Background: Algebraic Structure Of Musical Intervals And Pitches, From Notated Music to Audible Sounds
Previously: EDO Generators, Reinventing 7-Limit Just Intonation
There's a youtube channel I just found called Quartertone Harmony that explores... quartertone harmony. It's pretty good and it makes me want to learn about quartertone music, i.e. music written in 24-EDO.
A quick refresher on interval space and EDO tuning systems:
Tuning-systems map interval-space to frequency-ratios. For rank-2 interval space, we we can make a tuning system that breaks an Octave into logarithmically Equal Divisions (an EDO) by "tempering out" a second basis interval, B, that's independent of the octave, i.e. we set the tuned value of the basis interval to unity, so that t(P8) = 2 and t(B) = 1. That fully defines an EDO tuning system and we can use it to figure out the logarithmic frequency ratio for any other interval using those two basis intervals. There are other tuning systems, like just tuning systems and the meantone temperaments, but this post is mainly about 24-EDO.
How does the tuning work though? Well first, prior to the choice of any tuning system, the rank-2 intervals themselves can each be represented as a pair of integers in a coordinate system with two intervals as a coordinate basis. The classic basis for representing rank-2 intervals is the Pythagorean basis, (P5, P8), in which P5 has coordinates (1, 0) and P8 has coordinates (0, 1). With combinations of those you can get other intervals like a perfect fourth (which is a perfect fifth below an octave, P4 = (-1, 1)), or a major sixth (which is one octave below three perfect fifths, M6 = (3, -1)). The names like P4 and M6 are more fixed by history than fixed by the math, but history found a pretty good system and you can read about how intervals are regularly named in Algebraic Structure Of Musical Intervals And Pitches. (P5, P8) is historically important as a basis of rank-2 interval space, but my favorite rank-2 basis is (A1, d2), which Lilley also talks about in that post Algebraic Structure post.
Once you pick basis vectors for your tuning system and their tuned values, then you can tune any other interval. To tune a rank-2 interval (m, n) in a tuning system, we start with our two basis intervals (a, b) and (c, d), and tuned values for them, t(a, b) = basis_value1 and t(c, d) = basis_value2. Then
frequency_ratio = t(m, n) = (basis_value1)^x * (basis_value2)^y
where
y = (a * n - b * m) / (a * d - b * c)
This simplifies a lot for EDOs. Since one of the basis vectors has a value of one, and one raised to a power is one, the frequency ratio formula just becomes:
frequency_ratio = 2^y
where
y = (a * n - b * m) / (a * 12 - b * 7)
.
In the EDO-generator post we found out which intervals can be tempered out to make EDOs that preserve the same order of primitive intervals (P1, m2, M2, m3, M3, P4, P5, m6, M6, m7, M7) as 12-TET. We also found out that there is no interval that can be tempered out to produce a 24-EDO that preserves the 12-TET order of primitive intervals.
A lesser man might say, "I guess 24-EDO isn't well behaved, let's write music in a different tuning system.". That used to be me. Then I found an exuberant gesticulating youtuber with some really cool chord progressions and now I need to go deeper.
I'm going to try understanding 24-EDO music from several different perspectives. The first perspective is as a subset of an EDO with more divisions, namely 72-EDO. Next, I want to look at different 24-EDOs that don't preserve the 12-TET order and see if there's something musically interesting/important/insightful about how the usual order of primitive intervals is not preserved - maybe it's good that the usual primitive order is broken -- maybe 12-TET is wrong. Third, I want to try analyzing 24-EDO music as 12-TET music equipped with just-borrowings, e.g. pitch bends and tremolos that hint at harmonic intervals and other justly tuned intervals, including intervals that aren't rank-2. That last perspective is where I'll try to understand more deeply the stuff in the Quartertone Harmony videos.
Okay, first up, 24-EDO as a subset of a higher EDO. We can't use 48-EDO because it also isn't well behaved. The first even multiple of 12 that's well behaved is 72-EDO. The natural way to produce a 72-EDO is by tempering out dddddddddd14, which has coordinates of (12, 13) in the (A1, d2) basis, if you were wondering.
Here are the names of the 72-EDO intervals that have the same frequency ratios as 24-EDO:
2^(1/24): dd4
2^(2/24): dddd7
2^(3/24): dddddd10, AAAA-3
2^(4/24): A0
2^(5/24): m3
2^(6/24): dd6
2^(7/24): dddd9, AAAAA-4
2^(8/24): AAA-1
2^(9/24): A2
2^(10/24): d5
2^(11/24): ddd8
2^(12/24): ddddd11, AAAAA-2
2^(13/24): AAA1
2^(14/24): A4
2^(15/24): d7
2^(16/24): ddd10
2^(17/24): AAAA0, ddddd13
2^(18/24): AA3
2^(19/24): M6
2^(20/24): d9
2^(21/24): dddd12 AAAAAA-1
2^(22/24): AAAA2
2^(23/24): AA5
2^(24/24): P8
Analyzing 24-EDO as being secretly 72-EDO is about as crazy as analyzing 12-EDO as if it were secretly 72-EDO: no one would do it if there were a better option. You might notice that almost none of the primitive intervals are present above, e.g. if we try to analyze 24-EDO as 72-EDO, then we're looking at music that doesn't have any major thirds or perfect fifths. In this system, when someone plays what would be called a major triad in 12-TET, (P1, M3, P5), we should now call it (P1, AAA-1, A4). I confess, my knowledge of music theory is not so well developed that I know a standard name for that chord.
That didn't work so well. Next, let's look at how different 24-EDOs disrespect 12-TET.
The interval with the smallest absolute (A1, d2) coordinates that can be tempered out to get a 24-EDO is AAA6, with coordinates (12, 5) in (A1, d2). If we temper that out, we get... a weird mess.
In the 24-EDO made by tempering out AAA6, diminishing an interval increases its frequency by a factor of 2^(5/12), e.g.
m2 : 2^(7/24)
d2 : 2^(12/24)
dd2 : 2^(17/24)
.
Here's a fuller list:
.
And here are just the natural intervals:
2^(2/24): M2
2^(4/24): M3
2^(7/24): m2
2^(9/24): m3
2^(11/24): P4
2^(13/24): P5
2^(15/24): M6
2^(17/24): M7
2^(20/24): m6
2^(22/24): m7
2^(24/24): P8
.
I contemplated a respelling where every major is swapped with a minor and every augmentation is swapped with a diminution. That respelling still doesn't give us 12-TET order over primitive intervals, but it's a lot closer. The major triad in the respelled system actually looks pretty reasonable. It has frequency ratios of
2^(9/24): M3
2^(13/24): P5
whereas the frequency ratios for the elements of the major triad in 12 TET are now named:
2^(8/24): d5
2^(14/24): A3
in the respelled AAA6-based 24-EDO.
So the 8/24 becomes a 9/24, and the 14/24 becomes a 13/24 - that is to say, the third widens slightly and the fifth narrows slightly. Those aren't better approximations to harmonic/just intervals than we had in 12-TET, but they're also not too far off. They're a little spicy. If you're looking for that good ear spice, respelled 24-EDO has got you covered.
I want to figure out what that respelling is doing mathematically before I use it any more.
...
Want to try making a different 24-EDO by tempering out another interval besides AAA6? Sure! My EDO-finder program says that these should all work:
(24, 10): AAAAAAA11
(36, 15): AAAAAAAAAA16
(36, 19): AAA20
(36, 23): ddd24
(48, 20): AAAAAAAAAAAAA21
(60, 25): AAAAAAAAAAAAAAAAA26
(60, 37): ddd38
(72, 30): AAAAAAAAAAAAAAAAAAAA31
(72, 38): AAAAAAA39
(72, 46): ddddddd47
(84, 35): AAAAAAAAAAAAAAAAAAAAAAAA36
(84, 47): AAA48
(96, 40): AAAAAAAAAAAAAAAAAAAAAAAAAAA41
Those are not all linearly independent, e.g. (24, 10) is just (12, 5) times two. I think (36, 23) = ddd24 looks pretty interesting. Let's do that one.
...
Oh man, this one's even crazier.
2^(1/24): d8
2^(2/24): dd15
2^(3/24): M-1
2^(4/24): m6
2^(5/24): d13
2^(6/24): A-3
2^(7/24): P4
2^(8/24): d11
2^(9/24): A-5
2^(10/24): M2
2^(11/24): m9
2^(12/24): A-7, d16
2^(13/24): M0
2^(14/24): m7
2^(15/24): d14
2^(16/24): A-2
2^(17/24): P5
2^(18/24): d12
2^(19/24): A-4
2^(20/24): M3
2^(21/24): m10
2^(22/24): AA-6
2^(23/24): A1
2^(24/24): P8
.
There's no (m2, m3, M6, M7) above. Those first two are smaller than P1, the second two are larger than P8. Crazy.
...
If I ever figure out a good tuning system that maps rank-2 interval space to a 24-EDO, the pitch names will follow naturally. For example, no matter what rank-2 tuning system you're in, C2 + dd5 = Gbb2. The only question is, which of the 24 frequency ratios in 24-EDO should be called Gbb2? That's what you need the tuning system for.
In the mean time, until I figure out how to do it right, here's a notation that's definitely wrong: introduce new accidentals for pitch classes that are half sharp (+), half flat (-), three-halves sharp (#+), and three-halves flat (b-). Then we can name the 24 pitch classes of a chromatic scale incorrectly:
C+/Db-
C#+/D-
D+/Eb-
D#+/E-
E+/F-
F+/Gb-
G-/F#+
Ab-/G+
A-/G#+
A+/Bb-
B-/A#+
B+/C-
.
It's no worse than pretending that Bb is the same as A#, which lots of musicians and music teachers do. But it's still pretty gross.
What if we want to make circles that visit all 24 pitch classes by regular intervals, like the circle fifths? To do that, we can step by any integer number of divisions that's coprime with 24. In the range (1, 24), they happen to be:
[1, 5, 7, 11, 13, 17, 19, 23]
.
All of these have pretty clear interpretations.
* 19 divisions of the octave in 24-EDO, 2^(19/24), approximates an octave-reduced 7th harmonic (7/8), and 5 divisions is its octave-complement.
* 11 divisions, 2^(11/24), is close to a reduced 11th harmonic, (11/8), and 13 divisions is its complement.
* 17 divisions, 2^(17/24) is close to a reduced 13th harmonic, (13/8), and 7 divisions is its complement.
* 1 division and 23 divisions are obviously the chromatic scales in either direction. The frequency ratios are not close to any small harmonics. The octave-reduced 33rd harmonic, (33/32), is pretty close to 1 division, i.e. 2^(1/24), I suppose.
What about other small harmonics? We don't do any better approximating them in 24-EDO than we do in 12-TET:
* 14 divisions in 24-EDO approximates a reduced 3rd harmonic (3/2, the perfect fifth), just as 7 divisions did in 12-TET.
* 8 division in 24-EDO approximates a reduced 5th harmonic (5/4, the major third), just like 4 division did in 12 TET.
* 4 division in 24-EDO approximates an octave reduced 9th harmonic (9/8, the major second), just as 2 division did in 12-TET.
.
Generally, to find the number of EDO divisions that best approximates a frequency ratio, say 13/8, you start with
13/8 = 2^(x/24)
and invert
x = log_2(13/8) * 24
x ~ 16.81
.
So, you can see, 17 steps in 24-EDO isn't a stunning approximation to the 13th harmonic, but it's better than the 12-TET approximation (and will be whenever x rounds to an odd number).
...
Okay, so, the space of rank-2 musical intervals doesn't collapse down to a 24-EDO. And that's a damn shame, because we'd like to analyze stuff like Wyschnegradsky's 24-EDO preludes or King Gizzard and The Lizard Wizard's "Melting" with the full machinery of music theory.
What if we go up to rank-3 musical intervals? I've neve tried making EDOs from that. It could be interesting. Rank-3 interval space is good for representing 5-limit just intonation, so that's a start, but if we want to talk about how 24-EDO has 7th, 11th, and 13th harmonics, then maybe we should skip up to rank-6 interval space right away and see if we can make an EDO from that.
I don't know much about rank-6 interval space. I've written at length on this blog about my attempts to make/find a good rank-4 interval space; I never found a system as cool as I'd like, but I did talk about two fairly functional systems. In both systems, you just take 5-limit just intonation notationa and prepend new adjectives (sub and super) for ratios that differ from the 5-limit versions by a septimal comma - and the two systems just used different septimal commas.
If we continue on in that way from rank-4 to rank-5 and rank-6, we'll need some undecimal and tridecimal commas and some new adjectives to prepend onto the interval names.
A common staff notation for microtonal music is the Extended Helmholtz-Ellis Just Intonation Pitch Notation and it comes with a set of commas for different prime limits. Want to just try those? I do.
EHEJIN uses 64/63 as a septimal comma, 33/32 as a undecimal comma, and 27/26 as a tridecimal comma. All of those are super-particular ratios, which is kind of nice. We need some principle for picking commas, and super-particulars are about as good as any.
So ... I guess we also need adjectives. Adding or substracting a septimal comma makes something super or sub, notated Sb and Sp. How about we associate the undecimal comma with (ascendant | descendant), notated As and De. And the tridecimal comma can pair with (prominent | recessed), notated as Pr and Re, why not. Putting the HEJI 7th, 11th, and 13th commas together with the tuned basis intervals from Small Intervals in 5-Limit Just Intonation, we get:
.
There's another staff notation for just intonation made by the great composer Ben Johnston that uses some different commas for higher prime limit accidentals. His commas are also super-particular ratios, just not always the same ones. Again, importing his 7th, 11th, and 13th harmonic commas, we get:
.
In my post on reinventing 7-limit just intonation, I argued for the use of the fraction 21/20 (the Palatine comma) as a septimal accidental, so it's a little bit sad to me that neither HEJI nor Johnston use it, but also the fact that those two didn't agree on the septimal and tridecimal accidentals makes me think that it's all kind of meaningless and you can do whatever you want.
The HEJI accidental fractions only have 2 and 3 as factors, other than the prime limit that they're introducing:
t(As1) = 33/32 = 3 * 11 / 2^5
t(Pr1) = 27/26 = 3^3 / 2 * 13
So they're they're kind of Pythagorean under the hood. At least they are at first. The accidental associated with 17 in HEJI isn't super-particular and it also includes 5 as a factor. The accidental associated with 19 is pseudo-Pythagorean again, but not super-particular.
In contrast, Johnston includes 5 as a factor in his ratios from the start, and all of them are super particular:
t(As1) = 33/32 = 3 * 11 / 2^5
t(Pr1) = 65/64 = 5 * 13 / 2^6
. Johnston's commas continue on (51/50, 96/95, 46/45, 145/144, 31/30) for the 17th, 19th, 23rd, 29th, and 31s harmonics, respectively.
Since Pythagorean tuning kind of sucks, and since Johnston's system regularly uses super particular ratios, and since I really love his music, I'm particular to his system on all accounts. The one drawback is that most of my microtonal friends use HEJI.
Why didn't Johnston use the septimal Palatine comma though? Just too large, maybe? It's 84.4 cents, which is large, but his 11-limit and 13-limit commas are also more than a 50 cents, so....I dunno. Of course, if we only cared about the size of the numerator and denominator, then 8/7 (at 231 cents) would be an even simpler super-particular septimal accidental, and 17/16 (at 105 cents) would be the simplest accidental for the seventeenth harmonic. What if we limit ourselves to 70 cents for the accidental associated with the 7th harmonic and higher? Nope, we still don't recover the Johnston commas that way. The simplest super-particular septimal comma smaller than 70 cents is 28/27 at 63 cents.
So, I don't know how Johnston picked his accidentals, but it's not "take the simplest fraction that includes 2, 3, 5, and the prime in question as factors below a given cent value". Even if I limit myself to 57cents (to exclude simpler septimal ratios that he didn't include while still allowing for the 31th harmonic accidental that he did include), then we get this:
11th - 33/32 : 53 cents
13th - 40/39 : 44 cents
17th - 51/50 : 34 cents
19th - 76/75 : 23 cents
23th - 46/45 : 38 cents
31th - 31/30 : 57 cents
37th - 37/36 : 47 cents
39th - 40/39 : 44 cents
. which is great, but not Johnston. Finally, here's what we get if we set the cents cutoff so that I can include my preferred septimal accidental:
11th - 33/32 : 53 cents
13th - 26/25 : 68 cents
17th - 51/50 : 34 cents
19th - 76/75 : 23 cents
23th - 24/23 : 74 cents
31th - 31/30 : 57 cents
37th - 37/36 : 47 cents
39th - 26/25 : 68 cents
.
You can see that the 13 and the 39th are both about 68 cents. That seems like a waste. Maybe one criterion for choosing between different sets of accidentals is how evenly they're distributed from 0 to 100 cents.
Oh, I just had the best idea! Forget 11 and 13 and the rest. I hereby propose that 24-EDO should be thought of as a projection of a rank-4 interval space that approximates 7-limit jut intonation. The pure octave (2/1) and the perfect fifth (3/2) and the major third (5/4) all lie on the 12-EDO subset of 24-EDO, i.e. the even divisions,, and 7/4 lives at 19 divisions, on the odd subset, and we can and will and perhaps even should think about all the odd divisions as being septimal deviations from the even ones. Yes, yes, yes. And I should probably try projecting a rank-3 space down to an EDO before trying rank-4, but when have I ever done what's in my best interest?
Whoa! Out of my Palatine septimal comma (21/20 = 1.05) and the Leipzig comma used by HEJI (64/63 ~ 1.016) and the Johnston's septimal comma (36/35 ~ 1.0286), out of those three, Johnston's is by far the closest to the 24-EDO division: 2^(1/24) ~ 1.0293. Most impressive. If you want to make a septimal tuning system that explains 24-EDO, that might be the comma to use.
Okay, for my first sanity check: if I make a septimal tuning system by adding 36/35 to the standard Lilley commas for 5-limit JI, do I get a sensible short name for 7/4? Yeah, ,it's a subminor seventh.
Okay, I've now got a program that turns 4-tuple interval coordinates in the (36/35, 81/80, 25/24, 128/125) basis into septimal interval names. My next step is to convert those coordinates into the 7-limit reduced prime basis (2/1, 3/2, 5/4, 7/4). Then I can associate each of those to a number of empirical 24-EDO divisions, namely (24, 14, 8, 19), and then I can say "5 divisions in 24-EDO goes by all of these names".
Okay, here are my current commas, in no particular order:
81/80 : Ac1 :: (0, 1, 0, 0)
25/24 : A1 :: (0, 0, 1, 0)
128/125 : d2 :: (0, 0, 0, 1)
. And here are the reduced-prime basis intervals that I want to use instead, expressed in terms of those commas:
3/2 : P5 :: (0, 2, 7, 4)
5/4 : M3 :: (0, 1, 4, 2)
7/4 : Sbm7 :: (-1, 3, 10, 6)
And here are the septimal Johnston-Lilley commas expressed in terms of the reduced primes:
25/24 : A1 :: (0, -1, 2, 0)
128/125 : d2 :: (1, 0, -3, 0)
36/35 : Sp1 :: (0, 2, -1, -1)
where of course I'm using the order (P8, P5 M3, Sbm7) in the coordinates.
And that should do it, yeah? Those are the matrices I need for interconverting between the two interval bases.
Okay! Here are the intervals (up to four characters in length) that I think should logically correspond to the frequency ratios (expressed here as numbers of divisions) in 24-EDO:
Cleaning that up a bit,
1 : Sp1, Sbm2
2 : m2
3 : SbM2, Spm2
4 : M2
5 : Sbm3, SpM2
6 : m3
7 : SbM3, Spm3
8 : M3
9 : Sb4, SpM3
10 : P4
11 : Sp4
12 : A4, d5
13 : Sb5
14 : P5
15 : Sp5, Sbm6
16 : m6
17 : SbM6, Spm6
18 : M6
19 : Sbm7, SpM6
20 : m7
21 : SbM7, Spm7
22 : M7
23 : Sb8, SpM7
24 : P8
.
It's really, really good, right? I think so. Sb5 is a sub-fifth or sub-perfect fifth. Spm2 is a super minor second. If you see "spm" and you think "sperm", try thinking "spem" instead. It means "hope" in Latin, in the accusative singular inflection.
So, I didn't spell out the math behind what I was doing very well just now. I didn't even know what I was doing. But I think it's still mathematically well founded.
I took some basis vectors and I tuned them to exponential frequency ratios. And that gave me an EDO. It's not the usual route for projecting rank-2 interval space into an EDO, but it works. No trickery to be ashamed of here. In particular, I think I did this tuning:
t(P5) = 2^(14/24)
t(M3) = 2^(8/24)
t(Sbm7) = 2^(19/24)
The interval of 11 division in 24-EDO was a good approximation to the reduced 11th harmonic, 11/8. In my rank-4 septimal interpretation of 24-EDO, 11 divisions is associated to Sp4 primarily, but also SbA4, Sbd5, SpA3, among other more complicated intervals. The just frequency ratios that correspond to these in the Johnston-Lilley septimal tuning system are:
Oh. I think the matrix of the new basis I wanted to use, (P8, P5, GrA1), doesn't have a determinant of -1 or 1, so the coordinates of intervals in the new system won't generally be integers. Like M3 in the (P8, P5, GrA1) basis would be (-2/3, 5/3, 1/3). Kind of sad. That means that the tuning system
puts M3 at
(24 * -2/3) + (14 * 5/3) + (1 * 1/3) = 23/3 divisions in 24-EDO.
Tarek Yamani on youtube did a cool thing. Start with a dominant 13 sharp 11 chord, which is a normal chord in jazz, and which outlines a scale (spelled by thirds rather than by steps). This chord has intervals of
which we could spell with pitch classes over a C natural tonic as:
.
The scale intervals, expressed in the (A1, d2) basis, have A1 components of:
0 4 7 10 14 18 21
which are of course also the number of division in 12-EDO you need to climb above a tonic to get the next pitch in the chord.
Now divide all the the A1 components by 2, giving:
0 2 3.5 5 7 9 10.5
and interpret these as numbers of 12-EDO divisions again. Then we have a scale with two quarter-tone intervals. If we interpret this one as moving by step, rather than by thirds, this new scale has quarter-tone intervals in the third and seventh scale degrees (which before were the P5 of the chord and the M13 of the chord).
This new scale isn't actually new at all: it's called rast, and it's probably the most common mode (or dastgahs or maqam) in Iranian, Arabic, and Turkish music. So if you can improvise over a dominant 13 sharp 11 jazz chords on a keyboard, and you can switch your keyboard from 12-EDO to 24-EDO, then surprise, you can probably also play some pretty fluent 24-EDO middle eastern music. Congratulations.
Does this procedure of halving the A1s suggest a new way to interpret 24-EDO? I'm not sure. My mathematical experimentations haven't yielded promising results yet. It's not as easy as saying that two stacked octaves (a P15) should have a frequency ratio of 2 and that two stacked diminished seconds (a dd3) should be tempered out. That gives us a 34-EDO, usually, depending a little bit on which things you break along the way.
Does the halved major scale correspond to a maqam? By thirds, we can spell it
P1, M3, P5, M7, M9, P11, M13
. Translating to 12-EDO divisions, these become
0, 4, 7, 11, 14, 17, 21
. Dividing by 2, we get
0, 2, 3.5, 5.5, 7, 8.5, 10.5
. Reinterpreting by step, we have microtonal intervals on scale degrees ^3, ^4, ^6, and ^7.That's not a maqam in root position that I know. After a little investigation, it's not a permutation of any of the maqams that I know, either.
Using 24-EDO divisions instead of 12-TET divisions, these are the differences between successive scale degrees of the halved major scale:
4, 3, 4, 3, 3, 4, 3
.
I'll be using maqams as spelled out on wikipedia. The halved major scale is microtonal, so we know it's not a permutation of
Kurd: (Tonal. Phrygian)
Nahawand: (Tonal. Minor)
Nawa Athar: (C, D, Eb, F#, G, Ab, B, C) :: 4, 2, 6, 2, 2, 6, 2. (Tonal. Shad ‘Araban permutation. This would be a C.minor-major9#11b13 chord)
None of those. The halved major scale also doesn't have any separations of 2 divisions, so we know it's not a permutation of:
Jiharkah: (F, G, A, Bb, C, D, E-, F) :: 4, 4, 2, 4, 4, 3, 3. (Bayati permutation)
. And finally, the halved major scale looks pretty close to a permutation of our old friend Rast, but compare them closely (or find the lexicographically first permutation of each) and you'll see that they're distinct:
Husayni ‘Ushayran: (A, B-, C, D, E-, F, G, A) :: 3, 3, 4, 3, 3, 4, 4. (Rast permutation)
Apparently, the Arabic (et cetera) music is derived from Greek Pythagorean tuning though, so maybe reading more on that will give me better names for the microtonal intervals. First insight, two stacked Pythagorean commas are very close to a quarter tone:
Okay, let's suppose that the maqam quarter tones derive from the addition or subtraction of AAA-1 to the pitches a diatonic twelve tone scale. That means we're playing with these ugly suckers now:
Does something good happen if we tune the maqam comma to exactly 2^(1/24)? Kind of, I suppose? Here are a bunch of tuned intervals represented in cents, rounded to the nearest integer cent.
25 - A0 : (0, -1)
40 - dddd4 : (1, 3)
50 - AAA-1 : (0, -2)
90 - m2 : (1, 1)
115 - A1 : (1, 0)
140 - AA0 : (1, -1)
154 - ddd4 : (2, 3)
179 - d3 : (2, 2)
204 - M2 : (2, 1)
229 - AA1 : (2, 0)
244 - dddd5 : (3, 4)
254 - AAA0 : (2, -1)
269 - dd4 : (3, 3)
294 - m3 : (3, 2)
319 - A2 : (3, 1)
344 - AAA1 : (3, 0)
358 - ddd5 : (4, 4)
383 - d4 : (4, 3)
408 - M3 : (4, 2)
433 - AA2 : (4, 1)
448 - ddd6 : (5, 5)
458 - AAAA1 : (4, 0)
473 - dd5 : (5, 4)
498 - P4 : (5, 3)
523 - A3 : (5, 2)
538 - dddd7 : (6, 6)
548 - AAA2 : (5, 1)
563 - dd6 : (6, 5)
588 - d5 : (6, 4)
613 - A4 : (6, 3)
638 - AA3 : (6, 2)
65 - dd3 : (1, 2)
652 - ddd7 : (7, 6)
663 - AAAA2 : (6, 1)
677 - d6 : (7, 5)
702 - P5 : (7, 4)
727 - AA4 : (7, 3)
742 - dddd8 : (8, 7)
752 - AAA3 : (7, 2)
767 - dd7 : (8, 6)
792 - m6 : (8, 5)
817 - A5 : (8, 4)
842 - AAA4 : (8, 3)
856 - ddd8 : (9, 7)
881 - d7 : (9, 6)
906 - M6 : (9, 5)
931 - AA5 : (9, 4)
946 - ddd9 : (10, 8)
956 - AAAA4 : (9, 3)
971 - dd8 : (10, 7)
996 - m7 : (10, 6)
1021 - A6 : (10, 5)
1046 - AAA5 : (10, 4)
1060 - dd9 : (11, 8)
1085 - d8 : (11, 7)
1110 - M7 : (11, 6)
1135 - AA6 : (11, 5)
1150 - ddd10 : (12, 9)
1160 - AAAA5 : (11, 4)
1175 - d9 : (12, 8)
1200 - P8 : (12, 7)
.
On the one hand, yeah, t(P5) = 702 cents is really close to 700 cents like we'd expect (and even closer to the pure frequency ratio of 3/2), and t(m6) = 792 cents is pretty close to 800 cents like we'd expect (and even closer to the Pythagorean m6), and in between them we've got an interval interpretable as a quarter tone, namely t(AAA3) = 752. On the other hand, if Arabic music notates the pitch-class that's ~750 cents above a C as G-, isn't it kind of bullshit to say "Nope, wrong, that's an E###."? They use it by step between Fs and As as if it were some kind of a G, so maybe we should call it some kind of a G. Although honestly, even if we should technically call it an E###, that doesn't really help me to compose music, because I don't know how to use E###s. They're not going to show up by any reasonable modulations from C major, I don't think. Although I'll sooner use this AAA3 than the craziness from the 72-edo thing from the start, which had things like
2^(7/24): dddd9, AAAAA-4
, so maybe this is a triumph? It doesn't feel like a triumph. One day I'll try analyzing 24-EDO music as being tridecimal. That'll be a triumph.
I've got another idea for how to analyze quartertones. It also starts with Pythagorean tuning. I don't know what involvement, if any, Pythagoras actually had in early music theory, but here's part of the myth as I think of it.
Below we have some intervals in the (P5, P8) basis, their frequency ratios in Pythagorean tuning, and their modern names. Pythagoras wrote this table down, exactly as it appears below, on a goat skin parchment after eating some bad spanakopita:
with a conspicuous gap at 600 cents. Add in an interval at 600 cents, and you've got a nice 12 note chromatic scale with fairly even psycho-acoustic spacing. Make the spacing exactly even and you've got 12-TET. What if you forget about exactly even spacing? Then you do some more math and you'll eventually find that the more complicated fractions of the set, which were all made from powers of (3/2) and (2/1), are very close to simpler fractions made from powers of (5/4). and (3/2) and (2/1), and these simpler ratios with factors of five sound fantastic. That's 5-limit just intonation. And a lot of the melodiousness of music comes from a combination of there being (roughly) even spacing of intervals and of frequency ratios being (roughly) simple fractions. You can't get really get both at once, but you can favor one or the other in a principled way. Zheanna Erose writes some nice music that completely disregards logarithmically equal spacing, e.g. by just using frequency ratios with the same denominator. *Some* of it is nice. Some of it is like a seizure.
To get the 11 or 12 chromatic frequency ratios, we raised t(P5) = 3/2 to different powers, form -5 to 5, and then offset by octaves. But if we use use powers with larger magnitude than five, then we can get new intervals in between the 11 or 12 chromatic notes. Using powers of P5 between -10 and 10, we get:
90 cents - 256/243 - m2
114 cents - 2187/2048 - A1
180 cents - 65536/59049 - d3
204 cents - 9/8 - M2
294 cents - 32/27 - m3
318 cents - 19683/16384 - A2
384 cents - 8192/6561 - d4
408 cents - 81/64 - M3
498 cents - 4/3 - P4
588 cents - 1024/729 - d5
612 cents - 729/512 - A4
702 cents - 3/2 - P5
792 cents - 128/81 - m6
816 cents - 6561/4096 - A5
882 cents - 32768/19683 - d7
906 cents - 27/16 - M6
996 cents - 16/9 - m7
1020 cents - 59049/32768 - A6
1086 cents - 4096/2187 - d8
1110 cents - 243/128 - M7
. You can see that the intermediate intervals don't fall on the 50 cent marks, the quarter tones: most of them are like 20 cents off from the 100 cents marks, plus or minus. But what if they weren't? We could monkey with t(P5) or t(P8) to get things more quarter-tone-esque. For example, between m2 and M2, we now have intervals of A1 and d3. What if we keep t(P8) = 2/1, but we adjust t(P5) so that A1 = d3?
To do that, we start with the coordinates for A1 and d3 in the (P5, P8) basis:
(-10, 6) = d3
These coordinates, of course, relate the given intervals linearly to the basis intervals, e.g.:
(7 * P5 + -4 * P8) = A1
which is equivalent to a statement about the tuned values of the intervals:
t(A1) = t(P5)^(7) * t(P8)^(-4)
. Now we can solve for t(P5) = x in the equation t(A1) = t(d3):
x^7 * (2/1)^(-4) = x^(-10) * (2/1)^6
giving
x = 2^(10/17)
. The {x} is just a new variable here because I like single letter variables. It's not the same as any other {x} that I've used in the post.
That 17 in the denominator of the power looks kind of 17-EDO-esque. Did we just reinvent 17-EDO? We absolutely did, yes. But weren't we trying to explain 24-EDO? We absolutely were, yes. So does 17-EDO explain 24-EDO? Lol, I don't know. Probably not. Let's look.
2^(1/17): m2
2^(2/17): A1=d3
2^(3/17): M2
2^(4/17): m3
2^(5/17): A2=d4
2^(6/17): M3
2^(7/17): P4
2^(8/17): A3=d5
2^(9/17): A4=d6
2^(10/17): P5
2^(11/17): m6
2^(12/17): A5=d7
2^(13/17): M6
2^(14/17): m7
2^(15/17): A6=d8
2^(16/17): M7
2^(17/17): P8
. 17-EDO gives us tones between (m2 and M2), between (m3 and M3), between (m6 and M6) and between (m7 and M7), all of which feel kind of like quarter-tones, maybe, but it doesn't give us intermediate tones in a lot of other places, and there are 2 intermediate two between P4 and P5, which is fine I guess, but you might have hoped for 3 intermediate tones if you grew up with 12-TET. So maybe quarter tones are mostly used in practice between the major and minor versions of the 2nd, 3rds, 6ths, and 7ths? If so, then 17-EDO might help you to name and understand the intervals that makes Arabic/Turkish/Persian microtonal music work. Or maybe that's all wrong.
If instead we keep the perfect fifth pure, t(P5) = (3/2), and then adjust {t(P8) = y} so thatA1=d3, then our equation becomes:
(3/2)^7 * y^(-4) = (3/2)^(-10) * y^6
giving
y = (3/2)^(17/10) ~ 1.9923
. I don't suppose there's a name for that tuning system, but one day I'll post some music here that's tuned like that.
If the 12-note chromatic scale is better represented with 5-limit ratios than with 3-limit Pythagorean ratios, maybe we should try messing with a higher-rank tuning systems again to figure out which intermediate intervals between the 12 chromatic notes can be related to produce a tuning system that explains quarter tone music intervallically. Since there are three basis intervals in 5-limit just intonation tuning systems for us to monkey with, ... ... ...
I'm tired. Maybe some other day. Also, it's tridecimal, right? It' not 5-limit, it's 13-limit. Just analyze it as 13-limit.
-
Okay, if you design a tuning system for rank-2 interval space on the principles of 1) pure octaves and 2) "let's tune these two intervals to the same frequency", then you're always going to get an EDO, in particular, the EDO that tempers out the interval between the two intervals that you're tuning to the same frequency. Like, when we equated the tunings of A1 and d3, well they're separated by dd3, so we're also tuning dd3 to a frequency ratio of 1/1. Looking at the interval you're tempering out is the logical way to figure out what EDO scale you're producing. You may recall, I gave a formula in the "EDO-generators" post, abs(a * 7 - b * 12) / gcd(a, b), that uses the coordinates (a, b) of the tempered out interval in the (A1, d2) basis to calculate the number of EDO divisions.
I've talked a lot about different ways to interpret 24-EDO with musical intervals, with mixed success. But I haven't talked about is how to use the interval for harmonic analysis. Let's do that a little.
The ways to analyze harmony in terms of intervals, rather than frequency ratios, are chordally and contrapuntally. The blog post directly after this one, which I'm writing at the same time, is where I'm figuring out the rules of microtonal counterpoint, so let's focus on chords here.
Most western chord theory is based on stacked thirds drawn from scales. Here's the major scale spelled by thirds:
P1, M3, P5, M7, M9, P11, M13
. If you're on scale degree ^1 and you see the first three of those, or the first, five, or the first however many, then don't sweat, you've got a normal diatonic major chord made of thirds.
Here are all 7 diatonic modes spelled by step:
[P1, M2, m3, P4, P5, M6, m7] - Dorian (ii)
[P1, m2, m3, P4, P5, m6, m7] - Phrygian (iii)
[P1, M2, M3, A4, P5, M6, M7] - Lydian (IV)
[P1, M2, M3, P4, P5, M6, m7] - Mixolydian (V)
[P1, M2, m3, P4, P5, m6, m7] - Aeolian (vi)
[P1, m2, m3, P4, d5, m6, m7] - Locrian (vii)
and here they are spelled by thirds:
[P1, m3, P5, m7, M9, P11, M13] - Dorian (ii) = .m13
[P1, m3, P5, m7, m9, P11, m13] - Phrygian (iii) = .m11b9b13
[P1, M3, P5, M7, M9, A11, M13] - Lydian (IV) = .Maj13#11
[P1, M3, P5, m7, M9, P11, M13] - Mixolydian (V) = .13
[P1, m3, P5, m7, M9, P11, m13] - Aeolian (vi) = .m11b13
[P1, m3, d5, m7, m9, P11, m13] - Locrian (vii) = .m11b5b9b13
All of those are intervals I mention in my post Small Intervals in 5-Limit Just Intonation, except for shruti ^12, the AcAcA4. This scale is almost symmetric: almost every shruti has another one whih is its octave complement, like
for which (17 + 5 = 22), and (27/16 * 32/27 = 2/1) and (AcM6 + Grm3 = P8). The exceptions are shruti ^10, ^11, and ^12, right in the middle. The ^11 shruti can't be a 5-limit frequency ratio and be its own octave complement, because to get (11 + 11 = 22), you need (sqrt(2) * sqrt(2) = 2/1). The 5-limit AcA4 is a decent approximation for sqrt(2). It's closer than the 5-limit t(d5) = 36/25 or t(A4) = 25/18 for example. But the other ones? Just on the basis of mathematical aesthetics, I'm tempted to say that Sambamoorth got one wrong. If ^10 is an acute fourth with a simple ratio, then ^12 should be Gr5 -> 40/27. This isn't a tiny change: the Gr5 is about 68.7 cents sharper than Sambamoorth's shruti ^12 of AcAcA4, more than a quarter tone. But also, Sambamoorth's shruti ^12 of AcAcA4 was really close to his ^11, so I think there's an improvement in distinguishability, even though it's a big change. I should try to find his original writings and see what he says about it. But also, I'm not super sure I care what he says: it should be a Gr5. If we make shruti ^12 into a Gr5, then we get these changes in the commas between scale degrees:
Ac1 = ^13 - 12
. Righteous. I'm going to post all 22 again with ^12 fixed so that I have something correct to copy when I look back at this post in the future.
The 22 shrutis have all of the intervals of a normal chromatic scale with some extra bits. Here's a list with some bullet points where the Indian system makes additions:
[P1 • m2 M2 • • m3 M3 • P4 • • • P5 • m6 M6 • • m7 M7 • P8]
. It doesn't look much like 24-EDO, does it? Still cool though. I don't at all regret figuring this out. The frequency ratios are also not 24-EDO-esque. Here they are in cents, truncated to integers:
90: Grm2
111: m2
182: M2
203: AcM2
294: Grm3
315: m3
386: M3
407: AcM3
498: P4
519: Ac4
590: AcA4
680: Gr5
701: P5
792: Grm6
813: m6
884: M6
905: AcM6
996: Grm7
1017: m7
1088: M7
1109: AcM7
1200: P8
. Nothing close to 50 cents. Which his fine. I'd like to learn more about how Indian music uses these, but not tonight. And probably not in this post, since I don't think it will be relevant to quarter-tones after all. Goodnight.
...
Some microtonal composers really like 31-EDO. It provides nice neutral thirds, among other things. Want to try analyzing 24-EDO as an approximation to a subset of 31 EDO? I do. Here are some interval names: Here's a table with cents rounded to integers values on the left for different 31 EDO divisions, and also there are interval names n the right.
39 : 2^(1/31) - d2
77 : 2^(2/31) - A1
116 : 2^(3/31) - m2
155 : 2^(4/31) - AA1=dd3
194 : 2^(5/31) - M2
232 : 2^(6/31) - d3
271 : 2^(7/31) - A2
310 : 2^(8/31) - m3
348 : 2^(9/31) - AA2=dd4
387 : 2^(10/31) - M3
426 : 2^(11/31) - d4
465 : 2^(12/31) - A3
503 : 2^(13/31) - P4
542 : 2^(14/31) - AA3=dd5
581 : 2^(15/31) - A4
619 : 2^(16/31) - d5
658 : 2^(17/31) - AA4=dd6
697 : 2^(18/31) - P5
735 : 2^(19/31) - d6
774 : 2^(20/31) - A5
813 : 2^(21/31) - m6
852 : 2^(22/31) - AA5=dd7
890 : 2^(23/31) - M6
929 : 2^(24/31) - d7
968 : 2^(25/31) - A6
1006 : 2^(26/31) - m7
1045 : 2^(27/31) - AA6=dd8
1084 : 2^(28/31) - M7
1123 : 2^(29/31) - d8
1161 : 2^(30/31) - A7
.
, we get a set that has good approximations to the neutral quartertone 2nds, 3rds, 6th, and 7ths. The 600 cent tritone interval kind of has two approximations, which isn't great, but it's better than having no options, and then there are also pretty good quartertone-esque tones between (P4 and A4) and (d5 P5).
I think this is a success. The most melodically and harmonically important neutral tones are represented here and the corresponding intervals have short simple names. Some quarter tones are missing - like between M2 and m3, but there we have two intermediates:
232 : 2^(6/31) - d3
271 : 2^(7/31) - A2
310 : 2^(8/31) - m3
And maybe that's better. Maybe musicians who play microtonal instruments like fretless strings and voice, maybe their actual frequencies favor the d3 or A2 in practice, and 250 cents is more a convenient notional fiction than a musically useful frequency ratio. That could all be wrong, but it feels right.
Let's talk about which maqamat you can play with this quartertone subset of 31-EDO.
I can see right away that the Rast maqam is playable in the quartertone subset of 31-EDO. It's pitch classes were
Rast: (C, D, E-, F, G, A, B-, C)
and its quartertones E- (neutral third) and B- (neutral seventh) correspond to
348 : 2^(9/31) - AA2=dd4
and
1045 : 2^(27/31) - AA6=dd8
respectively.
Several maqamat have no quarter tones, and those can of course all be played in this subset of 31 EDO. The main things I care about now are Bayati/Jiharkah, and Huzam/Rahat al-Arwah/Sikah, and maybe Saba, but how important can Saba be if it's not notated correctly on that wikipedia page? We'll check them all.
Saba and Bayati aren't equivalent, but they both only have a neutral 2nd for quarter tones, which is
155 : 2^(4/31) - AA1=dd3
, so those are both doable, along with their modal permutations.
Huzam/Sikah starts on a quarter tone for some reason and has no other quarter tones, which means if you transpose it to start on C, then all six of the other chord tones of the 7 note scale become quartertones. We could do transposition and test all six, but there's an easier way to test Huzam for playability. Here's the Huzam maqam:
(E-, F, G, Ab, B, C, D, E-)
. We can just permute it so that it starts on C:
(C, D, E-, F, G, Ab, B, C)
and now the only microtone is a neutral third, which is
348 : 2^(9/31) - AA2=dd4
which we already saw in Rast. So we can play a permutation of Huzam in the quartertone subset of 31-EDO, which means we can also play Huzam. Great success! You don't need 24 EDO to play Arabic music, you just need a neutral 2nd, 3rd, and 7th, apparently? But having more would help with transposition.
I hear that some Turkish music uses eighth-tones, and I would be very pleased to learn that they're used in in the places where 31-EDO has good eighth tones but not good quarter tones, like those two eight tones between (M2 and m3):
194 cents : 2^(5/31) - M2
232 cents : 2^(6/31) - d3
271 cents : 2^(7/31) - A2
310 cents : 2^(8/31) - m3
. If I ever find out, I'll put it here. Time for 13-limit just intonation analysis, I think.
If you have a 7-limit frequency ratio, you can express it with a sequence of integers (a, b, c, d) which are the powers of the primes (2, 3, 5, 7). If you want to re-express this in an octave-reduced prime basis, (2/1, 3/2, 5/4, 7/4), then you just need to know the original basis vectors (the primes) in the new basis (the reduced primes), namely:
From this you can change any frequency ratio in the 7-limit prime basis, (a, b, c, d), to one in the reduced prime basis, (m, n,, o, p):
Suppose you want to now transform a frequency ratio (m, n, o, p) into the septimal Johnston-Lilley basis? I posted a function before for naming frequency ratios in the 7-JL basis, so there's a good reason to do this. The Johnston-Lilley basis has four intervals tuned to these frequency ratios: [(36/35), (81/80), (25/24), (128/125)]. The first of these is the septimal comma in the staff notation of the great microtonal composer Ben Johnston, and the next three are the tuned values of the basis vectors in Lilley's system of naming 5-limit just intervals. The specific intervals tuned to those frequency ratios in 7-JL are the septimal super first, the acute first, the augmented first, and the diminished second:
To make the conversion, first we need the frequency ratios of the old basis expressed in the new basis:
and then we just turn rows into columns again:
. Now the frequency-ratio (q, r, s, t) in the frequency-ratio basis ((36/35), (81/80), (25/24), (128/125)) can be reinterpreted as an interval in the interval basis (Sp1, Ac1, A1, d2), and then it's really easy to name the interval from there using the functions in septimal-interval-converter.py.
So that's one way to start with a 7-limit frequency ratio and figure out the name of the corresponding interval. You could also go straight from the prime factorization to the tuned septimal Johnston Lilley basis, but I had other reasons for changing bases twice.
I don't generally have names for 13-lmit frequency ratios; I haven't settled on an 11-limit comma or a 13-limit comma. Maybe one day.
...
Neutral intervals to focus on justifying:
n2: 155 cents - AA1=dd3 : 2^(4/31)
n3: 348 cents - AA2=dd4 : 2^(9/31)
n6: 852 cents - AA5=dd7 : 2^(22/31)
n7: 1045 cents - AA6=dd8 : 2^(27/31)
.
* 4 steps in 31-EDO can be used like a neutral second. There are infinitely many intervals that the 31-EDO tuning system tunes to a frequency ratio of 2^(4/31), but the simplest ones are the intervals AA1 and dd3. 31-EDO tunes both of those two intervals to a frequency ratio of 2^(4/31). Which justly tuned intervals are close to that exponential value, 2^(4/31)? 4 steps in 31-EDO is less than a cent flat relative to the reduced 35th harmonic. (35/32), but that's kind of crazy. People can't hear the 35th harmonic, so we shouldn't use that as a justification of why 4-steps in 31-EDO is a useful frequency ratio. It's still a fairly nice simple ratio, I'm just saying that we should just give it a septimal interval name, the sub acute major second, SbAcM2, instead of calling it a harmonic. 4 steps in 31-EDO is also 4 cents sharp of (12/11). I don't have names for fractions with 11 and 13 in their factors yet, so we'll just call that (12/11).
* 9 steps in 31-EDO can be used like a neutral 3rd. It is 6 cents sharp of (39/32), 5 cents sharp of 128/105 (the super grave minor third, SpGrm3), and 11 cents flat of 16/13 (the octave complement of the reduced 13th harmonic).
* 22 steps in 31-EDO can be used like a neutral 6th. It is 5 cents flat of 105/64 (the sub acute major sixth, SbAcM6). It's also 11 cents sharp of the reduced 13th harmonic, 13/8.
* 27 steps in 31-EDO can be used like a neutral 7th. It is less than a cent sharp of the octave complement of the reduced 35h harmonic, 64/35, but that's kind of crazy. It's also 4 cents flat of 11/6.
Let's go back to the neutral 2nd for a moment. What are the 5-limit justly tuned frequency ratios of AA1 and dd3, and how do they compare to their shared value when tuned in 31-EDO? And what theoretically "should" a neutral second be?
Here are the 5-limit versions of AA1 and dd3 in Lilley's system for naming 5-limit just intonation:
2^(4/31) ~ 1.09356
?dd3 : (1, 1, 2) :: [0, 3, -5] = (144/125) * (24/25) = (3456/3125) ~ 1.10592
They're fairly close. I did the ones with question marks by hand. I'll recheck them with a computer soon. I think they're right. So if you want to analyze 4-stepsin 31-EDO as being an AA1, I mean, yeah, that's the value to which 31-EDO tunes AA1, and the value is fairly close to the justly tuned 5-limit version of AA1, so go ahead.
What about the theoretical value for a neutral second? Here are the 5-limit minor second and major second:
M2 : (0, 2, 1) :: [1, -2, 1] = 10/9
.
The geometric mean of those frequency ratios is sqrt(32/27) = (4/3 * sqrt(2/3)) = (4 * sqrt(6) / 9) ~ 1.08866. That's logarithmically equally spaced between the justly tuned m2 and M2, so maybe that's the ideal value for a neutral third. It's much closer to the justly tuned AA1 than the justly tuned dd2. So if you're trying to choose an interval to use like a neutral third in some other tuning system, maybe try AA1 to start.
If the ideal neutral third is the geometric mean between
M3 : (1, 4, 2) :: [0, 0, 1] = 5/4
then it should have a frequency ratio of sqrt((6/5) * (5/4)) = sqrt(3/2) = sqrt(6)/2. That's kind of nice, right? Two neutral thirds stack to form a perfect fifth.
The ideal neutral sixth is the geometric mean of:
M6 : (2, 9, 5) :: [1, -1, 1] = 5/3
.
This has a frequency ratio of sqrt(8/3) = (2 * sqrt(6)/3).
Finally, the neutral 7th is the geometric mean of
M7 : (3, 11, 6) :: [0, 1, 1] = 15/8
so it has a frequency ratio of sqrt((9/5) * (15/8)) = sqrt(27/8) = (3 * sqrt(6) / 4).
All four of these neutral frequency ratios have sqrt(6) times a 3-limit fraction. That's kind of nice, right?
The theoretical tritone is kind of like a neutral tone, but it doesn't have a sqrt(6). From
P4 : (1, 5, 3) :: [1, -1, 0] = 4/3
P5 : (2, 7, 4) :: [0, 1, 0] = 3/2
we see that it's just sqrt((4/3) * (3/2)) = sqrt(2).
What about the other 24-EDO values, besides the n2, n3, n6, n7? If we find the frequency ratio between M2 and m3, will it also have a sqrt(6) factor? The justly tuned ratios are:
m3 : (1, 3, 2) :: [0, 1, -1] = 6/5
So, between the justly tuned M2 and m3, we have a frequency ratio sqrt((10/9) * (6/5)) = sqrt(4/3) = (2 * sqrt(3)/3). So it doesn't have a factor of sqrt(6), but you can stack two of them to get a justly tuned P4. That's pretty cool. Here's a partial table of frequency ratios that are logarithmically between the frequency ratios of justly tuned successive chromatic intervals:
(4 * sqrt(15) / 15)
m2
n2 = (4 * sqrt(6) / 9)
M2
(2 * sqrt(3) / 3)
m3
n3 = (sqrt(6) / 2)
M3
(4 * sqrt(15) / 15
P4
d5
P5
??
m6
n6 = (2 * sqrt(6) / 3)
M6
??
m7
n7 = (3*sqrt(6)/4)
M7
??
P8
.
Yep.
...
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