# 'Just Intonation' explained and demonstrated on video: Hansford Rowe with JI bass



## ixlramp (May 23, 2012)

Another JI Warwick bass, owner unknown:


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## Winspear (May 23, 2012)

Have understood this for a while now but never seen it explained so well! Awesome.

I just wish I understood why these instruments can't play in multiple keys. If they were as flexible as 12 tone I would most definitely try, they sound fantastic. That said, I rather mean the chords individually sound fantastic. I've not heard a piece I like..


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## ixlramp (May 24, 2012)

The precise JI intervals are unequally spaced through the octave, so each tonic you choose creates a new set of available JI intervals.

The thing to do, for freedom of modulation, would be choose an equal temperament that closely approximates some of the JI intervals you prefer.
I like 24ET because it approximates an exotic / Arabic 15 tone JI tone system (constructed from the 3rd and 11th harmonics) to within 4 cents, which is close enough and also creates a slow shimmer, which is preferable to perfect tuning.
19ET is roughly close to Just major and minor thirds, which is why ZIA use it a lot, it's good for space pop:


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## Mr. Big Noodles (May 24, 2012)

ixlramp, could you post some music that uses modern JI? Like EtherealEntity, I hear a lot of comparisons of "This is a poopy major triad! Booooo! This is a just major triad! Whoopee!", but not much music. I tried following Mr. Rowe's trail of crumbs, but he only succeeded in angering me with his smooth jazz.


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## EJA (May 24, 2012)

Here's some music using Just Intonation:



Just Intonation Blues:


Their new CD is amazing! Their first two albums are on iTunes...and maybe the new one--I haven't checked.

Michael Harrison's _Revelation
_
This one is on iTunes--get it now! 

Ambient Just Intonation
some of my stuff:

much of my music is free for download if you're interested: Eric Jackson - Ambient Composer - Science Fiction & Fantasy Writer - Fretless & Microtonal Guitarist and all in Just Intonation

Someone infinitely better than myself:



Classical Guitar in Just

Steel String guitar

The collection of these Lou Harrison guitar songs is iTunes. Por Guitaro by John Schneider. This is a wonderful, wonderful album.

Terry Riley's _The New Harp of Albion
_
another wonderful piano piece in Just. Also on iTunes.

And of course...there's Harry Partch. Great BBC documentary on youtube if you've got the time. Partch may be difficult for some people to get into at first, so I recommend a natural discovery of his music.


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## ixlramp (May 24, 2012)

SchecterWhore said:


> ixlramp, could you post some music that uses modern JI?


Sure i'll post in this thread as they come to mind.


To start with here's a free download EP by Cipher, a sorta new-wave / darkwave / post punk / deathrock / art-rock guitar band.

"The band Cipher (Los Angeles-late 70s to mid 80s) played in a 7-limit 22-tone scale of Erv Wilson. The intonation was done under the guidance of Jose Garcia who refretted all the guitars and bass. Co-composer, Marsha Mann, who was the lead singer and lyricist for the band, also sang in the same tuning. They appeared on New Wave Theater. Cipher is listed and pictured (above The Clash) in the 1985 illustrated encyclopedia, 'Who's New Wave in Music', by David Blanco, who refers to them as a 'microtonal dance band'."

'7 limit' meaning JI intervals derived from harmonics 2 3 5 and 7.
More info at 'goutroy' blog: A Viable Commercial: Cipher: Selftitled EP
30 MB download: Cipher EP.zip


One of my favourite microtonal artists is Jacky Ligon, here's an awesome electronic track in a fairly complex and original JI system:
Jacky Ligon - Other Time by Xen-Arts on SoundCloud - Create, record and share your sounds for free
This is from a free download EP, currently unavailable but keep an eye on:
Xen-Arts: Xen-Arts EP Release: Jacky Ligon | Other Time (2011)


Now various fractal generated music in JI by Billy Stiltner. These are long explorations of fractals using the cursor, the fractal orbits generate the pitches, i don't expect anyone to listen to the full tracks:










Matthew Grasso and Nada Brahma Music Ensemble using a 14 tone JI guitar:









Mark Allan Barnes, the microtonal Bard of Cholsey, England, using a DIY exchangeable neck guitar, this track's in Pythagorean tuning (not sure this is 'modern JI'), only a subtle difference from 12ET:


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## ixlramp (May 24, 2012)

EtherealEntity said:


> I just wish I understood why these instruments can't play in multiple keys. If they were as flexible as 12 tone I would most definitely try, they sound fantastic. That said, I rather mean the chords individually sound fantastic. I've not heard a piece I like..


... so yeah you can play multiple keys but changing the tonic note usually forces you to change the scale too, you end up with new sets of JI intervals usually more complex ('dissonant'). You can of course modulate to any of the modes. It's down to careful design of a JI fretboard, with plenty of frets some modulation while staying in the same scale is possible. Then there's the possibilities resulting from retuning the open strings.

But then, with an equal step system there are only ever a low number of intervals possible, for example the 19 intervals of 19ET are the same whichever tonic you choose. With a JI system every possible tonic generates many new JI intervals, so it is more subtle, complex, varied, but with limitations you have to work within.

The history of tuning systems in Europe is one of moving from JI to ET through various compromise stages, purely for the freedom of modulation. An ET is an abstract system brought about by compositional convenience alone, the intervals are not chosen artistically by ear, or derived from the natural harmonics as JI is. So that's the price you pay. Not saying ETs are 'bad', just abstract structures of pitch.

Microtonal guitarists are a minority within a minority, so it is rare to find something you like. Most of my favourite microtonal music is electronic for that reason alone.

Concerning your difficulty in choosing an ET ... Depending on your soundcard, free download SCALA can play 3900+ microtonal scales from your computer keyboard: Scala Home Page


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## ixlramp (May 24, 2012)

ixlramp said:


> I like 24ET because it approximates an exotic / Arabic 15 tone JI tone system (constructed from the 3rd and 11th harmonics) to within 4 cents



just interval frequency (root=1) / just interval cents / 24EDO interval cents / error cents / interval name

2/1 ........ 1200 ..... 1200 ..... 0 ..... Octave
64/33 ..... 1147 ..... 1150 ..... 3 ..... Supermajor seventh
121/64 .... 1103 ..... 1100 ..... 3 .... Major seventh
11/6 ....... 1049 ..... 1050 ..... 1 ..... Neutral seventh
16/9 ....... 996 ....... 1000 ..... 4 ..... Minor seventh
18/11 ..... 853 ....... 850 ....... 3 ..... Neutral sixth
3/2 ......... 702 ....... 700 ....... 2 ..... Fifth
16/11 ..... 649 ....... 650 ....... 1 ..... Sub fifth
11/8 ....... 551 ....... 550 ....... 1 ..... Super fourth
4/3 ......... 498 ....... 500 ....... 2 ..... Fourth
11/9 ....... 347 ....... 350 ....... 3 ..... Neutral third
9/8 ........ 204 ........ 200 ....... 4 ..... Major second
12/11 ..... 151 ....... 150 ....... 1 ..... Neutral second
128/121 .. 97 ........ 100 ....... 3 ...... Minor second
33/32 ..... 53 ......... 50 ........ 3 ...... Subminor second
1/1 ......... 0 .......... 0 .......... 0 ...... Unison


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## in-pursuit (May 25, 2012)

ixlramp, I'm listening to the first Billy Stiltner video and I have to say I am both amazed at how interesting it sounds but not at all surprised at the same time. I took a look through one of the links and he posted a link to the program he wrote which was used to compose the fractal music. After reading it I'm convinced that he is nothing short of a genius.

it's a bit of a shame there aren't many microtonal artists around, I think other tuning systems have a lot to offer even if a lot of them are difficult to implement with traditional western instruments.


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## EJA (May 25, 2012)

in-pursuit said:


> ixlramp, I'm listening to the first Billy Stiltner video and I have to say I am both amazed at how interesting it sounds but not at all surprised at the same time. I took a look through one of the links and he posted a link to the program he wrote which was used to compose the fractal music. After reading it I'm convinced that he is nothing short of a genius.
> 
> it's a bit of a shame there aren't many microtonal artists around, I think other tuning systems have a lot to offer even if a lot of them are difficult to implement with traditional western instruments.



Microtonal systems have their own unique rules which require effort to understanding. Some pitches and intervals do take a little while to get used to, but it's not terribly long. I used to think microtones were absolute nonsense and that it was just a bunch of people trying to be different and not actually listening to their music. Nothing could be further from the truth.

Just intonation can be combined with 12 tone equal temperament. The system is called the 12 Tone Ultra Plus. This is a pic of my Ultra Plus 7 string guitar:





I've been collaborating on a downtempo/ambient piece with someone who doesn't use any microtones with this guitar. We're able to completely communicate musically and it doesn't sound strange--he has never heard of just intonation before and hasn't once said it sounds strange.

I also teach regularly on this guitar (not microtonal stuff, regular guitar lessons) and I have no trouble playing traditional western stuff and including microtones at the same time.


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## ixlramp (May 25, 2012)

Good to see a photo, beautiful guitar.


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## Solodini (May 26, 2012)

That first Stiltner track is wonderful. I'll be listening to more of that!


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## ixlramp (May 26, 2012)

You can download Billy's music from here: billystiltner


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## Explorer (May 27, 2012)

When someone says "using a new tonic note winds up with a new set of intervals"... do they appear on the instrument like magic, or are you still working with the same intervals which are on the instrument? It sounds like one is just using another mode of whatever just scale is "hard fret-wired" on the fretboard.

On a equal-tempered fretboard, one can freely change modes and the intervallic relationships are the same for all scales. Some of the talk in this topic seems to be made up of odd statements which avoid addressing the inharmonicity of the relationships in some scales and modes. 

Justly intonated instruments don't go outside of certain keys and scales, and that's okay. Putting up vid after vid of instruments staying inside their limits is great for demonstrating how good it sounds in that narrow range. I would find it more convincing (not that it will happen) to see a vid where JI is put up against equal temperament across all keys and scales. Otherwise, I don't feel the "sweetness" of JI to overcome the inherent limit put on one's ability to play with other musicians. 

Then again, I feel the same way about people who insist on Esperanto instead of a more common language. The only people you'll be able to talk to in a given country are the language geeks there who also learned Esperanto. I'd rather have a real second language, because I'm going to find more people in a given country speaking that second language, even if it's not common, than are speaking Esperanto. 

I can find more French speakers in Tokyo than Esperanto speakers, for example.


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## EJA (May 27, 2012)

Explorer said:


> When someone says "using a new tonic note winds up with a new set of intervals"... do they appear on the instrument like magic, or are you still working with the same intervals which are on the instrument? It sounds like one is just using another mode of whatever just scale is "hard fret-wired" on the fretboard.
> 
> On a equal-tempered fretboard, one can freely change modes and the intervallic relationships are the same for all scales. Some of the talk in this topic seems to be made up of odd statements which avoid addressing the inharmonicity of the relationships in some scales and modes.
> 
> ...



Explorer, thanks for your question.

I think the best way for me to address these points is through a video. I'm going to put something together on youtube later today or tomorrow explaining these aspects through musical example.

For now, I'll try my best to answer these questions--and they are good questions and raise excellent points.

I have to point out that not being able to change keys in just intonation is a myth. It is plenty possible--you just can't do it the same way you do in equal temperament. Different theories are required which require musical use to discover. Additionally, explaining it often requires a lot of vocabulary which most people don't have. If you want to change key in just intonation, you're just going to need more notes in your system.

Firstly, lets start by analyzing what I feel are the two main reasons for using just intonation--these are the reasons I've adopted them in my music.

1. Being harmonically in tune.
2. Access to new harmonies/notes/chords.

Okay, so number one is all about harmony. If two notes are relatable by whole number divisions of a vibrating force, they are more consonant and more 'in tune'. Additionally, vibrating forces have a natural division point (the harmonics/nodes) at whole number measurements. This is physics--I'm not making it up.

So in regards to changing key, if I'm singing something, or using a fretless instrument--there are No limits. I can adapt to whatever key/root/fundamental I please. The issue comes when we want to assign frets or specifically tuned keys (like piano keys) to something. The 3rd harmonic and the 5th harmonic are more "microtonal" in a sense because of how close they are to our 12 tone notes. Our 12 notes are based on the sonic flavor of these two harmonics. What we've done is tempered them so that we have an approximation on each of the 12 pitches. 

Think of it like Taco Bell approximating traditional Mexican food. It's easier to hop in a car and go through the drive thru to get a taco than it is to catch a flight to the Mexico. 

The issue is that the 3rd harmonic and the 5th harmonic, physically, are very close to each other-they cause problems. Our 12 tone system favors the 3rd harmonic more, so the 5th suffers. An older system, meantone temperament, favored the 5th harmonic, and thus you got a "wolf" 3rd harmonic that was usually avoided. 

Here's an experiment you can try on your guitar right now to see the difference. Play the 3rd harmonic (located over the 7th fret) on your B string (low B on a 7 string or regular B). Look at your tuner. Unless you have a strobe tuner, most likely it will read as an 'F Sharp' (or G flat). This harmonic however is 2 cents _sharper_ than the F sharp you have a fret for. Though it's such a small difference that usually it's hard to hear. (If you wanted sustained drone music however, you will hear it! This is why classical Indian music is in Just intonation). Now on to the second part of the experiment:
Play the 5th harmonic on your D string. This is located a tad shy of the 4th fret--closer to the nut. Plug into a tuner and it will tell you that it's a 'mistuned' F sharp, or G flat. In reality, this F sharp is about 14 cents _flatter_ than the F sharp you have a fret for.

It is also 12 cents flatter than the 3rd harmonic you played on the B string. 

Now play the two together. 
First the 3rd harmonic (or 3/2 in JI jargon) of the B string.
and now the 5th harmonic (or 5/4) of the D string.

You'll hear a beating because the notes are close, but not exact. This difference in pitch is called a 'comma'. They are 12 cents apart. Imagine placing two frets 12 cents apart from eachother. Keep in mind all your frets on a normal guitar are _100_ cents apart!
This is where people assume you can't change key. From the 12 tone equal world, we want that F sharp to be the same note to serve multiple purposes, not only as the perfect 5th of B and the major third of D, but as the 9th of E, or the major 6th of A. In just intonation, each pitch is an individual force that serves a particular purpose--some are shared--but many (especially if we're talking about the 3rd and 5th harmonics) are not.

In practice, you could use the 2 F sharps we found on the D and B string. You could just refer to the harmonic ratios (the 5/4 of D, the 3/2 of B) which clearly indicate a difference. Or you can make slight indications such as writing a 'prime' symbol next to the higher F sharp in the musical notation. This is also another point which makes JI difficult--especially for guitarists--you _need_ a form of musical notation. Tab will not work. It cannot present the data necessary. You need to understand the harmonic relationships on a deeper level than you would even with traditional western music. 

No onto to number 2: Just Intonation as a portal for new notes.

When you take a single string and look at every odd number division, you'll get a new note. When you reach every new Prime number, you get a new note exclusive to the original pitch (the fundamental).

Let's say I take a string and tune it to 'A'.
My 1/1 is now A.

Now I find the 3rd harmonic (the 3/2) which to A is about an E. (this E is two cents sharper than the E your used to). 

So I tune another string to the pure 3/2.
Now I have A and E. The E is vibrating at 3 times the original A. Remember, it's all about whole numbers.

Let's say I now find the 3rd harmonic of this E. This will be the 9th harmonic in relation to the first pitch. Why? because 3 times 3 is nine. 3/2 x 3/2 = 9/8. This note will be about a B. But remember that 2 cents difference! This B is now 4 cents sharper than the B we're used to on a piano or traditional fret. 

Now if you took the original A string and divided its length by 1/9, surprise, you'll find this B, the 9/8. You can actually find the 9th harmonic almost directly over your second fret on a traditional guitar. It's a tough one to get to ring--distortion helps to make it clearer.

So we can find all the pitches based on the 3rd harmonic from constantly multiplying integers by 3, or integers that can be broken down into three. We have two names for this in the tuning world: either Pythagorean Tuning, or we call it 3 Limit--as 3 will be the highest prime number we multiply by.

We can do the same for 5. We'll increase the limit to 5 then.

3 and 5 is it for most Western ears. Even though we don't have them purely on a keyboard or fretted guitar, we have their 'taco bell' equivalent flavors.

Remember our original A, the 1/1. If we split that by a new prime number, say 7, we get a note that we don't even have a representation for in 12 tone music. For A, we'll get a G flatted by about 30 cents--we can call this a G Half Flat. We can go into higher primes as well, say 11--for A, this will give us a D Quarter Tone Sharp. (it's about 51 cents sharper than the D we usually use. 13 is also a kind of quarter tone at around 40 cents. These are new harmonies and melodic resources for us to play with. And they are naturally occurring in a vibrating force--this is not something that some mathematician made up.

If we want to freely change Key, than we need to draft up harmonics for whatever pitch we want to change to the key of. Say a simple I IV V progression where all notes are dominant 7th chords.

A
D
E

lets replace our usual minor 7th for the 7th harmonic (which is what Hansford Rowe was talking about in the video). 

A7
D7
E7

each chord must consist of the following ratios:
A7 (1/1, 5/4, 3/2, 7/4)
D7 (1/1, 5/4, 3/2, 7/4)
E7 (1/1, 5/4, 3/2, 7/4)

or in terms of notes:
A, C#, E, G half flat
D, F#, A, C half flat
E, G#, B, D half flat.

Notice how the 'D' for the E7 chord is not the same as the 'D' for the D7 chord. This just means that if you want to solo over this, or right a melody, you have to change key with the chord....wait, isn't that a Jazz concept?

Changing key in Just Intonation is Jazz theory on steroids. 

I hope this helped in some small way. I'll be posting a video very soon to make things a bit clearer.


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## Winspear (May 27, 2012)

Nice post, cheers!
I understand the concept of harmonics on a vibrating string. I also remember learning about the different ways instruments produce these harmonics, and their equivalents in saw waves, triangle waves etc. I recall that some instruments only produce the 1, 3, 5 etc. Other produce all..and so on.

What is the case with a guitar string? All of them? At which point do you think they start to become inaudible (In other words, what 'limit' do you think is recognizable in an open string?) That seems logically to me a good limit to work with - unless of course it produces a difficult fretboard system.

EDIT: I think that's actually a silly question. I seem to recall the 1/2 is produced at half the volume, the 1/3 at a third the volume, and so on. So the answer depends entirely on how loud the fundamental is. I'd be curious what you have to answer anyway


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## Explorer (May 27, 2012)

So apparently you need to have frets on your fretboard to correspond to each of the justly intonated pitches in each key you want to play in, no? Or, you need to compromise on some of those pitches, and use some kind of 31 frets per octave system. 

I just wanted to clarify, as that particular aspect always seems to get glossed over. 

Yes, I understand that the human voice, fretless instruments and the musical saw can play any pitch within their ranges. Since this topic has been about fretted instruments with just intonation, and since it's on a guitar forum, the versatility of all those instruments doesn't really matter to the paradigm under discussion... at least as far as I'm aware.

----

Here's an idea for free.

If I were seriously interested in following just intonation in multiple keys, I'd write a Max routine which would automatically do the pitch variations from normal straight frets for a given key, with the key selectable by a MIDI pedalboard. I'd run a Roland-ready instrument through that routine and be able to play in all keys, just having to push a button/pedal to choose a new key center. 

I used to be able to do this kind of thing using my Yamaha G10 guitar controller and a TX802, so I imagine there's no patent possibilities due to prior art. Having known the guy who was selling the interchangeable fretboard though, I can assure you that there's not a lot of money in that stuff.


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## Winspear (May 27, 2012)

I'm also interested about the fretting systems. Also the string to string tuning of the guitar and how that can create variables. 
I understand the basics of the creation of this system but applying it to an instrument with multiple strings and frets is really confusing to me!


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## EJA (May 27, 2012)

Explorer said:


> So apparently you need to have frets on your fretboard to correspond to each of the justly intonated pitches in each key you want to play in, no? Or, you need to compromise on some of those pitches, and use some kind of 31 frets per octave system.
> 
> I just wanted to clarify, as that particular aspect always seems to get glossed over.
> 
> ...



To comment on your first statement:
yes. You would need frets that correspond with specific ratios of various keys. However, if you're designing a JI fretboard, you still need a master 1/1. You can then select harmonics of various harmonics throughout the system. If you want the 7th harmonic of the third harmonic, to the 1/1, that would be the 21st harmonic for the 1/1 (21/16). We can think of it either has the 3/2's 7th harmonic (which will work for a dom 7 chord on the 3/2) or the 1/1's 21st. Either way. The point is that it always goes back to 1/1. It's kind of like the various modes in regular diatonic theory. Viewing a piece as being identified with Dorian, but still coming from Ionian. But you can add more notes to make Dorian its own Ionian, if that makes sense.
In JI, you can't have everything. You need to sacrifice some pitches for others. Equal temperament sacrifices the harmonic purity for ease of changing key, in Just, you will have to weed out certain ratios that will make a fretting or keyboard design difficult. This is why some JI composers have had 29, 43, 64, etc tuning systems. This would work well if guitarists embraced JI because now we'd have a real reason to justify buying more guitars. "But this one has the 14/9 of my 63/32!!!" 

And on your second point on the midi programming is dead on. JI is more viable with today's technology than it was in the past. Essentially what has happened is that we've forgotten about the harmonic series because we were so distracted by the awesome music we could create in 12 tone. 



EtherealEntity said:


> I'm also interested about the fretting systems. Also the string to string tuning of the guitar and how that can create variables.
> I understand the basics of the creation of this system but applying it to an instrument with multiple strings and frets is really confusing to me!



Your tuning of the strings still has to connect back to an original 1/1. This again is where people get confused about changing key. Yes there is a master 1/1 (let's say A). But there's nothing stopping you from playing in the key of the 7th harmonic!

Normal guitar tuning presents an issue when it comes to JI. The problem lies with the major 3rd between the G and B strings. Lets say your 1/1 is 'A'. Your G then would be a ratio of 16/9 (slight tuning variation from the 'normal' G) and your B would be 9/8. But the difference between 16/9 and 9/8 is not a pure JI major 3rd (5/4). Remember the issue I pointed out earlier between your D string's 5th harmonic and your B string's 3rd harmonic? Same thing. So you would end up with an impure interval between the G and B which would be called a Comma. If you tune them purely to a major third interval, then you end up with the comma somewhere else. Say you detune the B string to be a 5/4 from the G. Then in relation to the master 1/1, you'd have 160/81 on the B string, which is not a pure 9th.

If this is confusing to you--you're normal. This stuff _is_ confusing from the start. It took awhile for me. Nearly a year of constant use and practice to be able to rattle this stuff off like I can. Most people won't devote such time because this stuff is regarded as either archaic theory, or out of tune, or wrong. Or they'll just avoid it because nearly all universities find this stuff to be not worth the effort and don't teach any of it--or worse, claim its impossible or again not worth the effort.


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## All_¥our_Bass (May 27, 2012)

Explorer said:


> Or, you need to compromise on some of those pitches, and use some kind of 31 frets per octave system.


31 equal pitches to the octave gives you everything that's great about meantone without any wolves (BEAUTIFUL maj and min chords for one) and very closely approximates interval ratios 3 5 7 11 13, in addition to the pure octave (2).


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## ixlramp (May 27, 2012)

Explorer said:


> "using a new tonic note winds up with a new set of intervals"... do they appear on the instrument like magic, or are you still working with the same intervals which are on the instrument? It sounds like one is just using another mode of whatever just scale is "hard fret-wired" on the fretboard.


The latter, no magic ... yes, whatever JI system of frets is used, moving the tonic means you are then limited to the modes of that system. However since that system may have 24+ tones per octave there are still many possibilites within that mode.

Essentially it's just simple relativism due to the unequally spaced pitches in the octave. For analogy standing on an infinite grid of equally spaced lines, whichever vertex you stand on, the grid looks the same relative to you. Standing on a grid of unequally spaced lines, how this looks relative to you depends on where you stand.


Explorer said:


> odd statements which avoid addressing the inharmonicity of the relationships in some scales and modes.


The inharmonicity of the intervals of the modes is not a problem. I need to clarify some things since Hansford's video is extremely basic ...

JI is not about consonant intervals, it is not a single scale or a few harmonic-derived notes up to the 13th harmonic. There are an infinite number of JI intervals per octave creating a continuous spectrum if frequency, ranging from extremely consonant (the octave 2/1, the natural fifth 3/2) to the extremely dissonant, dissonance far beyond what 12ET is capable of.

See this page Stichting Huygens-Fokker: List of intervals which lists JI intervals in order from consonant to dissonant.

The more consonant intervals are often considered to be up to the '13 limit', meaning derived from the prime harmonics 3 5 7 11 13 and all possible intervals between those harmonics. These ones:
1/1 unison, perfect prime
2/1 octave
3/2 perfect fifth
4/3 perfect fourth
5/3 major sixth, BP sixth
5/4 major third
6/5 minor third
7/3 minimal tenth, BP tenth
7/4 harmonic seventh
7/5 septimal or Huygens' tritone, BP fourth
7/6 septimal minor third
8/5 minor sixth
8/7 septimal whole tone
9/4 major ninth
9/5 just minor seventh, BP seventh
9/7 septimal major third, BP third
9/8 major whole tone
10/7 Euler's tritone
10/9 minor whole tone
11/5 neutral ninth
11/6 21/4-tone, undecimal neutral seventh
11/7 undecimal augmented fifth
11/8 undecimal semi-augmented fourth
11/9 undecimal neutral third
11/10 4/5-tone, Ptolemy's second
12/7 septimal major sixth
12/11 3/4-tone, undecimal neutral second
13/7 16/3-tone
13/8 tridecimal neutral sixth
13/9 tridecimal diminished fifth
13/10 tridecimal semi-diminished fourth
13/11 tridecimal minor third
13/12 tridecimal 2/3-tone

When you see a JI interval expressed as for example 13/12, that means the frequencies of the 2 notes are in the exact ratio 13:12, but also, that is the interval between (and derived from) the 12th and 13th harmonics.


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## ixlramp (May 28, 2012)

EtherealEntity said:


> I recall that some instruments only produce the 1, 3, 5 etc. Other produce all..and so on.
> 
> What is the case with a guitar string? All of them? At which point do you think they start to become inaudible (In other words, what 'limit' do you think is recognizable in an open string?)


Yeah strings produce all harmonics, clarinets are odd harmonic only.

A subcontra bass string tuned to say 20Hz, picked hard will momentarily produce hundreds of harmonics, frequencies perhaps up to 20 000 Hz, which corresponds to the 1000th harmonic. This is the rough range of human hearing. More importantly though, inharmonicity of a string due to stiffness tends to sharpen the higher harmonics. At some point a harmonic will be more than 4 cents sharp and will not be compatible with a mathematically designed JI system. Where that point is i don't know, but i have a microtonal tuner and can find out.

For my theory work i tend to stop at harmonic 1024 (being 2^10).


EtherealEntity said:


> I understand the basics of the creation of this system but applying it to an instrument with multiple strings and frets is really confusing to me!


It's a total nightmare for me too.


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## ixlramp (May 28, 2012)

Explorer said:


> So apparently you need to have frets on your fretboard to correspond to each of the justly intonated pitches in each key you want to play in, no?


Yes  The limitation being keeping the fret spacing playable. Designing JI fretboards is a nightmare. My approach is to forget about what modulations i will want in future (cus you never know really) and just design a JI fretboard by cramming in as many JI intervals as possible while keeping a playable fret spacing ... then simply accept the limitations as they arise, work within them and let them shape the music i compose. This is how i designed this 29 tone per octave fretboard for a root-fith-root-fifth etc. open tuning:







The thread for that design is somewhere in this forum ...

Since i am a lousy musician i like limits.


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## ixlramp (May 28, 2012)

Check out the latest post in this thread ... http://www.sevenstring.org/forum/general-music-discussion/198590-microtonal-guitar-demo-video-slimeguitar-26-equal-steps-per-octave-more-videos.html#post3027261


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## All_¥our_Bass (May 29, 2012)

Or you could have adjustable fretlets, like this guy.


26 edo (*E*qually *D*ivided *O*ctave)
This isn't just tuning obviously, but I thought you guys would really dig this.


This thread also isn't helping my urge to buy another guitar to defret for some glissy micro goodness.


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## All_¥our_Bass (May 31, 2012)

Dante Rosati Guitar


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## ixlramp (Jun 6, 2012)




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## ixlramp (Jun 24, 2012)




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## tamahome (May 12, 2021)

I happened to come upon this link through google. Hansford Rowe's "Steel Blue" album on Bandcamp is an amazing JI album with Jon Catler on guitar.

Also some more good JI songs on Hansford Rowe's Soundcloud, look for:
"Tunes in Just Intonation (tuning system)". There's also a "‎Jon Catler Group by Hansford Rowe" on apple music.

You can easily try out JI yourself on electric guitar, tuning an open D7th chord low to high: D A D F# C D. Then just detune the F# (15 cents) and C (30 cents) until they're perfectly in tune.

Nice instrumental guitar JI songs on youtube or spotify (another Jon Catler band): Willie McBlind - "13 O'Clock Blues" or "Chicken". "Canonballer" is mostly instrumental.

Robert Rich's Neurogenesis or Electric Ladder albums is on spotify or Filaments on bandcamp. They're more "active" than his usual ambient albums, although the JI isn't as in your face as other artists.


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