Yahoo Tuning Groups Ultimate Backup tuning-math A 5-limit, "geometric" temperament list

The following is a complete list of all 5-limit temperaments satisfying
the requirements that rms error be less than 15, geometric complexity
less than 40, and the badness calculated from these less than 3000. If
people feel something valuable has been left off (e.g, 135/128, 25/24, or
16875/16384) we could raise the error limit.

81/80 (3)^4/(2)^4/(5) meantone
[[1, 0, -4], [0, 1, 4]]

comp 4.132030727 rms 4.217730828 bad 297.5565312
generators [1200., 1896.164845]

128/125 (2)^7/(5)^3 augmented
[[3, 5, 7], [0, 1, 0]]

comp 4.828313736 rms 9.677665780 bad 1089.323984
generators [400.0000000, 91.20185550]

256/243 (2)^8/(3)^5 quintal
[[5, 8, 12], [0, 0, -1]]

comp 5.493061445 rms 12.75974144 bad 2114.877638
generators [240.0000000, 84.66378778]

250/243 (2)*(5)^3/(3)^5 porcupine
[[1, 2, 3], [0, -3, -5]]

comp 5.948285733 rms 7.975800816 bad 1678.609846
generators [1200., 162.9960265]

2048/2025 (2)^11/(3)^4/(5)^2 diaschismic
[[2, 3, 5], [0, 1, -2]]

comp 6.271198982 rms 2.612821643 bad 644.4088670
generators [600.0000000, 105.4465315]

648/625 (2)^3*(3)^4/(5)^4 diminished
[[4, 6, 9], [0, 1, 1]]

comp 6.437751648 rms 11.06006024 bad 2950.938432
generators [300.0000000, 94.13435693]

3125/3072 (5)^5/(2)^10/(3) diesic (small diesic)
[[1, 0, 2], [0, 5, 1]]

comp 7.741412273 rms 4.569472316 bad 2119.954991
generators [1200., 379.9679493]

15625/15552 (5)^6/(2)^6/(3)^5 kleismic
[[1, 0, 1], [0, 6, 5]]

comp 9.338935129 rms 1.029625097 bad 838.6315482
generators [1200., 317.0796753]

32805/32768 (3)^8*(5)/(2)^15 shismic
[[1, 0, 15], [0, 1, -8]]

comp 9.459947973 rms .1616933186 bad 136.8857747
generators [1200., 1901.727514]

20000/19683 (2)^5*(5)^4/(3)^9 quadrafifths (minimal diesic)
[[1, 1, 1], [0, 4, 9]]

comp 9.785568434 rms 2.504205191 bad 2346.540676
generators [1200., 176.2822703]

78732/78125 (2)^2*(3)^9/(5)^7 hemisixths (tiny diesic)
[[1, -1, -1], [0, 7, 9]]

comp 12.19218236 rms 1.157498409 bad 2097.803242
generators [1200., 442.9792975]

393216/390625 (2)^17*(3)/(5)^8 wuerschmidt
[[1, -1, 2], [0, 8, 1]]

comp 12.54312332 rms 1.071949828 bad 2115.395301
generators [1200., 387.8196733]

2109375/2097152 (3)^3*(5)^7/(2)^21 orwell
[[1, 0, 3], [0, 7, -3]]

comp 12.77234114 rms .8004099292 bad 1667.723301
generators [1200., 271.5895996]

1600000/1594323 (2)^9*(5)^5/(3)^13 amt
[[1, 3, 6], [0, -5, -13]]

comp 13.79419993 rms .3831037874 bad 1005.555381
generators [1200., 339.5088256]

6115295232/6103515625 (2)^23*(3)^6/(5)^14 semisuper
[[2, 4, 5], [0, -7, -3]]

comp 21.20762522 rms .1940180530 bad 1850.624306
generators [600.0000000, 71.14606343]

1224440064/1220703125 (2)^8*(3)^14/(5)^13 parakleismic
[[1, 5, 6], [0, -13, -14]]

comp 21.32267248 rms .2766026501 bad 2681.521263
generators [1200., 315.2509133]

10485760000/10460353203 (2)^24*(5)^4/(3)^21
[[1, 0, -6], [0, 4, 21]]

comp 21.73304916 rms .1537673823 bad 1578.433204
generators [1200., 475.5422333]

274877906944/274658203125 (2)^38/(3)^2/(5)^15 hemithird
[[1, 4, 2], [0, -15, 2]]

comp 24.97702150 rms .6082244804e-1 bad 947.7326423
generators [1200., 193.1996149]

68719476736000/68630377364883 (2)^39*(5)^3/(3)^29 trichotififths
[[1, 0, -13], [0, 3, 29]]

comp 30.55081228 rms .5750010064e-1 bad 1639.596150
generators [1200., 634.0119851]

19073486328125/19042491875328 (5)^19/(2)^14/(3)^19 enneadecal
[[19, 30, 44], [0, 1, 1]]

comp 30.57932033 rms .1047837215 bad 2996.244873
generators [63.15789474, 7.292252126]

9010162353515625/9007199254740992 (3)^10*(5)^16/(2)^53
[[2, 1, 6], [0, 8, -5]]

comp 31.25573660 rms .1772520822e-1 bad 541.2283791
generators [600.0000000, 162.7418923]

7629394531250/7625597484987 (2)*(5)^18/(3)^27 ennealimmal
[[9, 13, 19], [0, 2, 3]]

comp 33.65327154 rms .2559250891e-1 bad 975.4269093
generators [133.3333333, 84.32451333]

50031545098999707/50000000000000000 (3)^35/(2)^16/(5)^17
[[1, -1, -3], [0, 17, 35]]

comp 38.84548584 rms .2546649929e-1 bad 1492.763207
generators [1200., 182.4660891]

450359962737049600/450283905890997363 (2)^54*(5)^2/(3)^37 monzismic
[[1, 2, 10], [0, -2, -37]]

comp 39.66560308 rms .5738429624e-2 bad 358.1254995
generators [1200., 249.0184479]

--- In tuning-math@y..., "hs" <straub@d...> wrote:
> >81/80 (3)^4/(2)^4/(5) meantone
> >[[1, 0, -4], [0, 1, 4]]
>
> Newbie question: what is the meaning of this vector notation? From
what I read so
> far, all I can imagine are unison vectors determining a periodicity
block. Something
> like this?
>
>
> Hans Straub

Luckily there are no wedgies here, so i think i can explain this one.

Meantone temperament takes the just lattice and modifies it
by "tempering out" the interval 81/80. that is, all the consonant
intervals, the "rungs" of the just lattice, are detuned slightly so
that 81/80 comes out to a unison.

(3)^4/(2)^4/(5) is simply the prime factorization of 81/80. 81 = (3)
^4; 1/80 = 1/(2)^4/(5), get it?

Then the next line tells you the mapping of the primes in terms of
the generators. You left out this line, which was included with the
ones you report below:

generators [1200., 1896.164845]
(these are in cents)

So . . . [1, 0, -4], [0, 1, 4] would be better arranged like this:

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

the first row tells you that the 2/1 is represented by [1]*1200 + [0]
*1896; the second row tells you that the 3/1 is represented by [0]
*1200 + [1]*1896; the third row tells you that the 5/1 is represented
by [-4]*1200 + [4]*1896.

Making sense?

p.s. i tried to reply directly to the freelists list, but i got the
following error message:

The original message was received at Mon, 29 Jul 2002 21:21:32 GMT
from user10.acadian-asset.com [208.253.47.10] (may be forged)

----- The following addresses had permanent fatal errors -----
<tuning-math@freelists.org>

----- Transcript of session follows -----
... while talking to turing.freelists.org. [206.53.239.180]:
>>> RCPT To:<tuning-math@freelists.org>
<<< 554 Service unavailable; [199.171.54.106] blocked using
relays.osirusoft.com, reason:
http://www.appliedinfogroup.com/html/emarketing.html (services
(appending, also look for 'opt-out')) 554 <tuning-
math@freelists.org>... Service unavailable

anyone know what's up?

A 5-limit, "geometric" temperament list

🔗Gene W Smith <genewardsmith@juno.com>

🔗Gene W Smith <genewardsmith@juno.com>

🔗hs <straub@datacomm.ch>

🔗wallyesterpaulrus <perlich@aya.yale.edu>

🔗Hans Straub <straub@datacomm.ch>