7-limit 3D temperaments

Hi there,

I'm just surprised ... You see, I was experimenting with 3D temperaments some time ago and now I'm realizing I've rediscovered many tunings which Gene Ward Smith already knew about much earlier ... It's amazing, I'm out of words -- why didn't I know about this webpage sooner?
http://lumma.org/tuning/gws/hermbas.htm

I'm only not sure if they have names -- I just know about starling (126/125), marvel (225/224) and breed (2401/2400)

Petr

Petr Pařízek wrote:
> Hi there,
> > I'm just surprised ... You see, I was experimenting with 3D temperaments > some time ago and now I'm realizing I've rediscovered many tunings which > Gene Ward Smith already knew about much earlier ... It's amazing, I'm out of > words -- why didn't I know about this webpage sooner?
> http://lumma.org/tuning/gws/hermbas.htm
> > I'm only not sure if they have names -- I just know about starling > (126/125), marvel (225/224) and breed (2401/2400)

There's a list of commas here:

http://lumma.org/tuning/gws/commalist.htm

Every comma corresponds to a rank 3 temperament but the names don't match up. I also found this list of commas on my hard drive, which is probably ordered in terms of some kind of temperament badness:

4375:4374
2401:2400
184528125:184473632
52734375:52706752
200120949:200000000
5250987:5242880
3136:3125
225:224
6144:6125
5120:5103
1029:1024
10976:10935
16875:16807
126:125
245:243
19683:19600
15625:15552
235298:234375
1600000:1594323
4096000:4084101
81:80
823543:820125
26873856:26796875
1728:1715
64:63
283435200:282475249
321489:320000
156250000:155649627
50:49
4000:3969
100442349:100000000

There are some 11-limit names here:

/tuning-math/message/14061

If you want higher limits, I can dust off my scripts.

Graham

Petr Pařízek wrote:
> Hi there,
> > I'm just surprised ... You see, I was experimenting with 3D temperaments > some time ago and now I'm realizing I've rediscovered many tunings which > Gene Ward Smith already knew about much earlier ... It's amazing, I'm out of > words -- why didn't I know about this webpage sooner?
> http://lumma.org/tuning/gws/hermbas.htm
> > I'm only not sure if they have names -- I just know about starling > (126/125), marvel (225/224) and breed (2401/2400)
> > Petr

I dug up a list of 7-limit rank 3 temperaments (or planar temperaments, as we were calling them back then) that Gene posted on July 22, 2002 in the tuning-math list. I don't know of any names for most of these. Some of them are the same commas as 5-limit rank 2 temperaments (81/80 is meantone, for an obvious example), but there's already a 7-limit rank 2 version of meantone, so calling the rank 3 "meantone" (when it only includes meantone as a subset) would be inappropriate. I might suggest a prefix like "super-" for the higher rank temperaments ... 81/80 rank 3 could be "supermeantone", but note that "superkleismic" is already taken (kleismic was a former name of what we now call "hanson", so a rank 3 kleismic temperament could be "superhanson" if we adopt this convention). I do have a list of 11-limit rank 3 temperaments with names, so it's possible the 7-limit ones also have names, but I don't know of a convenient list of them. Here's the list from Gene's 2002 tuning-math post.

4375/4374 (5)^4*(7)/(2)/(3)^7
250047/250000 (3)^6*(7)^3/(2)^4/(5)^6
2401/2400 (7)^4/(2)^5/(3)/(5)^2
225/224 (3)^2*(5)^2/(2)^5/(7)
64/63 (2)^6/(3)^2/(7)
81/80 (3)^4/(2)^4/(5)
32805/32768 (3)^8*(5)/(2)^15
126/125 (2)*(3)^2*(7)/(5)^3
5120/5103 (2)^10*(5)/(3)^6/(7)
2460375/2458624 (3)^9*(5)^3/(2)^10/(7)^4
28/27 (2)^2*(7)/(3)^3
65625/65536 (3)*(5)^5*(7)/(2)^16
420175/419904 (5)^2*(7)^5/(2)^6/(3)^8
6144/6125 (2)^11*(3)/(5)^3/(7)^2
703125/702464 (3)^2*(5)^7/(2)^11/(7)^3
1029/1024 (3)*(7)^3/(2)^10
49/48 (7)^2/(2)^4/(3)
50/49 (2)*(5)^2/(7)^2
3136/3125 (2)^6*(7)^2/(5)^5
245/243 (5)*(7)^2/(3)^5
4000/3969 (2)^5*(5)^3/(3)^4/(7)^2
10976/10935 (2)^5*(7)^3/(3)^7/(5)
5250987/5242880 (3)^7*(7)^4/(2)^20/(5)
2048/2025 (2)^11/(3)^4/(5)^2
875/864 (5)^3*(7)/(2)^5/(3)^3
1728/1715 (2)^6*(3)^3/(5)/(7)^3
128/125 (2)^7/(5)^3
525/512 (3)*(5)^2*(7)/(2)^9
19683/19600 (3)^9/(2)^4/(5)^2/(7)^2
321489/320000 (3)^8*(7)^2/(2)^9/(5)^4
4096000/4084101 (2)^15*(5)^3/(3)^5/(7)^5
686/675 (2)*(7)^3/(3)^3/(5)^2
33075/32768 (3)^3*(5)^2*(7)^2/(2)^15
405/392 (3)^4*(5)/(2)^3/(7)^2
3645/3584 (3)^6*(5)/(2)^9/(7)
2100875/2097152 (5)^3*(7)^5/(2)^21
1323/1280 (3)^3*(7)^2/(2)^8/(5)
67108864/66976875 (2)^26/(3)^7/(5)^4/(7)^2
15625/15552 (5)^6/(2)^6/(3)^5
16875/16807 (3)^3*(5)^4/(7)^5
102760448/102515625 (2)^21*(7)^2/(3)^8/(5)^6
250/243 (2)*(5)^3/(3)^5
458752/455625 (2)^16*(7)/(3)^6/(5)^4
589824/588245 (2)^16*(3)^2/(5)/(7)^6
256/243 (2)^8/(3)^5

Herman wrote:

> I do have a list of 11-limit rank 3 temperaments with
> names, so it's possible the 7-limit ones also have names, but I don't
> know of a convenient list of them.

Perhaps it’s the one that Graham suggested in the preceding message?

> Here's the list from Gene's 2002
> tuning-math post.

To be honest, I’m not sure how I could make a 3D temperament from commas like 703125/702464. If I try to temper it out using three different intervals, I get two occurrences of 3/1, three occurrences of 2/7, and seven occurrences of 5/4, but none of them is there only once, which means I don’t know how to get two generator sizes out of that.

Petr

Petr Pařízek wrote:
> Herman wrote:
> >> I do have a list of 11-limit rank 3 temperaments with >> names, so it's possible the 7-limit ones also have names, but I don't >> know of a convenient list of them.
> > Perhaps it’s the one that Graham suggested in the preceding message?

I tracked down Gene's original list to

/tuning-math/message/12298

> >> Here's the list from Gene's 2002 >> tuning-math post.
> > To be honest, I’m not sure how I could make a 3D temperament from commas like 703125/702464. If I try to temper it out using three different intervals, I get two occurrences of 3/1, three occurrences of 2/7, and seven occurrences of 5/4, but none of them is there only once, which means I don’t know how to get two generator sizes out of that.

What I've done is search for mappings that temper it out, which gives the following:

<12, 19, 28, 34]
<19, 30, 44, 53
<31, 49, 72, 87]
<50, 79, 116, 140]
<121, 192, 281, 340]
<140, 222, 325, 393]
<152, 241, 353, 427]
<159, 252, 369, 446]
<171, 271, 397, 480]
<183, 290, 425, 514]
<190, 301, 441, 533]
<202, 320, 469, 567]

Then taking three linearly-independent mappings gives

[ 7.17354686 13.17099221 7.13807445] cent steps

wedgie:
(3, 7, -2, -11)

mapping by steps:
[[ 12 19 28 34]
[ 19 30 44 53]
[121 192 281 340]]

tuning map (cents):
[[ 1200.03842294 1901.93745132 2786.18189018 3368.90849387]]

scalar complexity: 0.261
RMS weighted error: 0.038 cents/octave
max weighted error: 0.057 cents/octave

There's a also a linear algebra solution. Here, you can take the defining comma along with those for 16:15, 25:24, and 225:224.

array([[ 4, -1, -1, 0],
[ -3, -1, 2, 0],
[ -5, 2, 2, -1],
[-11, 2, 7, -3]])

The adjoint of that is

array([[ 7., 5., 9., -3.],
[ 11., 8., 15., -5.],
[ 16., 12., 21., -7.],
[ 19., 15., 26., -9.]])

You ignore the last column because it corresponds to not tempering out the given comma. The other commas give you equal temperaments of 12, 7, and 9 steps. Presumably they build up to give the same temperament class -- I haven't checked it.

Graham