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Encyclopedia of Microtonal Music Theory

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[Gene Ward Smith, Yahoo tuning-math, message 12302 (Tue Jun 14, 2005 12:40 pm PDT)]

The triprime commas of a rank two temperament are the reduced commas of the temperament (meaning they are greater than one and not a power) which factor into three primes or less. The coefficients of the monzos of the triprime commas can be taken from the coefficients of the wedgie for the temperament up to determination of the sign. Hence, they may be computed very quickly and easily from the wedgie. They have various uses; for instance they can be used to find a comma basis for the temperament.

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[Gene Ward Smith, Yahoo tuning-math, message 12304 (Tue Jun 14, 2005 10:54 pm PDT)]

Rank n temperaments always come equipped with n+1 prime factor commas. For instance, 5-limit 12-et has 3^12/2^19, 2^7/5^3 and 3^28/5^19 as two-prime commas, which you can see directly derive from the val, <12 19 28|. A rank three temperament likewise has four prime commas, etc.

[Gene Ward Smith, Yahoo tuning-math, message 12304 (Tue Jun 14, 2005 10:54 pm PDT)]

The commas in the above example appear in monzo format thus:

   2  3    5

|-19 12,   0 >
|  7  0,  -3 >
|  0 28, -19 >
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[Gene Ward Smith, Yahoo tuning-math, message 11084]

Suppose W is a bival in some prime limit q.

Suppose p1 < p2 < p3 are three primes <= q;

Then C = p1^(e(p2,p3,p1)*w_[p2, p3]) * p2^(e(p1,p3,p2)*w_[p1, p3]) * p3^(e(p1,p2,p3)*w_[p1, p2])

If C<1 we take the inverse, and if it is a power we reduce it by taking the gcd of the exponents and dividing it out; in other words if it is an nth power we take the nth root of it.

The result is the {p1, p2, p3} -triprime comma of the wedgie W; taken together these are the triprime commas of W.

It should be noted that sometimes the comma is simply a 1; however any comma of W is a product of its triprime commas.

For trivals you would similarly get quadprime commas, and so forth; and obviously for vals we can easily define the biprime commas.

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[Gene Ward Smith, Yahoo tuning-math, message 11090]

From the 7-limit meantone wedgie <<1 4 10 4 13 12||, we get the four triprime commas:

factors      monzo

{3,5,7}    |  0 12 -13 4 >
{2,5,7}    |  6  0 -5  2 >
{2,3,7}    |-13 10  0 -1 >
(2,3,5}    | -4  4 -1  0 >

Going up to the 11-limit would add six more such commas; for {2,3,11}, {2,5,11}, {2,7,11}, {3,5,11}, {3,7,11} and {5,7,11}, which of course would be different for the two different versions of 11-limit meantone [meanpop and huygens].

[from Monzo: Note the diagonal of zeros which runs down the matrix of monzos in the example above.]

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[Gene Ward Smith, Yahoo tuning-math, message 11092]

... we think of 7-limit temperaments sharing a 5-limit comma as belonging to the same extended family, and the name family if they have corresponding periods and generators. This is a stronger relationship than merely sharing a comma, since it means that if you reduce down to the {2,3,5} temperament, they are the same.

However, sharing any tricomma will do something similar.

For instance:

Meantone and hemiwuerschmidt share the same {2,5,7} comma, 3136/3125.

They define the same {2,5,7} linear temperament, whose generator is a whole tone which is half of a major third, and where 5/2 of a major third, or five tones, is a 7/4.

If you took a piece in meantone and tossed out all of the notes which came out to an odd number of fifths in terms of the circle of fifths, you'd get something which could be retuned to hemiwuerschmidt (99&130), a much more accurate temperament; whereupon you could go stick notes back in again, trying to come close to what you started with. I haven't tried it but I wonder what would happen if I did!

Hemithirds is another temperament you could play this game with, boosting your meantone piece to 118-equal.

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