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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

A very small unit of interval measurement, used mainly in connection with Sagittal notation, suggested first by Gene Ward Smith, then advocated by George Secor and Dave Keenan (the developers of Sagittal notation) in 2004. The name is an abbreviation of "schismina".

A mina divides the 12-edo semitone into 205 logarithmically equal parts, and also divides one degree of 41-edo into 60 logarithmically equal parts. It therefore divides the octave into 12 * 205 or 41 * 60 = 2,460 equal parts, which is a very close and consistent approximation of just-intonation to the 27 limit.

The mina is therefore calculated as the 2460th root of 2, or 2(1/2460), with a ratio of approximately 1:1.000281807. It is an irrational number.

A mina is:

The formula for calculating the mina-value of any ratio r is: minas = log10(r) * [ 2460 / log10(2) ] or minas = log2(r) * 2460

A mina represents one degree of 2460-edo tuning.

The 12-edo semitone is exactly 205 minas.

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[Gene Ward Smith, Yahoo tuning, message 71115 (Wed Apr 11, 2007 11:54 am)]

Mina is an excellent unit; divisible by 12, it gets the 27-limit diamond with an error of less than 1/6 cent, and is consistent through the 27-limit. Since the 27 odd limit is a good cutoff (otherwise you get 29 and 31, and even 23 is pretty much of a stretch) I think there's a fine case for saying that 2460 level accuracy is all the majority of microtonalists need.

It's an "atomic" system, tempering out the atom, and it also eliminates the glum comma (|91 -12 -31>), the landscape comma, the gauss comma (9801/9800) and on and on of course.

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[George Secor, Yahoo tuning, message 71116 (Wed Apr 11, 2007 11:56 am)]

--- In tuning@yahoogroups.com, "Ozan Yarman" wrote:
> Do JI fanatics need any more precision than 2460-EDO?

I hope not. But just in case, there's always 6079-EDO (29-limit consistent, but not a multiple of 12), which Sagittal just might be able to notate (with a few tricks).

Beyond that, there's 11664-EDO (27-limit consistency that's better than 2460), which is a multiple of 12 (and also 72), but I have no idea how I would notate it (which doesn't really matter, since I figure the above is a rhetorical question. ;-)

All kidding aside, Sagittal *actually does* offer more precision than 2460-EDO for those who need it. For example, 98:99 and 99:100 equate to the same number of minas (36) [= ~0.1766 cents = 28 bingo-card 14mus = ~28.941930192 floating-point 14mus], but they may be notated with separate symbols, if necessary.

Also, 45:46 and 1664:1701 (both 78 minas) differ by ~0.0226 cents [= 2 bingo-card 14mus = ~3.705627270 floating-point 14mus], but they can also be notated with separate symbols.

Finally, 40960:41553 and 6561:6656 (both 51 minas) differ by only ~0.0033 cents [= 5 bingo-card 14mus = ~0.548294006 floating-point 14mus], but they also can have separate symbols.

These distinctions are more useful than they may seem, inasmuch as these pairs of intervals will not always map to the same number of degrees in a given division of the octave. The 51-mina pair, e.g., maps differently in each of the following EDOs: 217, 270, 301, 311, 342, 364, 525, 653, and 742.

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minas calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = minas

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