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My Make Noise Morphagene has the following comment in its configuration file:

//Ratio Table - Equal Temperament
//1.00000: Unison
//1.05946: minor 2nd
//1.12246: Major 2nd
//1.18920: minor 3rd
//1.25992: Major 3rd
//1.33484: Perfect 4th
//1.41421: Tritone
//1.49830: Perfect 5th
//1.58740: minor 6th
//1.68179: Major 6th
//1.78179: minor 7th
//1.88774: Major 7th
//2.00000: Octave
//Multiply these numbers by 2, 4 or 8 for higher octaves, divide for lower octaves.

As noted, these are the ratios of the 12 chromatic notes in equal temperament tuning.

I am trying to convert these ratios to Just Intonation. So far, I've calculated the following numbers:

1.00000: Unison       (1/1)
         minor 2nd
1.12500: Major 2nd    (9/8)
         minor 3rd    (6/5?)
1.25000: Major 3rd    (5/4)
1.33333: Perfect 4th  (4/3)
         Tritone
1.50000: Perfect 5th  (3/2)
         minor 6th
1.66667: Major 6th    (5/3)
         minor 7th
1.87500: Major 7th    (15/8)
2.00000: Octave       (2/1)

My understanding of the Wikipedia page on Just Intonation is that there are different ways to calculate the remaining ratios. However, I do not quite understand the explanation.

Could someone enlighten me on how to calculate the missing numbers?

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    Could you say more about how you're looking to use the revised temperament? "Just intonation" is not a single set of ratios. There are a variety of different just intonations and each has its own method of calculation. Which set of calculations you use depends on your purpose.
    – Aaron
    Nov 30, 2020 at 1:06
  • The Morphagene plays one of these notes together with the root note. By changing its tuning to just intonation I'm looking to reduce the "beating" that sometimes occurs.
    – Jaap Joris Vens
    Nov 30, 2020 at 7:08
  • 1
    Which ratios to use depends on which chords you plan to use. Even the ratios you've specified in the question make some chords badly out of tune, specifically the fifth between the major second and the major sixth.
    – phoog
    Nov 30, 2020 at 23:10

2 Answers 2

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In order to reduce "beats" against the tonic, and to keep the ratios already calculated, use five-limit tuning. The intervals in the OP are correct, with the following additions.

minor 2nd (16/15)
minor 3rd (6/5)
Tritone   (64/45)
minor 6th (8/5)
minor 7th (9/5)

The complete tuning is:

1.00000: Unison       (1/1)
1.06667: minor 2nd    (16/15)
1.12500: Major 2nd    (9/8)
1.20000: minor 3rd    (6/5)
1.25000: Major 3rd    (5/4)
1.33333: Perfect 4th  (4/3)
1.40625: Tritone      (45/32)
1.50000: Perfect 5th  (3/2)
1.60000: minor 6th    (8/5)
1.66667: Major 6th    (5/3)
1.80000: minor 7th    (9/5)
1.87500: Major 7th    (15/8)
2.00000: Octave       (2/1)

(SOURCE)

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  • Is the tritone 64/45 or 45/32?
    – Jaap Joris Vens
    Nov 30, 2020 at 10:21
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    @hedgie - there's no definitive answer to that question, because it depends on what version of just temperament you are using. The concept of a "tritone" doesn't come from the harmonic series, but from tonality. Other candidates for a "just tritone" include 7/5 and 11/8.
    – Scott Wallace
    Nov 30, 2020 at 15:35
  • @hedgie It can be either one. In 5-limit intonation both are possible and the choice is arbitrary.
    – Aaron
    Nov 30, 2020 at 15:48
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    @ScottWallace it depends not only on which version of just intonation you're using but also on whether the tritone is an augmented fourth or a diminished fifth.
    – phoog
    Dec 1, 2020 at 6:08
  • @hedgie to answer that question, you first need to decide whether the note a tritone above C is G♭ or F♯. Then you need to make a few more decisions.
    – phoog
    Dec 1, 2020 at 6:09
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Mostly these are done by combining or inverting other intervals. A major third is 5/3 so a minor sixth should be 3/5 (and moving to the same octave) 6/5; likewise a minor second is the inverse of a major seventh giving 8/15 becoming 16/15 in the 1.0 to 2.0 octave.

A minor seventh is the inverse of a major second so 8/9 going to 16/9 but this isn't necessarily the minor seventh which is a major third above the fifth; that would be 3/2*6/5 which gives 9/5 (both are close to each other.) There's also the 7/4 major seventh.

Just Intonation need on-the-fly adjustments. For example, 5/4 is the just major third but two whole tones give 81/64; an adjustment is often made making another major second being 10/9. A tritone has different frequencies depending on definition: half an octave, three whole tones, two minor thirds, etc.

This all makes Just Intonation difficult to use consistently.

There are some adjustments that can be made in performance by looking both forward and backward in the score. I'll try to post a bit more later.

I didn't find the article I wanted, but here a link to a bunch of pretty good articles on just and other intonations.

I did provide the computation of everything except the tritone. A problem arises in that there may be more than one relation that we wish tones to satisfy. For example, the ratio of 4-5-6 is desired for a major chord. Just intonation gives 1-5/4-3/2 which exactly that relation. Just Intonation as usually described uses ratios of 2,3, and 5 and their powers. No seven or larger prime is needed. However, as no powers of 2, 3, and 5 (or any other prime powers) are close (except for the zeroth power = 1), things become a problem. Seven octaves do not exactly equal twelve fifths. (For a guitar, four perfect fourths at 4/3 and one third at 5/4 give a ration of 4/34/34/35/44/3 which is 320/81 whereas 2 octaves is a ration of 320/80 or 4/1. One either has to temper the notes or play with some chords slightly out of tune or move dynamically (which addressed in the paper I can't find.)

One can adapt in that for some chords one plays "pure" intervals based on the root and in other places on the third or fifth. Doing this nicely shows up in good choirs or string quartets.

Back to the tritone. Three Just Major Seconds gives 9/89/89/8 or 729/512; using two minor thirds gives 6/56/5 gives 36/25. Two semi-tones don't equal one whole-tone 16/1516/15=256/225 which is neither 9/8 nor 10/9. An augmented fourth would be a semi-tone above a fourth 4/316/15=64/45; a diminished fifth is 3/215/16 or 32/45; these are n to equal either.

In performing a piece, one must (if trying to use Just Intonation) choose at each note which version to use so that the preceding and following harmonies (and melodies) make sense.

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  • Honestly, I only understood half of your answer. Would it be possible to calculate the missing numbers, and if so, what would they be?
    – Jaap Joris Vens
    Nov 29, 2020 at 21:08
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    There a bunch of mistakes in the interval examples – don't really matter for the point your making with the answer, but would still be nice to fix them.
    – leftaroundabout
    Nov 29, 2020 at 23:55
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    Could you point out the mistakes or perhaps correct them yourself?
    – Jaap Joris Vens
    Nov 30, 2020 at 7:18
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    "Two semi-tones don't equal one whole-tone": they do once you realize that there are several different sizes of semitone, just as there are two different sizes of whole tone. @hedgie if you don't understand this answer then you should probably not plan on using just intonation (except as a learning exercise, after which you will be able to understand this answer).
    – phoog
    Nov 30, 2020 at 23:16
  • @JaapJorisVens one correction (to the first paragraph): the major third is 5/4; the minor sixth is 8/5 (5/3 is the major sixth, and 3/5 is also the major sixth, but descending instead of ascending). Another correction (to the second paragraph): the minor seventh is a minor third above the fifth, not a major third.
    – phoog
    May 18, 2022 at 9:31

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