th of a modern semitone) https://www.kylegann.com/Octave.html
There are an infinite number of them because there are an infinite number of mathematically possible frequency ratios, however only the simpler ratios sound consonant, as the ratio gets more complex the interval gets more dissonant.

If we list JI intervals in order of consonance it would go something like:
1/1 Unison.
2/1 Octave.
3/2 Fifth.
5/3
5/4
6/5
The above intervals are roughly represented in modern 12 tone equal temperament.
7/4
7/5
7/6
The above three are the 'blue notes' used in jazz and blues using note bending.
...
...

Here's a good intro to JI with many music examples https://www.kylegann.com/tuning.html
Here's a page with information about how tuning has changed through history https://www.kylegann.com/histune.html

Here's a video that does a fairly good job of explaining the movement from Just Intonation, through Pythagorean, Meantone, Well temperament, to Equal temperament:


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## ixlramp (Nov 15, 2018)

My list of JI intervals in order of decreasing consonance missed out the fourth 4/3, can't edit it now.
I found a webpage with a large list in this order http://www.huygens-fokker.org/docs/intervals.html
Here's an edited version:

1/1 unison
2/1 octave
3/2 fifth
4/3 fourth
5/3 major sixth
5/4 major third
6/5 minor third
7/4 harmonic seventh
7/5 septimal tritone
7/6 septimal minor third
8/5 minor sixth
8/7 septimal tone
9/5 minor seventh
9/7 septimal major third
9/8 major tone

All these intervals are roughly represented in 12TET except the 'septimal' intervals that contain the number 7 in the frequency ratio. 12TET essentially approximates JI intervals that are combinations of the prime numbers 2, 3 and 5. Primes 7, 11, 13 are not present.

About the word 'temperament': It comes from 'tempering a JI interval', where 'tempering' means altering a JI interval slightly in the process of creating a tuning system.
So any JI tonal system is not a temperament.
Any abstract mathmatical tonal system where the octave is simply divided into 'n' equal divisions is not a temperament, it is an 'EDO'.

So modern 12TET is a temperament because it uses a tempered fifth of 700 cents (2 cents flat from the perfectly tuned JI fifth that is a 3:2 frequency ratio). Then 11 of those intervals are stacked to get all 12 tones. The tones are equally spaced so we call it '12 tone equal temperament':

Cents
0
700
1400 = 200
900
1600 = 400
1100
1800 = 600
1300 = 100
800
1500 = 300
1000
1700 = 500


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## ixlramp (Nov 18, 2018)

Writing about Just Intonation intervals, i have not yet explained the fundamental aspect that these arise from the harmonic series of a vibrating string.

The Harmonic Series
---------------------------
When a string vibrates it contains many frequencies, the lowest frequency is called the 'fundamental' or '1st harmonic' and is the frequency we usually refer to when we talk about the 'frequency of a note', for example A4 being 440Hz (Hz = Hertz = vibrational cycles per second).
Then there are the 'harmonics' or more correctly, the 'higher harmonics' above the 1st harmonic. Also known as 'overtones'. These can be isolated and heard by lightly touching a string at certain points.
There are hundreds of these, the higher the frequency of the harmonic the quieter they are, which is why we tend to hear the frequency of the string as being the 1st or 2nd harmonic.

The fundamental plus the higher harmonics form the 'harmonic series' of a string.
For a vibrating string of, for example, fundamental frequency 100Hz, all the harmonics (including the fundamental) will have frequencies:
100, 200, 300, 400, 500, 600, 700, 800, 900 ... etc.
So for fundamental frequency 1, the harmonic series wil have frequencies 1, 2, 3, 4, 5, 6, 7, 8, 9 ... etc.
It's easy to remember as the 1st harmonic has frequency 1, the 2nd has frequency 2, etc.
Also, for example, the 5th harmonic has a frequency that is the frequecy of the 1st harmonic multiplied by 5.


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## ixlramp (Nov 18, 2018)

Deriving intervals from the Harmonic Series
--------------------------------------------------------
Playing the open note of a string and the harmonics, it sounds pleasantly musical, the intervals sound harmonius.
So what happens if we derive our muscal intervals from the harmonics?
Remember that the frequencies of the harmonic series are 1, 2, 3, 4, 5, 6, 7, 8, 9 ... etc.

The most harmonius interval is formed by the open string and the 2nd harmonic.
The frequency ratio is 2:1, so it's a JI interval, this interval is what we call the 'octave'.
It's so harmonius it sounds almost like the same note but higher in pitch.

The next most harmonius interval is between the 2nd harmonic and the 3rd.
The frequency ratio is 3:2, so it's a JI interval, this interval is what we call the JI 'fifth' (702 cents).

Continuing, taking various pair combinations of harmonics we end up deriving these JI intervals:
(Note that an interval with frequency ratio 3:2 is written as 3/2 for a reason i will explain later)
(Note that intervals larger than an octave are ignored because they are considered to be an octave plus a smaller interval, because we consider scales repeat in every octave. For example 3/1 (the 3rd harmonic) = 2/1 plus 3/2 = an 'octave' plus a 'fifth')

2/1
3/2
4/3
5/3
5/4
(6/4 is an interval identical to 3/2)
6/5
7/4
7/5
7/6
8/5
(8/6 is an interval identical to 4/3)
8/7
9/5
(9/6 is an interval identical to 3/2)
9/7
9/8
...
etc.

The convention for writing a JI interval is that we don't write the frequency ratio, instead we write what the frequency of the higher note would be if the lower note has frequency 1:
Ratio
3:2
Divide both sides by 2:
3/2 : 2/2
= 3/2 : 1
So write 3/2.

Also, the convention for writing the unison interval is '1/1' instead of '1'.

So for example the JI major scale is written:
1/1 9/8 5/4 4/3 3/2 5/3 15/8 (2/1)


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## The Omega Cluster (Nov 18, 2018)

This is why it's interesting to study the Gamelan scales, which arose because of the overtones of the drums and bells they use (which are fundamentally different from string overtones).


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## Bobro (Dec 6, 2018)

These are all excellent responses! 

I've been using only a 17-tone to the octave Just Intonation tuning system for many years now and studying microtonal music for even longer. I think one of the most important things is to be concerned about your own personal expressive and musical needs, and beware of any claims that some one tuning is the be all and end all of musical tuning. My own tuning is concerned with getting my own versions of all the middle eastern maqams and ancient Greek tetrachords, and I wound up reinventing the wheel of Ibn Sina's medieval Islamic tunings. By ear. It was a long difficult process, but worth the effort because I based it all on "what feels most natural for me to sing?". I highly recommend that any guitarist interested in microtonality invest in some kind of saz with moveable tied-on frets- then you can test whatever you please for playability and "earability". Then you can invest later in a custom fretting, with previous experience in the tuning and with great confidence that you will use it and not just have some weirdo curiosity guitar lying around unplayed. 

I hope it is not declassé and gauche to mention here that I have a lovely little cura saz, that I bought in Istanbul, for sale and you can just PM me! "Cura" means "little girl" and that describes it very well, but the sound relative to the size is pretty huge (middle eastern instruments mostly evolved outdoors, I believe). 3 double courses, holds a tuning very well and super easy to calculate where the frets should be moved to according to some tuning or other.


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## Stevie_B (Jul 22, 2019)

Hey guys.

I have a pretty nice reply to this thread, but the system won't let me post it because of links, allthough I removed those links.
Maybe someone can figure this out.

Stevie


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## ixlramp (Jul 23, 2019)

Maybe as a new member you are under temporary limitations? Maybe it's an anti-spam thing, stopping links being posted by very new members?
Anyway, i look forward to your input.


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