Reduced generators and special commas

For a p-limit linear temperament with octave period, we may call the
p-limit rational number q, 1 < q < sqrt(2), the reduced generator if
it defines a generator when approximated by the temperament, and if
it has minimal Tenney height given this condition. We may call the
comma c, c>1 the "special comma" for this temperament if it is the
comma defined by taking the p-limit consonance q^i for i>1 of the
temperament, and equating it to the corresponding JI version of this
consonace. In other words, it is C/q^i or q^i/C, i>1, where C is a
ratio of odd numbers less than or equal to p.

Below are some 7-limit reduced generators and special commas, along
with poptimal generators and TM reduced bases. Note particularly the
pairs of temperaments (Meantone and Flattone, Hemiwuerschmidt and
Hemithird) with the same reduced generator and special comma; these
are very closely related.

Dominant seventh poptimal 17/41
[36/35, 64/63] reduced 4/3 special 64/63

Pelogic poptimal 18/41
[36/35, 135/128] reduced 4/3 special 135/128

Negri poptimal 5/48
[49/48, 225/224] reduced 15/14 special 225/224

Kleismic poptimal 14/53
[49/48, 126/125] reduced 6/5 special 126/125

Hemifourth poptimal 4/19
[49/48, 81/80] reduced 8/7 special 49/48

Meantone poptimal 86/205
[81/80, 126/125] reduced 4/3 special 81/80

Porcupine poptimal 8/59
[64/63, 250/243] reduced 10/9 special 250/243

Superpythagorean poptimal 31/76
[64/63, 245/243] reduced 4/3 special 64/63

Beatles poptimal 19/64
[64/63, 686/675] reduced 49/40 special 2401/2400

Flattone poptimal 46/109
[81/80, 525/512] reduced 4/3 special 81/80

Magic poptimal 13/41
[225/224, 245/243] reduced 5/4 special 3125/3072

Nonkleismic poptimal 23/89
[126/125, 1728/1715] reduced 6/5 special 126/125

Semisixths poptimal 44/119
[126/125, 245/243] reduced 9/7 special 245/243

Orwell poptimal 26/115
[225/224, 1728/1715] reduced 7/6 special 1728/1715

Miracle poptimal 17/175
[225/224, 1029/1024] reduced 15/14 special 225/224

Quartaminorthirds poptimal 9/139
[126/125, 1029/1024] reduced 21/20 special 64827/64000

Supermajor seconds poptimal 53/274
[81/80, 1029/1024] reduced 8/7 special 1029/1024

Schismic poptimal 39/94
[225/224, 3125/3087] reduced 4/3 special 5120/5103

Squares poptimal 93/262
[81/80, 2401/2400] reduced 9/7 special 19683/19208

Octafifths poptimal 13/177
[245/243, 2401/2400] reduced 21/20 special 4000/3969

Catakleismic poptimal 52/197
[225/224, 4375/4374] reduced 6/5 special 15625/15552

Hemiwuerschmidt poptimal 37/229
[2401/2400, 3136/3125] reduced 28/25 special 3136/3125

Hemithird poptimal 24/149
[1029/1024, 3136/3125] reduced 28/25 special 3136/3125

Amity poptimal 155/548
[4375/4374, 5120/5103] reduced 128/105 special
4294967296/4254271875

Hemififth poptimal 70/239
[2401/2400, 5120/5103] reduced 49/40 special 2401/2400

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:

In other words, it is C/q^i or q^i/C, i>1, where C is a
> ratio of odd numbers less than or equal to p.

This should be, it is the octave reduction of C/q^i or q^i/C,
whichever one produces something between 1 and sqrt(2).

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:

Note particularly the
> pairs of temperaments (Meantone and Flattone, Hemiwuerschmidt and
> Hemithird) with the same reduced generator and special comma; these
> are very closely related.

To these we may add Beatles and Hemififth; Negri and Miracle.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith
<gwsmith@s...>"
> <gwsmith@s...> wrote:
>
> Note particularly the
> > pairs of temperaments (Meantone and Flattone, Hemiwuerschmidt and
> > Hemithird) with the same reduced generator and special comma;
these
> > are very closely related.
>
> To these we may add Beatles and Hemififth; Negri and Miracle.

And Kleismic and Nonkleismic, and Dominant Sevenths and
Superpythagorean. Do I need to make a computer do this?

>For a p-limit linear temperament with octave period, we may call the
>p-limit rational number q, 1 < q < sqrt(2), the reduced generator if
>it defines a generator when approximated by the temperament, and if
>it has minimal Tenney height given this condition. We may call the
>comma c, c>1 the "special comma" for this temperament if it is the
>comma defined by taking the p-limit consonance q^i for i>1 of the
>temperament, and equating it to the corresponding JI version of this
>consonace. In other words, it is C/q^i or q^i/C, i>1, where C is a
>ratio of odd numbers less than or equal to p.

*The* special comma -- couldn't there be more than one? In meantone,
i=4 gives 81:80, i=12 the Pythagorean comma, etc. Right?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> *The* special comma -- couldn't there be more than one? In
meantone,
> i=4 gives 81:80, i=12 the Pythagorean comma, etc. Right?

I defined it so that there is only one. Sometimes that may be a bit
arbitrary, as in the case of Amity, where there are other reductions
of the generator with height not too much higher, leading to
different commas. However, in your case 4 is less than 12, so 12 is
out of it; moreover, 12 isn't actually mapped to a 7-limit consonance
anyway.

By my definition, i=3 is the first value to consider, and
we get (12/5)/(4/3)^3 = 81/80 as the special comma. We get the same
for i=4 and (16/5)/(4/3)^4, but i=6 gives us (4/3)^6/(28/5) =
5120/5103, i=9 leads to (96/7)/(4/3)^9 = 59049/57344, and finally
i=10 to (128/7)/(4/3)^10 = 59049/57344. Hence, septimal meantone has
special commas [81/80, 5120/5103, 59049/57344] in that order, if we
want to extend the definition to secondary and tertiary special
commas (and etc.)

>>*The* special comma -- couldn't there be more than one? In
>>meantone, i=4 gives 81:80, i=12 the Pythagorean comma, etc. Right?
>
>I defined it so that there is only one.

What's Tenney height? just i? Did Tenney propose this?

>12 isn't actually mapped to a 7-limit consonance anyway.

Oh, bad example. :(

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>*The* special comma -- couldn't there be more than one? In
> >>meantone, i=4 gives 81:80, i=12 the Pythagorean comma, etc.
Right?
> >
> >I defined it so that there is only one.
>
> What's Tenney height? just i? Did Tenney propose this?

For p/q in reduced form, p*q; either that or log(p*q).