An interesting ratio from M. Ayomak

[Ozan Yarman]

Mildan Niyazi Ayomak was a curious figure from the era when the early Republic of Turkey moved to ban Turkish Music from the schools and the radio (the 1926-36 period). He propounded a 53 comma system to explain maqams, whose details still elude me. But I found a reference where a gruelling ratio emerges (Adnan Atalay's Master's Thesis "Tone-Systems of Traditional Turkish Art Music", 1989, p. 99):

2532014013864211267508803108601856

divided by

1283279475728998470512529213751296

yields

1.9730807370902884798124344941395187800715413376085021667504012674972386398490142838057281993673803865...

(Only the first 100 decimals shown)

which equals precisely 1,176.54 cents.

In comparison, the 52nd step of 53-EDO is precisely 1,177.3585 cents. The difference is a mere 0.82 cents. It appears he was trying to approximate 53-EDO via ratios, however crudely.

Oz.

✩ ✩ ✩

www.ozanyarman.com

[Robert Walker]

Just to say, your ratio is  2^20/3^12, so it's the inversion of the Pythagorean comma which is close in size to one step of 53-et (and also close in size to the syntonic comma).

Of course you must know that, must be just that the full expansion of the ratio, and the inversion makes it look momentarily a little different from usual.

Robert

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[Ozan Yarman]

This quote from C. Debussy:

"Music is the arithmetic of sounds as optics is the geometry of light."

is exactly what I should have said in retort to Alaeddin Yavaşça when he was chairman of a session of the Turkish music congress back in 2007 smugly objecting to my qanun & music theory approach to explaining makams with the words: "Music is Art, music is not math". :)

Yes, I see that the ratio could be simplified down to 1048576/531441 which arises from the Pythagorean chain of fifths (11 fifths down from F). The expansion was so cumbersome, I couldn't do the proper calculations with my machinery here.

Thanks!
Oz.

✩ ✩ ✩
www.ozanyarman.com

[Robert Walker]

Glad it helped. Just to say there are on-line bignum calculators which you might find help with this sort of work.

There's a lightning fast online factorizer here:

http://wims.unice.fr/wims/wims.cgi?session=9R8296D526.12&+lang=en&+module=tool/algebra/factor

handles these numbers in a fraction of a second.

Then if you want to work the other way to find the expansion of a number from it's prime factorization,

https://defuse.ca/big-number-calculator.htm

where you can check that 

2^65*3^29 = 2532014013864211267508803108601856

2^45*3^41 = 1283279475728998470512529213751296

I actually found the answer by putting it into Tune Smithy's calculator which told me it was 2^20/3^12. 

With numbers this large, however, it calculates it as a decimal, then converts the decimal to a ratio, and factorises it. So though good for getting a first idea of the solution, you have to check it's answers.

Tune Smithy can handle exact numbers up to 19 digits (64 bits in binary) - which is fine for most microtonal purposes so left it at that.

Thanks,

Robert