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I am looking for ratios of 21-limit just intonation tuning system to later compare it with the 31-EDO tuning system since they are to some extent similar and close. How could I calculate 21-limit just intervals and ratios?

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    "since they are to some extent similar and close": do you have a source for this? If so, what is it? It could help us to understand your question better.
    – phoog
    Jun 3, 2023 at 15:09

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You unfortunately seem to have a certain amount of misapprehension as to what is referred to by "[something]-limit just intonation". I will try to clarify this for you here.

Just tuning systems are based on the natural harmonics that occur when you have some object (– in our case, we are thinking of a musical instrument or some part of a musical instrument) resonating in a "one-dimensional" manner. The classical teaching example is a string of a fixed length and tension, but identical principles apply to, e.g., a key of a glockenspiel, or the length of tube of a brass instrument. A string of a fixed mass, length and tension has a corresponding "fundamental frequency" which can be calculated by a mathematical equation, which is the frequency it resonates at 'along its whole length'. But we can get the string to resonate at other frequencies, by dividing its length up into equal parts. For example, if we 'stop' the string exactly halfway along its length, the string will resonate with twice the fundamental freqency. If we stop it at one-third of its length, it will resonate with three times the frequency. Interestingly, if say we were to stop the string at two-fifths of its length, which is not an equal division of the string, since you can't make a whole up out of combining parts of size two-fifths of the whole, the physical forces will work out the 'simplest' equal division of the string corresponding to being stopped at two-fifths of its length: hopefully you can yourself see that a division into fifths is this simplest solution; and so the string will resonate with five times the fundamental frequency.

These exact divisions of the resonating body/exact multiples of the fundamental frequency have an interesting psychological effect on a human listener: we hear notes that feel 'similar' or 'related' to the note produced by the fundamental frequency. In fact, it is broadly accepted that when we double the frequency, a listener experiences this as being "the same note, but higher", or "a higher version of the same note". So, for the remainder of this explanation I will adopt the view that doubling or halving a frequency gives us "the same note", in a different octave, as this view is adopted by all the tuning systems in scope and does not mark a point of difference between just intonation and equal division of the octave.

Slightly more interesting is tripling the frequency. We generally experience this as a different note, but one that feels like a close companion or neighbour of the note given by the fundamental frequency, in a sense we have come to call consonant. To start making this concrete, take a fundamental frequency of 220Hz. The note with frequency 440Hz feels like "the same note but higher". But the note with frequency 660Hz feels like "a different but very consonant note". We express the relationship between this note and the fundamental note in terms of a ratio of frequencies: the ratio here is 3:1. Of course, I can take this note of frequency 660Hz and move it down an octave, by halving the frequency (to 330Hz) and it will sound like "the same note"; so I can look at the relationship between the fundamental frequency and the new note by looking at the ratio 3:2 (– that is, the ratio between 330Hz and 220Hz).

Multiplying the fundamental frequency by 4 gives us "the same note again", because 4 = 2x2, so we have doubled the note we got from doubling the fundamental frequency, and we are assuming that doubling any given note's frequency gives us the same note.

I am going to temporarily skip over multiplying by 5 to immediately observe something about multiplying by 6. This gives a frequency of 1320Hz. But if I want to move this note into the same octave as the fundamental, I can divide by 4 (that is, divide by 2 and then divide by 2 again), which again gives me a note of 330Hz. So multiplying by 6 gives us the same note, up to octave equivalence, as multiplying by 3. We note that the prime factorisation of 6 is 2x3; essentially what has happened is any multiple of 2 in the prime factorisation doesn't change the note as they can be divided out by moving to a different octave.

Multiplying by 5 gives us 1100Hz / a ratio of 5:1, but if we move this note down until it is in the same octave as the fundamental, we divide by 4 to give a frequency of 275Hz and a ratio of 5:4. This sounds like a different note to either of the ones we have encountered so far (that is, the fundamental (which you could express as 1:1, I suppose) and the 'multiplying by 3' note / the ratio 3:2): it again sounds like it 'matches'/'fits with' the fundamental, but maybe we feel it has a bit more content/a bit more personality/is a bit more distinctive/specific than the 3:2 note, which was quite neutral/plain/austere. We should pay attention to how this note feels against the fundamental.

I am again going to skip over multiplying by 7. Multiplying by 8 is just doubling then doubling then doubling again: it is 2^3 (two cubed; two to the power of three).

Multiplying by 9 gives 1980Hz; in the same octave as the fundamental this is 247.5Hz and 9:8; but let's (for purposes which will become clear) move this up one octave to 495Hz (9:4). If we play the 1:1 note then the 3:2 note, or if we play these two notes together, we hear the nature of the realtionship between them. If we give ourselves a bit of time to clear our heads, and then play the 3:2 note followed by this 9:4 note, it feels like the same kind of relationship – the notes feel "the same distance apart"/"separated by the same interval" as the 1:1 and 3:2 notes did. And then to our astonishment, we notice that we first applied a ratio of 3:2 to the fundamental (1:1) to get 3:2; and then if we apply that ratio again to the note 3:2 itself, we get 9:4. So applying a ratio feels like/sounds like moving by a (musical) interval. Mathematically, looking at 9:4, or the 9:8 that we started with, on the left hand side we have 3x3, and on the right hand side we just have repeated multiplication by 2 which just moves octaves, and what we have done is moved by a certain interval twice: that is, however many times we have multiplied by 3 on the left is how many times we have moved by this interval (- the interval that I'm sure we all recognise as a (just) perfect fifth).

What does it sound like if I apply a 5:4 ratio to the 3:2 note? I get a frequency of 412.5Hz / a ratio of 15:8. If I play this 15:8 note and the 3:2 note together, their relationship to each other sounds like/feels like the same relationship that the 5:4 note had to the fundamental: again, ratios correspond to moving by an interval. Looking mathematically at 15:8, on the left I have 3x5 and on the right repeated multiples of 2. So I have moved by a perfect fifth ('timesing by 3') once, and by this new different interval ('timesing by 5') once. So, multiplying by 5 on the left corresponds to moving by a just major third.

We can hopefully see that moving around by only intervals of perfect fifths and major thirds (and octaves) will gives us ratios m:n where the prime factors of m and n are taken only from 2, 3 and 5, and where the power of each of these primes in the factorisation gives, respectively, how many octaves, just perfect fifths and just major thirds we have moved by, upwards if on the left of the ratio and downwards if on the right – that is, the 'intervallic' content of a ratio is completely determined by its prime factorisation and we only get "new" intervals from (new) prime numbers.

The notes we get from doing what I say in the previous paragraph is called 5-limit just intonation, because 5 is the largest prime I use.

21-limit just intonation doesn't exist, because 21=3x7 is not a prime number.

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An X-limit system is one in which the interval ratios can all be expressed in terms of prime factorizations in which X is the largest prime. So a "21-limit" tuning would be equivalent to a 19-limit tuning (or even a 7-limit system), since any ratio involving 21 could be expressed in terms of 3 and 7, and any ratio involving 20 could be expressed in terms of 2 and 5.

Ratios for a 19-limit system and others can be found at Xenharmonic Wiki.

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