Temperaments and notations

Here is yet another twist on that useful device I've dubbed a "notation", this one finding a notation associated to a linear temperament.

If we have a temperament defined in terms of an octave and period, where the period is expressed in JI terms, then we may complete this to a basis for a notation. The notation mapping then becomes an extended mapping of generators to primes, which could be useful for planar temperaments.

For example, meantone temperament with generators <2, 3/2> and
kernel <81/80> leads to the notation basis <2,3/2,81/80>, the inverse of which is

[1, 0, 0]
[1, 1, 0]
[0, 4, -1]

The coumns of this matrix are vals giving mappings to primes, and the rows show the primes in terms of <2,3/2,81/80>.

For 7-limit Orwell, we have generators <2, 7/6> and a reduced basis <225/224, 1728/1715>. Putting these together to form the
notation basis <2,7/6,225/224,1728/1715> and taking the inverse gives us

[1, 0, 0, 0]
[0, 7, 1, 2]
[3, -3, 0, -1]
[1, 8, 1, 2]

The rows of this show us that

3 = (7/6)^6 (225/224) (1728/1715)^2
5 = 2^3 (7/6)^(-3) (1728/1715)^(-1)
7 = 2 (7/6)^8 (225/224) (1728/1715)^2

We can now measure the rms steps to 7-limit consonances for 225/224
and 1728/1715, leading to .8165 and 2.1213 respectively. This shows that keeping 1728/1715 as a comma leads to something closer to Orwell than 225/224, even though 225/224 is a more significant comma in general.

This conclusion is strengthed by the fact that it 7/6 is a 7-limit consonance which clearly is the generator; in the case of miracle, we might choose either 15/14 or 16/15 for our generator, leading to different maps. From <2,15/14,225/224,243/242,385/384> we get

[1, 0, 0, 0, 0]
[1, 6, -4, 1, 2]
[3, -7, 5, -1, -2]
[3, -2, 1, 0, 0]
[2, 15, -10, 2, 5]]

While from <2, 16/15, 225/224, 243/242, 385/384> we get

[1, 0, 0, 0, 0]
[1, 6, 2, 1, 2]
[3, -7, -2, -1, -2]
[3, -2, -1, 0, 0]
[2, 15, 5, 2, 5]

Similarly, from <2,15/14,225/224,1029/1024> we get

[1, 0, 0, 0]
[1, 6, -3, 1]
[3, -7, 4, -1]
[3, -2, 1, 0]

while from <2,16/15,225/224,1029/1024> we get

[1, 0, 0, 0]
[1, 6, 3, 1]
[3, -7, -3, -1]
[3, -2, -1, 0]

Even with the added uncertainty, it is clear 225/224 is the most significant for miracle.

Finally, there is no requirement that octaves rather than fractions of octaves be generators. For pajara, we have <2,7/5,50/49,64/63>,
leading to

[2, 0, 1, 0]
[2, 1, 1, 0]
[7, -2, 4, -1]
[8, -2, 4, -1]

Clearly 50/49 seems to be the more significant comma, so pajara would be more closely related to the 50/49 planar temperament than to the 64/63 planar temperament.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> If we have a temperament defined in terms of an octave and period, where the period is expressed in JI terms, then we may complete this to a basis for a notation. The notation mapping then becomes an extended mapping of generators to primes, which could be useful for planar temperaments.

In these terms, what George and Dave are up to is related to a basis for the 12-et version of meantone, for instance
<2,3/2,81/80,64/63,33/32>^(-1) gives us

[1, 0, 0, 0, 0]
[1, 1, 0, 0, 0]
[0, 4, -1, 0, 0]
[4, -2, 0, -1, 0]
[4, -1, 0, 0, 1]