Follow up to Archytas' Harmonic Means
On Wikipedia, I found a really interesting description of the fret placement on the Turkish Baglama, a kind of lute. The frets are almost 24-EDO quarter tones. I'm going to call the frequency ratios associated with the frets a scale. The scale is 17-limit and due to Yalçın Tura, but I haven't read the original source.
I like this scale because it's justly tuned, it has fairly small frequency ratios, the intervals between the scale steps are repeated a few times (suggesting a regular construction and the possibility of an intervallic analysis), and the subset of covered 24-EDO values seems like an important hint about which microtones are actually of use in middle eastern music. I'm also curious if the slightly deviations from 24-EDO are reproduced by players of fretless lutes like the oud. I also don't know much about the principled use of frequency ratios with factors of 17, so there's another interesting thing about the scale.
First, I tried making my own 17-limit scale that approximates 24-EDO in order to get a really good understanding of what choices Tura made and why. In comparing the two scales, I figured out how Tura made his, and it's very simple and actually non-intervallic. I feel a little sad deleting all the work I did on my scale, but the reader shouldn't have to suffer through my work just because I did.
So here directly is the 17-limit Baglama tuning given by Yalçın Tura, which is also approximately a subset of 24-EDO:
Fret 0: 1/1 ~ 0c
.
Fret 1: 18/17 ~ 100c
Fret 2: 12/11 ~ 150c
Fret 3: 9/8 ~ 200c
.
Fret 4: 81/68 ~ 300c
Fret 5: 27/22 ~ 350c
Fret 6: 81/64 ~ 400c
.
Fret 7: 4/3 ~ 500c
.
Fret 8: 24/17 ~ 600c
Fret 9: 16/11 ~ 650c
Fret 10: 3/2 ~ 700c
.
Fret 11: 27/17 ~ 800c
Fret 12: 18/11 ~ 850c
Fret 13: 27/16 ~ 900c
.
Fret 14: 16/9 ~ 1000c
.
Fret 15: 32/17 ~ 1100c
Fret 16: 64/33 ~ 1150c
Fret 17: 2/1 ~ 1200c
It has just four distinct step-wise interval differences: (34/33 and 33/32 at about 50 cents) and (256/243 and 18/17) at 90 to 100 cents.
0 to 1 : 18/17
1 to 2 : 34/33
2 to 3 : 33/32
3 to 4 : 18/17
4 to 5 : 34/33
5 to 6 : 33/32
6 to 7 : 256/243
7 to 8 : 18/17
8 to 9 : 34/33
9 to 10 : 33/32
10 to 11 : 18/17
11 to 12 : 34/33
12 to 13 : 33/32
13 to 14 : 256/243
14 to 15 : 18/17
15 to 16 : 34/33
16 to 17 : 33/32
Interestingly, the scale fails to respect octave-complementation in several places.
In total, Tura has frets for all of the 12-EDO pitches (in hundreds of cents), and also half-flat frets for (150c, 350c, 650c, 850c, and 1150c). If we root these on a C natural, the microtones are a neutral second (D-), a neutral third (E-), a half-flat fifth (G-) which is crazy to me, a neutral sixth (A-), and a half-flat octave (C-) rather than a neutral seventh, which is also crazy. I guess the open string isn't the tonic. Maybe some permutation of those makes sense. The thing I expected was was (150c, 350c, 850c, 1050c).
Ok, yes! There are two cyclic permutations of Tura's Baglama scale which give us neutral seconds, thirds, sixths, and sevenths.
If we start on fret-7 (500 c), we get a scale with neutral 2nds, 3rds, 6ths, 7ths, and also an interval at 650 cents that's like a half-flat P5. The frequency ratios rooted from here are:
[1/1, 18/17, 12/11, 9/8, 81/68, 27/22, 81/64, 4/3, 24/17, 16/11, 3/2, 27/17, 18/11, 27/16, 243/136, 81/44, 243/128, 2/1]
If we start on fret-14 (1,000 cents), we get a scale with neutral 2nd, 3rds, 6ths, 7ths, and also an interval at 550 cents (half sharp from P4). The frequency ratios rooted from here are:
[1/1, 18/17, 12/11, 9/8, 81/68, 27/22, 81/64, 729/544, 243/176, 729/512, 3/2, 27/17, 18/11, 27/16, 243/136, 81/44, 243/128, 2/1]
This has a pretty messed up P4 (729/544), but a lot in common with the scale rooted on fret-7 actually.
Okay, time to explain where the scale really comes from. We can construct a Major scale from major seconds and minor seconds. The usual format is
[M2, M2, m2, M2, M2 M2 m2]
These are the intervals between successive scale degrees. We can accumulate them to get the intervals for each scale step relative to the tonic:
[P1, M2, M3, P4, P5, M6, M7, P8]
This can be obtained by a cycle of pure fifths, starting on P4 and going up to M7. Then we add on P8 for closure. For some reason, Tura starts with a cyclic permutation of this, the mixolydian mode, but we'll just work with the major mode. In Pythagorean tuning, the M2 is tuned to 9/8 and the m2 is tuned to 256/243. So, we have
[9/8, 9/8, 256/243, 9/8, 9/8, 9/8, 256/243]
as the tuned intervals between steps of a Pythagorean major scale.
We can accumulate these frequency ratios (multiplicatively) to get a the frequency ratios for each step of the major scale in Pythagorean tuning:
P1: 1/1 - 0c
M2: 9/8 - 203c
M3: 81/64 - 407c
P4: 4/3 - 498c
P5: 3/2 - 701c
M6: 27/16 - 905c
M7: 16/9 - 996c
P8: 2/1 - 1200c
A nice familiar major scale. But now for something exotic. Instead of making a chromatic scale by continuing the cycle of fifths, we'll use harmonic means!
The 9/8 value for the tuned major second (that showed up so frequently between scale degrees) can be split, roughly, into rational multiplicative-halves (i.e. approximate square roots) using the harmonic mean. More precisely, we take the harmonic mean of the ratio, 9/8 in this case, with 1/1 (as Archytas did, see my recent post on Archytas means and complements). This harmonic mean and its Archytas complement (given by dividing 9/8 by its harmonic mean) gives us two frequency ratios that are roughly minor seconds, in the sense of being tuned to about 100 cents:
9/8 → 18/17 and 17/16
They're 99 cents and 105 cents, respectively. The 17/16 one is a little bigger. We'll split all the tuned major seconds (9/8s) of the major scale
[9/8, 9/8, 256/243, 9/8, 9/8, 9/8, 256/243]
in this way, with 18/17 coming first:
[(18/17, 17/16), (18/17, 17/16), 256/243, (18/17, 17/16), (18/17, 17/16), (18/17, 17/16), 256/243]
I've included parentheses just to help show the grouping. This gives us intervals that are like minor Nths below the major Nths of the major scale. Also, we happen to introduce below the P5 an interval that's like a diminished fifth, since there was a gap of 9/8 between P4 and P5 and this also became more fine-grained through division.
If we multiplicatively accumulate these interval between scale degrees, we get a running product that is a fine chromatic scale approximating 12-EDO:
P1: 1/1 - 0c
m2: 18/17 - 98c
M2: 9/8 - 203c
m3: 81/68 - 302c
M3: 81/64 - 407c
P4: 4/3 - 498c
d5: 24/17 - 596c
P5: 3/2 - 701c
m6: 27/17 - 800c
M6: 27/16 - 905c
m7: 243/136 - 1004c
M7: 243/128 - 1109c
P8: 2/1 - 1200c
It looks like the major seventh has the most deviation from 12-EDO, being 9 cents sharper, and that's just due to it being unmodified by this procedure: the Pythagorean M7 is a little sharp, relative to both 12-EDO and 5-limit just intonation.
So that's a nice chromatic scale. To get Turkish microtones, we're just going to break up one of the intervals of our (now chromatic) scale one more time. All of the 17/16 ratios that take us from a minor Nth to a major Nth are going to get split into harmonic means and complements also. This will give us neutral intervals on the 2nds, 3rds, 6ths, and 7ths. It will also add a new little guy between the diminished fifth and the perfect fifth. A half-flat fifth we might call it.
To do this, we first break up the ratio 17/16 into its Archytas mean and complement:
17/16 → 34/33 and 33/32
and then we perform that replacement all throughout our chromatic scale, again with the small harmonic mean (34/33) being placed first and the larger Archytas complement (33/32) being placed second:
[1/1, 18/17, (34/33, 33/32), 18/17, (34/33, 33/32), 256/243, 18/17, (34/33, 33/32), 18/17, (34/33, 33/32), 18/17, (34/33, 33/32), 256/243]
If we accumulate the frequency ratios between scale degrees multiplicatively, we get this for the tuned steps of each scale degree relative to the tonic:
P1: 1/1 - 0c
m2: 18/17 - 98c
n2: 12/11 - 150c
M2: 9/8 - 203c
m3: 81/68 - 302c
n3: 27/22 - 354c
M3: 81/64 - 407c
P4: 4/3 - 498c
d5: 24/17 - 596c
-5: 16/11 - 648c
P5: 3/2 - 701c
m6: 27/17 - 800c
n6: 18/11 - 852c
M6: 27/16 - 905c
m7: 243/136 - 1004c
n7: 81/44 - 1056c
M7: 243/128 - 1109c
P8: 2/1 - 1200c
where an "n7" is a neutral seventh and "-5" is a half flat fifth. This scale above is exactly Tura's Baglama scale rooted on the 7th fret (4/3 at 500 cents). The scale doesn't obey octave complementation, because it wasn't made from regular interval arithmetic, like the Pythagorean major and chromatic scales, but was rather made from subdivisions of frequency ratios into harmonic means.
Now I kind of want to define a system now where intervals are named based on subdivision by harmonic means. That could be cool.
If we take octave complements of Tura's scale, we find that there are multiple frequency ratios associated with a few of the 24-EDO steps:
^0: 1/1 ~ 0c
^1: 33/32 ~ 50c
^2: 18/17, 17/16 ~ 100c
^3: 12/11 ~ 150c
^4: 9/8 ~ 200c
^5: ?
^6: 81/68, 32/27 ~ 300c
^7: 27/22, 11/9 ~ 350c
^8: 81/64, 34/27 ~ 400c
^9: ?
^10: 4/3 ~ 500c
^11: 11/8 ~ 550c
^12: 24/17, 17/12 ~ 600c
^13: 16/11 ~ 650c
^14: 3/2 ~ 700c
^15: ?
^16: 27/17, 128/81 ~ 800c
^17: 18/11 ~ 850c
^18: 27/16, 136/81 ~ 900c
^19: ?
^20: 16/9 ~ 1000c
^21: 11/6 ~ 1050c
^22: 32/17 ~ 1100c
^23: 64/33 ~ 1150c
^24: 2/1 ~ 1200c
.
The commas between ratios on a single step are these (below 3/2):
(17/16) / (18/17) = 289/288, "semitonisma"
(81/68) / (32/27) = 2187/2176, "septendecimal schisma"
(27/22) / (11/9) = 243/242, "rastma"
(81/64) / (34/27) = 2187/2176
(17/12) / (24/17) = 289/288
You might wonder what EDO tempers out all three of those. I checked every EDO up to 100. The simplest one is 24-EDO, and then we get integer multiples of 24-EDO too, e.g. (48, 72, 96, ...)-EDO. That's it.
If we go the other way and just look for intervals on each step of 24-EDO that are justly tuned to low complexity frequency ratios, then comparing that to its octave complement we get:
^0: 1/1
^1: 21/20, 28/27
^2: 16/15
^3: 11/10, 12/11
^4: 9/8, 10/9
^5: 7/6, 8/7
^6: 6/5
^7: 11/9
^8: 5/4
^9: 9/7
^10: 4/3
^11: 7/5
^12: 25/18, 36/25
^13: 10/7
^14: 3/2
^15: 14/9
^16: 8/5
^17: 18/11
^18: 5/3
^19: 7/4, 12/7
^20: 9/5, 16/9
^21: 11/6, 20/11
^22: 15/8
^23: 27/14, 40/21
^24: 2/1
which has tempered commas below 3/2 of:
(21/20) / (28/27) = 81/80
(11/10) / (12/11) = 121/120
(9/8) / (10/9) = 81/80
(7/6) / (8/7) = 49/48
(36/25) / (25/18) = 648/625
The first three unique ones of these are square super particulars:
S9 = 81/80
S11 = 121/120
S7 = 49/48
and the last one is the justly tuned acute diminished second, Acd2. So there.
Here are just names for the Tura fractions and their octave complements:
~0c: P1 = [0, 0, 0, 0, 0, 0, 0] # 1/1
~50c: As1 = [-5, 1, 0, 0, 1, 0, 0] # 33/32
~100c: HuAcm2 = [1, 2, 0, 0, 0, 0, -1] # 18/17
~100c: ExA1 = [-4, 0, 0, 0, 0, 0, 1] # 17/16
~150c: DeAcM2 = [2, 1, 0, 0, -1, 0, 0] # 12/11
~200c: AcM2 = [-3, 2, 0, 0, 0, 0, 0] # 9/8
~250c: ?
~300c: HuAcm3 = [-2, 4, 0, 0, 0, 0, -1] # 81/68
~300c: Grm3 = [5, -3, 0, 0, 0, 0, 0] # 32/27
~350c: DeAcM3 = [-1, 3, 0, 0, -1, 0, 0] # 27/22
~350c: AsGrm3 = [0, -2, 0, 0, 1, 0, 0] # 11/9
~400c: AcM3 = [-6, 4, 0, 0, 0, 0, 0] # 81/64
~400c: ExGrM3 = [1, -3, 0, 0, 0, 0, 1] # 34/27
~450c: ?
~500c: P4 = [2, -1, 0, 0, 0, 0, 0] # 4/3
~550c: As4 = [-3, 0, 0, 0, 1, 0, 0] # 11/8
~600c: Hud5 = [3, 1, 0, 0, 0, 0, -1] # 24/17
~600c: ExA4 = [-2, -1, 0, 0, 0, 0, 1] # 17/12
~650c: De5 = [4, 0, 0, 0, -1, 0, 0] # 16/11
~700c: P5 = [-1, 1, 0, 0, 0, 0, 0] # 3/2
~750c: ?
~800c: HuAcm6 = [0, 3, 0, 0, 0, 0, -1] # 27/17
~800c: Grm6 = [7, -4, 0, 0, 0, 0, 0] # 128/81
~850c: DeAcM6 = [1, 2, 0, 0, -1, 0, 0] # 18/11
~900c: AcM6 = [-4, 3, 0, 0, 0, 0, 0] # 27/16
~900c: ExGrM6 = [3, -4, 0, 0, 0, 0, 1] # 136/81
~950c: ?
~1000c: Grm7 = [4, -2, 0, 0, 0, 0, 0] # 16/9
~1050c: AsGrm7 = [-1, -1, 0, 0, 1, 0, 0] # 11/6
~1100c: Hud8 = [5, 0, 0, 0, 0, 0, -1] # 32/17
~1150c: De8 = [6, -1, 0, 0, -1, 0, 0] # 64/33
~1200c: P8 = [1, 0, 0, 0, 0, 0, 0] # 2/1
Hu1 is the humbled unison, justly tuned to 50/51, and Ex1 is the exalted unison, justly tuned to 51/50.
The commas tempered out in the system that equates Tura's fretting with its octave complement are:
ExExGrA0 = [-5, -2, 0, 0, 0, 0, 2] # 289/288
HuAcAc1 = [-7, 7, 0, 0, 0, 0, -1] # 2187/2176
DeDeAcAcA1 = [-1, 5, 0, 0, -2, 0, 0] # 243/242
I should try starting with rank-7 space, tempering out those three commas to produce a rank-4 space, and then seeing which simple intervals have frequency ratios on 250c, 450c, 750c, 950c.
I notice that none of the frequency ratios have factors of 5, 7, or 13. Maybe that suggests a few more commas to temper out. Also, ratios in the complementized Tura fretting have factors of 11 or 17 or neither, but not both. Also, the powers of 11 and 17 are in (-1, 0, 1).
Using those constraints, we find that these are very natural additions to Tura's complementized Baglama scale:
~250c: AsAcM2 = [-8, 3, 0, 0, 1, 0, 0] # 297/256
~450c: De4 = [7, -2, 0, 0, -1, 0, 0] # 128/99
~750c: As5 = [-6, 2, 0, 0, 1, 0, 0] # 99/64
~950c DeGrm7 = [9, -3, 0, 0, -1, 0, 0] # 512/297
Now all the holes are filled in. These happen to just Pythagorean intervals raised or lowered by by Johnston's 11 limit-comma. No factors of 17 are involved.
...
The commas that arose when we tried mixing Tura's scale with its complement were
(17/16) / (18/17) = 289/288, "semitonisma"
(81/68) / (32/27) = 2187/2176, "septendecimal schisma"
(27/22) / (11/9) = 243/242, "rastma"
The second one contains the third one as a factor. If we remove it, then we get 1089/1088, with prime interval [-6, 2, 0, 0, 2, 0, -1].
I'm not sure this is an iron-clad proof, but here's an argument that tempering out those three comas and purely tuning the octave produces 24-EDO:
determinant of [[-6, 2, 2, -1], [-1, 5, -2, 0], [-5, -2, 0, 2], [1, 0, 0, 0]] = 24
I just removed the slots of the prime intervals corresponding to powers of 5, 7, and 13.
Whether you find that convincing or not, it seems probable to me and very nice that we can define 24-edo on the 2.3.11.17 JI subspace in terms of pure octaves and three intervals to temper out that are justly tuned to superparticular ratios.
Two of the ratios are also square super-particulars:
1089/1088 = (33/32) / (34/33)
289/288 = (17/16) / (18/17)
Isn't life wonderful?
Some more linear algebra that might not be convincing to you, perhaps in part because I don't know any linear algebra:
If we take the inverse of our comma + octave matrix in the 2.3.11.17 subspace
inverse of [[-6, 2, 2, -1], [-1, 5, -2, 0], [-5, -2, 0, 2], [1, 0, 0, 0]]
then WolframAlpha tells us we get
1/24 * [[0, 0, 0, 24], [4, 4, 2, 38], [10, -2, 5, 83], [4, 4, 14, 98]]
Suppose we take the 2.3.11.17 subsection of two intervals that were about (350c/50c = ) 7 steps of 24-EDO, e.g.
~350c: DeAcM3 = [-1, 3, 0, 0, -1, 0, 0] # 27/22
~350c: AsGrm3 = [0, -2, 0, 0, 1, 0, 0] # 11/9
Multiply the 2.3.11.17 parts by the inverse of the comma+octave matrix and we get:
[-1, 3, -1, 0] * (1/24 * [[0, 0, 0, 24], [4, 4, 2, 38], [10, -2, 5, 83], [4, 4, 14, 98]]) = (1/12, 7/12, 1/24, 7/24)
[0, -2, 1, 0] * (1/24 * [[0, 0, 0, 24], [4, 4, 2, 38], [10, -2, 5, 83], [4, 4, 14, 98]]) = (1/12, -5/12, 1/24, 7/24)
The 7/24ths at the end of each 4-component vector are powers of the octave, and the other coordinates are powers of the commas, but those comma powers become irrelevant when we temper out the commas. So tempering the three commas and tuning the octave purely means that intervals justly associated with 27/22 and 11/9 will both be tuned to a frequency ratio of 2^(7/24). As well they should in 24-EDO.
And all of this goes to show that Tura's 17-limit baglama tuning, when combined with its octave complement, implies 24-EDO. And this is a pretty cool way of defining 24-EDO in which simple just intervals before mis-tuning are basically already 24-EDO to begin with, so maybe those 17-limit just intervals have some explanatory merit in the interpretation of 24-EDO music.
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