Yahoo Tuning Groups Ultimate Backup tuning Linear temperaments with consonant generators

Below are 7-limit linear temperaments with 9-limit consonance
generators (if you don't think 10/9 is much of a consonace, think 9/5
instead.) As I've pointed out before, there are only a finite number
of linear temperaments with given generator below any fixed logflat
badness cutoff. I didn't quite use that, since I also cult off high
error temperaments. Herman will probably find some of his favorites
missing, such as mavila, but I didn't want to mess with things which
conflate two consonances and use that for generators (and one--
godzilla--made it to my list anyway.) But aside from that, this is
it. You don't have all that many consonant-generated temperaments
when you start pusing the complexity higher.

7/5

Liese <<3 12 11 12 9 -8||
Triton <<3 -7 -8 -18 -21 1||
Tritonic <<5 -11 -12 -29 -33 3||
Neptune <<40 22 21 -58 -79 -13||

4/3

Meantone <<1 4 10 4 13 12||
Pontiac <<1 -8 39 -15 59 113||
Dominant <<1 4 -2 4 -6 -16||
Garibaldi <<1 -8 -14 -15 -25 -10||
Superpyth <<1 9 -2 12 -6 -30||
Flattone <<1 4 -9 4 -17 -32||
Kwai <<1 33 27 50 40 -30||

9/7

Supermajor <<37 46 75 -13 15 45||
Sensi <<7 9 13 -2 1 5||

5/4

Magic <<5 1 12 -10 5 25||
Wuerschmidt <<8 1 18 -17 6 39||
Muggles <<5 1 -7 -10 -25 -19||
Grendel <<23 -1 13 -55 -44 33||

6/5

Catakleismic <<6 5 22 -6 18 37||
Parakleismic <<13 14 35 -8 19 42||
Keemun <<6 5 3 -6 -12 -7||
Myna <<10 9 7 -9 -17 -9||
Superkleismic <<9 10 -3 -5 -30 -35||
Countercata <<6 5 -31 -6 -66 -86||

7/6

Godzilla <<2 8 1 8 -4 -20|| (7/6 and 8/7)
Orwell <<7 -3 8 -21 -7 27||

8/7

Godzilla <<2 8 1 8 -4 -20||
Gamera <<23 40 1 10 -63 -110||
Mothra <<3 12 -1 12 -10 -36||
Rodan <<3 17 -1 20 -10 -50||
Guiron <<3 -24 -1 -45 -10 65||

10/9

Porcupine <<3 5 -6 1 -18 -28||
Mitonic <<17 35 -21 16 -81 -147||

Gene Ward Smith wrote:
> Below are 7-limit linear temperaments with 9-limit consonance > generators (if you don't think 10/9 is much of a consonace, think 9/5 > instead.) As I've pointed out before, there are only a finite number > of linear temperaments with given generator below any fixed logflat > badness cutoff. I didn't quite use that, since I also cult off high > error temperaments. Herman will probably find some of his favorites > missing, such as mavila, but I didn't want to mess with things which > conflate two consonances and use that for generators (and one--
> godzilla--made it to my list anyway.) But aside from that, this is > it. You don't have all that many consonant-generated temperaments > when you start pusing the complexity higher.

Interesting list. For those who don't have a convenient catalog of regular temperaments to refer to: since this isn't the tuning-math list, and some of these aren't as well known as others, it would be useful to give the generator mapping and a couple of suggested tunings. For example:

> 7/5
> > Liese <<3 12 11 12 9 -8||
19&36 |<1, 3, 8, 8|, <0, -3, -12, -11|>
TOP: P = 1202.625, G = 569.049, TOP-RMS: P = 1201.571, G = 568.338

This shows that Liese temperament has a period of an octave and an optimal generator somewhat flatter than a 7/5. The |<1, 3, 8, 8|, <0, -3, -12, -11|> bit shows how you can combine different multiples of the period and generator to reach the tempered equivalents of 2/1, 3/1, 5/1, and 7/1. 5/1 for instance in TOP-RMS liese is 8(1201.571) - 12(568.338), or 2792.520 cents (6.2 cents sharp). The 19&36 bit means that you can draw a line between 19-ET and 36-ET on a chart like the one on monz's equal-temperament page to represent the temperament. If you know something about the character of the ET's at the endpoints of the line, you can get some idea of what the temperament is like.

(http://tonalsoft.com/enc/e/equal-temperament.aspx for monz's page with Paul Erlich's ET charts, a very useful set of charts)

> Triton <<3 -7 -8 -18 -21 1||
|<1, 3, -1, -1|, <0, -3, 7, 8|>
TOP: P = 1202.901, G = 570.448; TOP-RMS: P = 1203.405, G = 570.480

> Tritonic <<5 -11 -12 -29 -33 3||
29&31 |<1, 4, -3, -3|, <0, -5, 11, 12|>
TOP: P = 1201.023, G = 580.752; TOP-RMS: P = 1201.356, G = 580.942

> Neptune <<40 22 21 -58 -79 -13||
35&68 |<1, 21, 13, 13|, <0, -40, -22, -21|>
TOP: P = 1200.053, G = 582.477; TOP-RMS: P = 1200.066, G = 582.484

> 4/3
> > Meantone <<1 4 10 4 13 12||
12&19 |<1, 2, 4, 7|, <0, -1, -4, -10|>
TOP: P = 1201.699, G = 504.134; TOP-RMS: P = 1201.242, G = 504.026

> Pontiac <<1 -8 39 -15 59 113||
53&118 |<1, 2, -1, 19|, <0, -1, 8, -39|>
TOP: P = 1200.075, G = 498.277; TOP-RMS: P = 1200.099, G = 498.284

> Dominant <<1 4 -2 4 -6 -16||
5&12 |<1, 2, 4, 2|, <0, -1, -4, 2|>
TOP: P = 1195.229, G = 495.881; TOP-RMS: P = 1195.412, G = 496.521

> Garibaldi <<1 -8 -14 -15 -25 -10||
12&29 |<1, 2, -1, -3|, <0, -1, 8, 14|>
TOP: P = 1200.761, G = 498.119; TOP-RMS: P = 1200.125, G = 497.967

> Superpyth <<1 9 -2 12 -6 -30||
5&22 |<1, 2, 6, 2|, <0, -1, -9, 2|>
TOP: P = 1197.596, G = 489.427; TOP-RMS: P = 1197.067, G = 488.512

> Flattone <<1 4 -9 4 -17 -32||
19&26 |<1, 2, 4, -1|, <0, -1, -4, 9|>
TOP: P = 1202.536, G = 507.138; TOP-RMS: P = 1203.646, G = 507.759

> Kwai <<1 33 27 50 40 -30||
14&70 |<1, 2, 16, 14|, <0, -1, -33, -27|>
TOP: P = 1199.680, G = 497.252; TOP-RMS: P = 1199.734, G = 497.274

> 9/7
> > Supermajor <<37 46 75 -13 15 45||
80&91 |<1, 15, 19, 30|, <0, -37, -46, -75|>
TOP: P = 1200.022, G = 435.090; TOP-RMS: P = 1200.007, G = 435.085

> Sensi <<7 9 13 -2 1 5||
19&27 |<1, -1, -1, -2|, <0, 7, 9, 13|>
TOP: P = 1198.390, G = 443.160; TOP-RMS: P = 1199.714, G = 443.277

> 5/4
> > Magic <<5 1 12 -10 5 25||
19&22 |<1, 0, 2, -1|, <0, 5, 1, 12|>
TOP: P = 1201.277, G = 380.796; TOP-RMS: P = 1201.082, G = 380.695

> Wuerschmidt <<8 1 18 -17 6 39||
|<1, -1, 2, -3|, <0, 8, 1, 18|>
TOP: P = 1201.136, G = 387.584; TOP-RMS: P = 1199.979, G = 387.376

> Muggles <<5 1 -7 -10 -25 -19||
16&19 |<1, 0, 2, 5|, <0, 5, 1, -7|>
TOP: P = 1203.148, G = 379.393, TOP-RMS: P = 1203.978, G = 379.734

> Grendel <<23 -1 13 -55 -44 33||
31&59 |<1, 9, 2, 7|, <0, -23, 1, -13|>
TOP: P = 1199.672, G = 386.766, TOP-RMS: P = 1199.735, G = 386.777

> 6/5
> > Catakleismic <<6 5 22 -6 18 37||
19&53 |<1, 0, 1, -3|, <0, 6, 5, 22|>
TOP: P = 1200.536, G = 316.906, TOP-RMS: P = 1200.597, G = 316.889

> Parakleismic <<13 14 35 -8 19 42||
19&80 |<1, 5, 6, 12|, <0, -13, -14, -35|>
TOP: P = 1199.738, G = 315.108, TOP-RMS: P = 1199.782, G = 315.124

> Keemun <<6 5 3 -6 -12 -7||
4&15 |<1, 0, 1, 2|, <0, 6, 5, 3|>
TOP: P = 1203.187, G = 317.834, TOP-RMS: P = 1202.646, G = 317.170

> Myna <<10 9 7 -9 -17 -9||
4&27 |<1, -1, 0, 1|, <0, 10, 9, 7|>
TOP: P = 1198.828, G = 309.893, TOP-RMS: P = 1199.344, G = 309.976

> Superkleismic <<9 10 -3 -5 -30 -35||
15&26 |<1, 4, 5, 2|, <0, -9, -10, 3|>
TOP: P = 1201.372, G = 322.373, TOP-RMS: P = 1200.768, G = 322.136

> Countercata <<6 5 -31 -6 -66 -86||
53&87 |<1, 0, 1, 11|, <0, 6, 5, -31|>
TOP: P = 1199.977, G = 317.092, TOP-RMS: P = 1199.917, G = 317.100

> 7/6
> > Godzilla <<2 8 1 8 -4 -20|| (7/6 and 8/7)
5&19 |<1, 2, 4, 3|, <0, -2, -8, -1|>
TOP: P = 1203.669, G = 252.480, TOP-RMS: P = 1203.853, G = 253.446

> Orwell <<7 -3 8 -21 -7 27||
9&22 |<1, 0, 3, 1|, <0, 7, -3, 8|>
TOP: P = 1199.533, G = 271.494, TOP-RMS: P = 1200.021, G = 271.513

> 8/7
> > Godzilla <<2 8 1 8 -4 -20||
see above

> Gamera <<23 40 1 10 -63 -110||
26&73 |<1, 6, 10, 3|, <0, -23, -40, -1|>
TOP: P = 1199.852, G = 230.314, TOP-RMS: P = 1199.848, G = 230.307

> Mothra <<3 12 -1 12 -10 -36||
5&26 |<1, 1, 0, 3|, <0, 3, 12, -1|>
TOP: P = 1201.699, G = 232.521, TOP-RMS: P = 1200.937, G = 232.375

> Rodan <<3 17 -1 20 -10 -50||
5&41 |<1, 1, -1, 3|, <0, 3, 17, -1|>
TOP: P = 1200.232, G = 234.380, TOP-RMS: P = 1200.216, G = 234.459

> Guiron <<3 -24 -1 -45 -10 65||
36&41 |<1, 1, 7, 3|, <0, 3, -24, -1|>
TOP: P = 1200.486, G = 233.998, TOP-RMS: P = 1200.340, G = 233.996

> 10/9
> > Porcupine <<3 5 -6 1 -18 -28||
15&22 |<1, 2, 3, 2|, <0, -3, -5, 6|>
TOP: P = 1196.906, G = 162.318, TOP-RMS: P = 1197.839, G = 162.587

> Mitonic <<17 35 -21 16 -81 -147||
46&125 |<1, -1, -3, 6|, <0, 17, 35, -21|>
TOP: P = 1200.073, G = 182.468, TOP-RMS: P = 1200.096, G = 182.473

Linear temperaments with consonant generators

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Herman Miller <hmiller@IO.COM>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>