Miller commas for some 7-limit temperaments

A Miller comma for a rank-two temperament is a triprime comma of
maximal absolute value of relative error for the TOP tuning
(similarly for other ranks of temperament.) Here are some examples,
along with the minimal comma, which is the one where the relative
error is least. I was hoping for a patterm to emerge, but I don't see
it. Do you?

Decimal
miller: 25/24 minimal: 50/49

Dominant
miller: 256/245 minimal: 81/80

Blackwood
miller: 2401/2187 minimal: 16807/16384

Pajara
miller: 50/49 minimal: 2048/2025

Mavila
miller: 243/224 minimal: 1071875/1048576

Negri
miller: 49/48 minimal: 16875/16384

Meantone
miller: 81/80 minimal: 3136/3125

Porcupine
miller: 250/243 minimal: 268435456/262609375

Superpyth
miller: 20480/19683 minimal: 245/243

Magic
miller: 3125/3072 minimal: 537824/531441

Myna
miller: 10077696/9765625 minimal: 40353607/40000000

Sensi
miller: 1291315424/1220703125 minimal: 78732/78125

Orwell
miller: 595749803514609375/558545864083284007 minimal:
134217728/133984375

Miracle
miller: 34171875/33554432 minimal: 823543/819200

Garibaldi
miller: 3125/3087 minimal: 32805/32768

Octacot
miller: 20000/19683 minimal: 1638400000000000/1628413597910449

Rodan
miller: 245/243 minimal: 1029/1024

Hemiwuer
miller: 393216/390625 minimal: 33397665693696/33232930569601

Hemififths
miller: 858993459200/847288609443 minimal:
1342177280000000000000/1341068619663964900807

Ennealimmal
miller: 40353607/40310784 minimal: 7629394531250/7625597484987

Dominant, superpyth, sensi, orwell, and miracle stand out to me.

-Carl

At 12:02 AM 3/9/2007, you wrote:
>A Miller comma for a rank-two temperament is a triprime comma of
>maximal absolute value of relative error for the TOP tuning
>(similarly for other ranks of temperament.) Here are some examples,
>along with the minimal comma, which is the one where the relative
>error is least. I was hoping for a patterm to emerge, but I don't see
>it. Do you?
>
>Decimal
>miller: 25/24 minimal: 50/49
>
>Dominant
>miller: 256/245 minimal: 81/80
>
>Blackwood
>miller: 2401/2187 minimal: 16807/16384
>
>Pajara
>miller: 50/49 minimal: 2048/2025
>
>Mavila
>miller: 243/224 minimal: 1071875/1048576
>
>Negri
>miller: 49/48 minimal: 16875/16384
>
>Meantone
>miller: 81/80 minimal: 3136/3125
>
>Porcupine
>miller: 250/243 minimal: 268435456/262609375
>
>Superpyth
>miller: 20480/19683 minimal: 245/243
>
>Magic
>miller: 3125/3072 minimal: 537824/531441
>
>Myna
>miller: 10077696/9765625 minimal: 40353607/40000000
>
>Sensi
>miller: 1291315424/1220703125 minimal: 78732/78125
>
>Orwell
>miller: 595749803514609375/558545864083284007 minimal:
>134217728/133984375
>
>Miracle
>miller: 34171875/33554432 minimal: 823543/819200
>
>Garibaldi
>miller: 3125/3087 minimal: 32805/32768
>
>Octacot
>miller: 20000/19683 minimal: 1638400000000000/1628413597910449
>
>Rodan
>miller: 245/243 minimal: 1029/1024
>
>Hemiwuer
>miller: 393216/390625 minimal: 33397665693696/33232930569601
>
>Hemififths
>miller: 858993459200/847288609443 minimal:
>1342177280000000000000/1341068619663964900807
>
>Ennealimmal
>miller: 40353607/40310784 minimal: 7629394531250/7625597484987

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

Here are normalized TOP tuning errors for the primes for these
temperaments. By normalized, I mean that I give TOP(p)/log2(p) - 1.
You can see that the Miller commas are precisely those which have the
primes having the largest errors, and there are three of these.

> Decimal
> miller: 25/24 minimal: 50/49

[.63814981124527923e-2, .63814981124527923e-2, -.638149811245279233e-
2, .10495794842535288e-2]
Biggest error on 2, 3, and 5

> Dominant
> miller: 256/245 minimal: 81/80

[-.397587378466822917e-2, -.387922586218883341e-
2, .39758737846682291e-2, .39758737846682292e-2]
Biggest error on 2, 5, and 7

...

> Ennealimmal
> miller: 40353607/40310784 minimal: 7629394531250/7625597484987

[.303139927736095e-4, .303139927736094e-4, .131297523225660e-4, -
.3031399277360949e-4]
Biggest error on 2,3, and 7.

The relative error on the Miller prime is the same for each of the
prime factors, and that error is the maximum error. So the Miller
prime gets the maximum relative error.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" > <genewardsmith@...> wrote:
> > Here are normalized TOP tuning errors for the primes for these > temperaments. By normalized, I mean that I give TOP(p)/log2(p) - 1.
> You can see that the Miller commas are precisely those which have the > primes having the largest errors, and there are three of these.

An exception is blacksmith, with only two primes having the largest errors. The "comma" is [0, -14, 0, 8> or 5764801/4782969, which hardly qualifies as a comma, but it produces the correct result. Augene (tripletone), with a Miller comma of [7, 0, -3, 0>, is another example.

Hmm, this could get ugly. Once you've found the TOP period for Augene, then you have to optimize the generator to have the smallest error for both 3 and 7...

No comment on this... those are the temperaments with
Miller commas more complex than their minimal commas.

-Carl

At 01:37 AM 3/9/2007, you wrote:
>Dominant, superpyth, sensi, orwell, and miracle stand out to me.
>
>-Carl
>
>At 12:02 AM 3/9/2007, you wrote:
>>A Miller comma for a rank-two temperament is a triprime comma of
>>maximal absolute value of relative error for the TOP tuning
>>(similarly for other ranks of temperament.) Here are some examples,
>>along with the minimal comma, which is the one where the relative
>>error is least. I was hoping for a patterm to emerge, but I don't see
>>it. Do you?
>>
>>Decimal
>>miller: 25/24 minimal: 50/49
>>
>>Dominant
>>miller: 256/245 minimal: 81/80
>>
>>Blackwood
>>miller: 2401/2187 minimal: 16807/16384
>>
>>Pajara
>>miller: 50/49 minimal: 2048/2025
>>
>>Mavila
>>miller: 243/224 minimal: 1071875/1048576
>>
>>Negri
>>miller: 49/48 minimal: 16875/16384
>>
>>Meantone
>>miller: 81/80 minimal: 3136/3125
>>
>>Porcupine
>>miller: 250/243 minimal: 268435456/262609375
>>
>>Superpyth
>>miller: 20480/19683 minimal: 245/243
>>
>>Magic
>>miller: 3125/3072 minimal: 537824/531441
>>
>>Myna
>>miller: 10077696/9765625 minimal: 40353607/40000000
>>
>>Sensi
>>miller: 1291315424/1220703125 minimal: 78732/78125
>>
>>Orwell
>>miller: 595749803514609375/558545864083284007 minimal:
>>134217728/133984375
>>
>>Miracle
>>miller: 34171875/33554432 minimal: 823543/819200
>>
>>Garibaldi
>>miller: 3125/3087 minimal: 32805/32768
>>
>>Octacot
>>miller: 20000/19683 minimal: 1638400000000000/1628413597910449
>>
>>Rodan
>>miller: 245/243 minimal: 1029/1024
>>
>>Hemiwuer
>>miller: 393216/390625 minimal: 33397665693696/33232930569601
>>
>>Hemififths
>>miller: 858993459200/847288609443 minimal:
>>1342177280000000000000/1341068619663964900807
>>
>>Ennealimmal
>>miller: 40353607/40310784 minimal: 7629394531250/7625597484987
>
>
>
>
>
>Yahoo! Groups Links
>
>
>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> No comment on this... those are the temperaments with
> Miller commas more complex than their minimal commas.

Now to figure out what that really means, I guess.