7-limit MT reduced bases for ets

9: [21/20, 27/25, 128/125]
10: [25/24, 28/27, 49/48]
12: [36/35, 50/49, 64/63]
15: [28/27, 49/48, 126/125]
19: [49/48, 81/80, 126/125]
22: [50/49, 64/63, 245/243]
27: [64/63, 126/125, 245/243]
31: [81/80, 126/125, 1029/1024]
41: [225/224, 245/243, 1029/1024]
68: [245/243, 2048/2025, 2401/2400]
72: [225/224, 1029/1024, 4375/4374]
99: [2401/2400, 3136/3125, 4375/4374]
130: [2401/2400, 3136/3125, 19683/19600]
140: [2401/2400, 5120/5103, 15625/15552]

For any prime limit, we could consider the most characteristic linear temperament of a particular et to be the one leaving off the last member of the MT reduced basis. It is interesting to note that the characteristic linear temperament of 99 and 130 is the same. Of course we can do the same for planar temperaments, etc.

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

> 140: [2401/2400, 5120/5103, 15625/15552]

I accidentally left off

171: [2401/2400, 4375/4374, 32805/32768]

Wouldn't want to do that--look at those three high-powered commas!

> 9: [21/20, 27/25, 128/125]
> 10: [25/24, 28/27, 49/48]
> 12: [36/35, 50/49, 64/63]
> 15: [28/27, 49/48, 126/125]
> 19: [49/48, 81/80, 126/125]
> 22: [50/49, 64/63, 245/243]
> 27: [64/63, 126/125, 245/243]
> 31: [81/80, 126/125, 1029/1024]
> 41: [225/224, 245/243, 1029/1024]
> 68: [245/243, 2048/2025, 2401/2400]
> 72: [225/224, 1029/1024, 4375/4374]
> 99: [2401/2400, 3136/3125, 4375/4374]
> 130: [2401/2400, 3136/3125, 19683/19600]
> 140: [2401/2400, 5120/5103, 15625/15552]

This is seriously cool.

>For any prime limit, we could consider the most characteristic
>linear temperament of a particular et to be the one leaving off
>the last member of the MT reduced basis.

Does it have to be prime (not odd) limit?

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> Does it have to be prime (not odd) limit?

Fraid so. It occurs to me another fun game to play with these is to find the corresponding Fokker blocks.

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> > 9: [21/20, 27/25, 128/125]
> > 10: [25/24, 28/27, 49/48]
> > 12: [36/35, 50/49, 64/63]
> > 15: [28/27, 49/48, 126/125]
> > 19: [49/48, 81/80, 126/125]
> > 22: [50/49, 64/63, 245/243]
> > 27: [64/63, 126/125, 245/243]
> > 31: [81/80, 126/125, 1029/1024]
> > 41: [225/224, 245/243, 1029/1024]
> > 68: [245/243, 2048/2025, 2401/2400]
> > 72: [225/224, 1029/1024, 4375/4374]
> > 99: [2401/2400, 3136/3125, 4375/4374]
> > 130: [2401/2400, 3136/3125, 19683/19600]
> > 140: [2401/2400, 5120/5103, 15625/15552]
>
> This is seriously cool.
>
> >For any prime limit, we could consider the most characteristic
> >linear temperament of a particular et to be the one leaving off
> >the last member of the MT reduced basis.
>
> Does it have to be prime (not odd) limit?
>
> -Carl

Yeah, Carl, these are _tuning system building_ considerations and not
_simultaneous consonance_ considerations.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "clumma" <carl@l...> wrote:
>
> > Does it have to be prime (not odd) limit?
>
> Fraid so. It occurs to me another fun game to play with these is to
>find the corresponding Fokker blocks.

Or better yet, the most compact blocks where ratio odd-limit measures
distance from a central 1/1 (this is what I refer to as the van
Prooijen metric). Remember, these are constrained to be periodicity
blocks, so will _not_ be ellipsoids.

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

> For any prime limit, we could consider the most characteristic
linear temperament of a particular et to be the one leaving off the
last member of the MT reduced basis.

Let's follow up on that:

9: 21/20 ^ 27/25 = [2, 3, 1, 0, -4, -6]
mapping: [[1,2,3,3], [0,-2,-3,-1]]

10: 25/24 ^ 28/27 = [2, 1, 6, -3, 4, 11]
mapping: [[1,1,2,1], [0, 2, 1, 6]]

12: 36/35 ^ 50/49 = [4, 4, 4, -3, -5, -2]
mapping: [[4,6,9,11], [0,1,1,1]]
Diminished

15: 28/27 ^ 49/48 = [0, 5, 0, 8, 0, -14]
mapping: [[5,8,12,14], [0,0,-1,0]]
Blackwood

19: 49/48 ^ 81/80 = [2, 8, 1, 8, -4, -20]
mapping: [[1,2,4,3], [0,-2,-8,-1]]
Hemifourth

22: 64/63 ^ 50/49 = [2, -4, -4, -11, -12, 2]
mapping: [[2,3,5,6], [0,1,-2,-2]]
Pajara

27: 126/125 ^ 64/63 = [3, 0, -6, -7, -18, -14]
mapping: [[3,5,7,8], [0,-1,0,2]]
Tripletone

31: 126/125 ^ 81/80 = [1, 4, 10, 4, 13, 12]
mapping: [[1,2,4,7], [0,-1,-4,-10]]
Meantone

41: 225/224 ^ 245/243 = [5, 1, 12, -10, 5, 25]
mapping: [[1,0,2,-1], [0,5,1,12]]
Magic

68: 245/243 ^ 2048/2025 = [4, -8, 14, -22, 11, 55]
mapping: [[2,3,5,5], [0,2,-4,7]]
Shrutar

72: 225/224 ^ 1029/1024 = [6, -7, -2, -25, -20, 15]
mapping: [[1,1,3,3], [0,6,-7,-2]]
Miracle

99 and 130: 2401/2400 ^ 3136/3125 = [16, 2, 5, -34, -37, 6]
mapping: [[1,-1,2,2], [0,16,2,5]]
Hemiwuerschmidt

140: 5120/5103 ^ 2401/2400 = [2, 25, 13, 35, 15, -40]
mapping: [[1,1,-5,-1], [0,2,25,13]]
Hemififth

171: 4375/4374 ^ 2401/2400 = [18, 27, 18, 1, -22, -34]
mapping: [[9,15,22,26], [0,-2,-3,-2]]
Ennealimmal