An Ethnographic Smattering

:: Intro

Microtonal music, i.e. music that isn't in 12-EDO, is really just normal music; music that isn't in 12-EDO has been practiced in all cultures in all of history.  I won't say that there is an endless number of complex microtonal musical traditions to characterize - some cultures have fairly simple music or they have tunings fairly similar to western music, but there's still a lot a variety out there.

:: Chopi Mbila Music

Near the south eastern shore of Mozambique, the Chopi people play xylophones in large ensembles, sometimes 10 people or more, and also there are choreographed dances and whistles and flutes in concert. The xylophones, in addition to being played expertly by large, talented, and well coordinated groups, have a cool tuning. We're going to talk about it for a bit. A single one of their xylophones is called an mbila and multiple are called timbila.

The Chopi music equi-heptatonic, meaning that there are 7 notes per octaves and they're equally spaced - i.e. 7-EDO. I've heard some people claim that it's not quite 7-EDO, and I'm going to argue for a while that it totally is. 

I investigated some 7 note scales that are approximately evenly spaced, like a major scale in the temperament [P8, m3, AcA1] tuned to [2/1, 6/5, 1/1], but I don't think these are better than 7-EDO theoretically. The Chopi people explicitly try to tune their xylophones equally, and even if the frequencies are off from 7-EDO by 20 cents in either direction, that's still within normal variation of human intonation, especially when 7-EDO doesn't have pure low-limit prime harmonics besides the octave, such that you could use e.g. the 3rd or 5th harmonic to tune one of the steps.

Here's a table of tunings for one octave of mbila notes from five different Chopi communities, as reported in Hugh Tracey's "Chopi Musicians. Their music, poetry, and instruments" (1948), reproduced in Eduardo Oliveira's "Assessing the geometry and tuning properties of historical timbilas through non-destructive reverse engineering techniques", (2020). The tuning is in hertz, to the nearest 2 cents:

"Zavala": [252, 276, 304, 336, 368, 408, 456, 504] hz

"Chisiko": [256, 288, 316, 348, 384, 424, 472, 512] hz

"Mavila": [260, 284, 320, 352, 384, 424, 464, 520] hz

"Banguza": [260, 288, 316, 356, 388, 432, 472, 520] hz

"Zandamela": [276, 308, 336, 368, 408, 448, 496, 552] hz

If we divide through by the lowest frequency, we get frequency ratios over unison, which we can convert to cents and then average, giving:

[0, 176, 345, 519, 680, 854, 1027, 1200] cents

If we take octave complements, the complement scale is very close in tuning to the previous one:

[0, 173, 346, 520, 681, 855, 1024, 1200] cents

which is somewhat encouraging. The tuning between communities is not so close; for example the second scale degree ranges between:

 [157, 204, 152, 177, 190] cents

so there's like 50 cents difference, but the averages hide that, and I think that's fine. Averaging the original and the octave complement, we get

[0, 175, 346, 519, 681, 854, 1025, 1200] cents

In comparison, the 7-EDO frequency ratios are

[0, 171, 343, 514, 686, 857, 1029, 1200] cents

which only differs from the Chopi mbila tuning by like 5 cents, and that's very near the limit of human perception. So in summary: the tuning of the Chopi mbila, realtive to its tonic pitch, can vary by like 50 cents from region to region, and their tonic pitch varies from 252 hz to 276 hz by region, but when you average out the human variation, you get a scale that's basically equal to itself under octave complementation and also it's equal to 7-edo up to the precision of human hearing. Also they explicitly will tell you that they're trying to tune their scales equally and we should take that seriously. The Chopi use 7-EDO proper and not some unequal temperament that has a tempered comma in common with 7-EDO, as some microtonalists will tell you.

And this is amazing! You thought 5-limit frequencies were the bread and butter of all folk melodies, world-over? And you thought that perhaps the virtuoso middle eastern lutenists were the only people that had ever really gotten away from 5-limit melodies, at the expense of losing their harmonies? Guess again! The Chopi of Mozambique play in Seven Expletive EDO!

There are lots of xylophones across south Africa, and some of the other ones also reportedly play equi-heptatonic or equi-pentatonic scales, but this is the one I have the best evidence about and I think it's pretty solid.

...

: An Intervallic Analysis Of The Polish Duda

My acquaintance Rafał shared some pages from a book showing that the notes on the chanter of the Polish duda, a kind of bagpipe, are quite microtonal.

The tones are conventionally notated [D, F#, G, A, B, C, D, E] and they are separated by roughly [350, 85, 190, 190, 120, 140, 180] cents. We can see that these are only precise to 5 or 10 cents.

I tried to find frequency ratios to make sense of this. I looked for ratios that

1) Match the interval sizes, 

2) Are simple (have small numerators and denominators), 

3) Multiply together with adjacent ratios to be even simpler. 

Here's the scale that I came up with. The ratios only have factors of 2, 3, 5, and 7.

D to F# :: ~(350 cents) -> 60/49  (351 cents)

F# to G :: ~(85 cents) -> 21/20 (84 cents)

G to A :: ~(190 cents) -> 10/9 (182 cents)

A to B :: ~(190 cents) -> 28/25 (196 cents)

B to C :: ~(120 cents) -> 15/14 (119 cents)

C to D :: ~(140 cents) -> 49/45 (147 cents)

D to E :: ~(180 cents) -> 10/9 (182 cents)

Here are some nice compound intervals produced by this tuning:

Low D to G: 9/7

F# to A: 7/6

A to C: 6/5

B to high D: 7/6

G to C: 4/3

Low D to A: 3/2

A to High D: 7/5

If there were an intermediate tone "E" between the low D and F#, it would probably be 15/14 (119 cents) over the low D, leaving 8/7 (231 cents) to reach F#. The interval from E to G would then be 6/5, a just minor third.

The reference from Rafał also showed how each scale tone deviates from 12-TET, with the {A natural} being the only note tuned the same. This makes me think that I should write all the intervals relative to {A natural}, perhaps a drone {A natural} an octave below the one in the scale. This starts out well but goes a little crazy at the end:

[Sbd5, P5, SpM6, m7, P8, Sbd10, m10, SbSbdd12, SbSbd13]

[7/5, 3/2, 12/7, 9/5, 2/1, 56/25, 12/5, 196/75, 392/135]

Note that I've included the "E" tone above low D because I like it. I don't have much faith that Polish bagpipes use  intervals like "SbSbdd12" and "SbSbd13".

If the second to last interval (between C and High D) were 120 cents instead of 140 cents, we could interpret it as 15/14 and we'd have this very nice scale:

[Sbd5, P5, SpM6, m7, P8, Sbd10, m10, SpM10, SpA11]

[7/5, 3/2, 12/7, 9/5, 2/1, 56/25, 12/5, 18/7, 20/7]

I'm still looking for other ways to make sense of the scale.

...

If you add up all the cents between A (in the scale, not the theoretical drone) and the high D, you get 

190 + 120 + 140 = 450 cents

which looks like 

Sb4 # 35/27 _ 449c

This would mean that the D is

(35/27) / (6/5) =  175/162 at  134c

over C, which is the just tuning for SbM2, an odd duck but fairly close to the 140 that the reference specified. If the ratio above that really is 10/9, then the E is a SbGr5 over A, which is justly tuned to 350/243 at 632c. If we could change that 10/9 to a 9/8, we'd have a fairly nice Sb5 at 35/24. Alas that interval is supposed to be about 180 cents, and 10/9 is 182c while 9/8 is a 204c.

For the tuning of the high E, the reference says that it's about 630 cents over A. The sub grave fifth justly tuned to 350/243 is 632c, but also a diminished 5th, justly tuned to 36/25, is 631 cents.

Anyway, I don't feel that I've fully solved this, but I've done pretty well with the data that I have.

...