Makams and Maqamat IV

: Yalçin Tura's Baglama And 24-EDO

A baglama is a long-necked Turkish lute. Wikipedia lists a tuning for the frets of a baglama due to modern Turkish composer Yalçın Tura. It has weird 17-limit frequency ratios that I found really interesting, so I played around with the math until I figured out how Tura must have constructed his tuning, which is like a western chromatic scale that has a few additional middle eastern microtones. This will be one more source of data on the intervals and frequencies that are used in middle eastern music.

We'll start with a rank-2 major scale, specified in terms of relative intervals between steps:

[M2, M2, m2, M2, M2 M2 m2]

These steps can be accumulated into our old friend

[P1, M2, M3, P4, P5, M6, M7, P8]

which we also could have constructed (without the closing octave) by a spiral of fifth, starting on P4 and cycling upward to M7.

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

Nothing new yet. But now Tura does thing inspired by a different ancient Greek music theorist. To get a minor second-like scale degree of ~100 cents, Tura splits the tuned M2 by taking the arithmetic mean of 9/8 with 1/1. 

((9/8) + (1/1)) / 2 = 17/16 at 105c.

We already were using a Pythagorean minor second as a relative-step in the construction of the Major scale, so coming up with a second m2 in a surprising turn. However, for super-particular ratios like 9/8, the arithmetic mean with 1/1 provides a good approximation for the square root. Consequently, 9/8 divided by 17/16 will also be a good approximation for the square root of 9/8:

(9/8) / (17/16) = 18/17 at 99c.

You can see that both frequency ratios are about 100c, which is half of the ~200c for 9/8 and also a good 12-EDO-ish minor second. We also could have constructed the 18/17 fraction as the harmonic mean of 9/8 with 1/1. This procedure of splitting a super-particular frequency ratio into parts using the arithmetic and harmonic means with 1/1 was used extensively by the ancient Greek music theorist Archytas.

If we expand all of the tuned M2s of the relative-step Pythagorean major scale into these parts, we get a chromatic scale specified by relative-steps:

[(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 of what used to be M2 intervals. This splitting 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 a M2 between P4 and P5 and this also became more fine-grained through division.

Why Tura chose to put the 18/17 before the 17/16 in each expansion of the tuned M2, I don't know, but it will turn out to not matter much in the end of my analysis.

To get middle-eastern 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 an arithmetic mean and a harmonic mean. 

17/16 → 33/32 and 34/33

When we accumulate all of the relative steps, this will give us neutral 2nds, 3rds, 6ths, and 7ths. It will also add a half-flat fifth  between the diminished fifth and the perfect fifth.

Here are the expanded relative steps:

[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

n5: 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

All of these frequency ratios are made of factors of 2, 3, 11, and 17. We could say that they lie on a four dimensional subspace of 17-limit just intonation. In the language of the Xenharmonic community, this is called the "2.3.11.17" just intonation subspace. Fine.

These just ratios are very close to 24-EDO frequency ratios. The largest deviation is the tuned M7 at 9 cents sharp of 24-EDO, and that's just because it's the unmodified Pythagorean frequency ratio that we started with. All of Tura's modifications to the Pythagorean major scale got us closer to 24-EDO.

This scale wasn't constructed with regular interval arithmetic - it was constructed in the manner of Archytas's rational approximations to square roots - and so it doesn't obey octave complementation. For example, the Archytas m2 + Pythagorean M7 != P8.

But what if we try to relate Tura's scale to its octave complement? With octave complementation, we find that e.g. the Pythagorean m2 and the Archytas m2 are separated by a small 9 cent frequency ratio

(18/17) / (256/243) = 2187/2176 at 9 cents.

that we might want to temper out so that we have this cool 17-limit scale from Tura in which regular interval arithmetic still works.

Another comma that shows up when we do octave complementation is the Archytas neutral third divided by the octave complement of the Archytas neutral sixth:

(27/22) / (11/9) = 243/242 at 7 cents.

We might want to temper that one out as well.

There's one more comma that shows up. If we take the octave complement of the Archytas d5 (24/17), we get 17/12, which is also at 12-steps of 24-EDO. The ratio is 

(17/12)/(24/17)  = 289/288

These three commas are not quite independent: we can simplify the first comma, 2187/2176, a little bit by dividing it by the second comma,  243/242:

 (2187/2176)  / (243/242) = 1089/1088

And now we've got three beautiful independent super-particular commas.

I don't know if you'll find it a surprise at this point, but if we start with the four dimensional 2.3.11.17 just intonation space, and then  we tune our octaves purely and temper out these three commas, we get 24-EDO.

Here's how that works.

Start with the intervals justly associated with the three Tura commas and the octave, all expressed in the rank-7 prime harmonic basis:

[-5, -2, 0, 0, 0, 0, 1] # 289/288 = (17/16) / (18/17)

[-1, 5, 0, 0, -2, 0, 0] # 243/242 = (27/22) / (11/9)

[-6, 2, 0, 0, 2, 0, -1] # 1089/1088 = (33/32) / (34/33)

[1, 0, 0, 0, 0, 0, 0] # 2/1

Since we're only working in the 2.3.11.17 subgroup, just remove all of the coordinates associated with 5, 7, and 13. They were all zero anyway:

[-5, -2, 0, 2] # 289/288 = (17/16) / (18/17)

[-1, 5, -2, 0] # 243/242 = (27/22) / (11/9)

[-6, 2, 2, -1] # 1089/1088 = (33/32) / (34/33)

[1, 0, 0, 0] # 2/1

This matrix has an absolute determinant of 24. You can swap round some rows to get +24 instead of -24 if you care about that, but it's still 24-EDO.

But maybe the determinant isn't convincing to you. How about we tune some intervals using the comma+octave matrix?

First find the inverse of the matrix. Wolfram Alpha gives

> inverse of [[-5, -2, 0, 2], [-1, 5, -2, 0], [-6, 2, 2, -1], [1, 0, 0, 0]]

= 1/24 * [[0, 0, 0, 24], [2, 4, 4, 38], [5, -2, 10, 83], [14, 4, 4, 98]]

Now let's look at two Tura intervals that were justly tuned to nearly the same 24-EDO step, say

~350c: DeAcM3 = [-1, 3, -1, 0] # 27/22

~350c: AsGrm3 = [0, -2, 1, 0] # 11/9

both at around (350c/50c = ) 7 steps of 24-EDO. Again, I've removed the coordinates above that were associated with prime harmonic 5, 7, and 13. Now we just multiply these by the inverse of the comma+octave matrix to see how they're tuned:

[-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)

Since we're tempering out the commas (i.e. tuning them to a frequency ratio of 1/1) it doesn't matter what coordinates show up in the first three slots: one raised to a real power is still one. The only coordinate that maters is the last one, associated with the octave, which shows that [-1, 3, -1, 0] and [0, -2, 1, 0] are both tuned to 2^(7/24), i.e. 7 steps of 24-EDO.

What does all of this show? It shows that 24-EDO is not such a bad scale for describing middle eastern music, if Tura's baglama tuning is actually used anywhere. It also shows that if you want a just analysis of music written in 24-EDO, you can do worse than using fractions in the 2.3.11.17 J.I. subspace. It also just shows how you can do cool temperament things in high dimensions. Like now you could try tuning one of the other commas justly to see what happens.

I had a dream of relating the half-flat and half-sharp interval names more directly to the 2.3.7.13 intervals, like so that I could say whether

~350c: DeAcM3 = [-1, 3, 0, 0, -1, 0, 0] # 27/22

~350c: AsGrm3 = [0, -2, 0, 0, 1, 0, 0] # 11/9

were each SbM3 or Spm3 or anything else. I think I'm going to give up on that though. I've got more advanced and fined grained analyses I want to do of middle eastern music than focus on the stuff related to 24-EDO.