5-limit

>135/128

>

>Map:

>

>[ 0 1]

>[-1 2]

>[ 3 1]

>

>Generators: a = 10.0215/23; b = 1

>

>badness: 46.1

>rms: 18.1

>g: 2.94

>errors: [-24.8, -17.7, 7.1]

Not on monz's chart. What's "g"?

>648/625

>

>Map:

>

>[ 0 4]

>[ 1 5]

>[ 1 8]

>

>Generators: a = 21.0205/64; b = 1/4

>

>badness: 385

>rms: 11.06

>g: 3.266

>errors: [-7.82, 7.82, 15.64]

>

>64-et, anyone? It could also be used to temper the 12-et.

diminished.

>250/243

>

>Map:

>

>[ 0 1]

>[-3 2]

>[-5 3]

>

>Generators: a = 2.9883/22; b = 1

>

>badness: 360

>rms: 7.98

>g: 3.559

>errors: [9.06, -1.29, -10.35]

>

>One way to cure those 22-et major thirds of what ails them.

porcupine.

>128/125

>

>Map:

>

>[ 0 3]

>[-1 6]

>[ 0 7]

>

>Generators: a = 11.052/27 (~4/3); b = 1/3

>

>badness: 142

>rms: 9.68

>g: 2.449

>errors: [6.84, 13.69, 6.84]

augmented

>3125/3072

>

>Map:

>

>[ 0 1]

>[ 5 0]

>[ 1 2]

>

>Generators: a = 12.9822/41 (=6.016/19); b = 1

>

>badness: 239

>rms: 4.57

>g: 3.74

>errors: [-2.115, -6.346, -4.231]

>

>Graham has named this one: Magic.

>81/80

>

>Map:

>

>[ 0 1]

>[-1 2]

>[-4 4]

>

>Generators: a = 20.9931/50; b = 1

>

>badness: 108

>rms: 4.22

>g: 2.944

>errors: [-5.79, -1.65, 4.14]

>

>Nothing left to say about this one. :)

>2048/2025

>

>Map:

>

>[ 0 2]

>[-1 4]

>[ 2 3]

>

>Generators: 14.0123/34 (~4/3); b = 1/2

>

>badness: 211

>rms: 2.613

>g: 4.32

>errors: [3.49, 2.79, -.70]

>

>A good way to take advantage of the 34-ets excellent 5-limit

>harmonies is two gothish 17-et chains of fifths a sqrt(2)

>apart.

diaschismic

>78732/78125 = 2^2 3^9 5^-7

>

>Map:

>

>[ 0 1]

>[ 7 -1]

>[ 9 -1]

>

>Generators: 23.9947/65 (~9/7); b = 1

>

>badness: 346

>rms: 1.157

>g: 6.68

>errors: [-1.1, 0.5, 1.6]

un-named on monz's chart!

>393216/390625 = 2^17 3 5^-8

>

>Map:

>

>[ 0 1]

>[ 8 -1]

>[ 1 2]

>

>Generators: a = 31.9951/99 (~5/4); b = 1

>Works with 31,34,65,99,164

>

>badness: 251

>rms: 1.072

>g: 6.16

>error: [.602, 1.506, .904]

wuerschmidt

>2109375/2097152 = 2^-21 3^3 5^7 Orwell

>

>Map:

>

>[ 0 1]

>[ 7 0]

>[-3 3]

>

>Generators: a = 19.01127197/84; b = 1

>

>badness: 305.93

>rms: .8004

>g: 7.257

>errors: [-.828, -1.082, -.255]

>

>ets: 22,31,53,84

>15625/15552 = 2^-6 36-5 5^6 Kleismic

>

>Map:

>

>[ 0 1]

>[ 6 0]

>[ 5 1]

>

>Generators: a = 14.00435233/53 (~6/5); b = 1

>

>badness: 97

>rms: 1.030

>g: 4.546

>errors: [.523, -.915, -1.438]

>

>ets: 19,34,53,68,72,87,140

>1600000/1594323 = 2^9 3^-13 5^-2 Acute Minor Third system

>

>Map:

>

>[ 0 1]

>[-5 3]

>[-13 6]

>

>Generators: a = 28.00947813/99 (~243/200); b = 1

>

>badness: 305.53

>rms: .3831

>g: 9.273

>error: [-.5009, .0716, -.4293]

not on monz's chart.

>6115295232/6103515625 = 2^23 3^6 5^-15 Semisuper

>

>Map:

>

>[ 0 2]

>[ 7 -3]

>[ 3 2]

>

>Generators: a = 52.00397043/118 (~3125/2304); b = 1/2

/.../

>badness: 190

>rms: .1940

>g: 9.933

>errors: [.0226, .2081, .2255]

not on monz's chart.

>32805/32768 Shismic

>

>Map:

>

>[ 0 1]

>[-1 2]

>[ 8 1]

>

>Generators: a = 120.000624/289 (~4/3); b = 1

>

>badness: 55

>rms: .1617

>g: 6.976

>errors: [-.2275, -.1338, .0937]

7-limit

//augmented

>When extended to the 7-limit, this becomes the

>

>[ 0 3]

>[-1 6]

>[ 0 7]

>[ 2 6]

>

>system I've already mentioned in several contexts, such as

>the 15+12 system of the 27-et. Both as a 5-limit and a

>7-limit system, it is good enough to deserve a name of its

>own.

Jeez- I just realized that the wholetone scale contains

4:5:7 chords. Here's the 4:5:6:7 in augmented in 27-et:

27 1200

0 0

9 400

16 711

22 978

This nonatonic looks interesting: 0 2 4 9 11 13 18 20 22 (27)

>(1) [6,10,10,-5,1,2] ets: 22

>

>[0 2]

>[3 1]

>[5 1]

>[5 2]

>

>a = 7.98567775 / 22 (~9/7) ; b = 1/2

>measure 3165

What is this? What's "measure"?

>(4) [10,14,14,-7,6,-1] ets: 26

>

>[0 2]

>[5 2]

>[7 3]

>[7 4]

>

>a = 3.026421762 / 26; b = 1/2

>measure 8510

This and the above look suspiciously like

the decatonic and double-diatonic systems.

But they're not, are they?

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Not on monz's chart. What's "g"?

An average number of generator steps to get to the consonances.

> >(1) [6,10,10,-5,1,2] ets: 22

> >

> >[0 2]

> >[3 1]

> >[5 1]

> >[5 2]

> >

> >a = 7.98567775 / 22 (~9/7) ; b = 1/2

> >measure 3165

>

> What is this? What's "measure"?

This I think is the steps^3 thing some people wanted in order to weight the low-rent temperaments more strongly. I like log-flat measures myself, I've concluded.

> >(4) [10,14,14,-7,6,-1] ets: 26

> >

> >[0 2]

> >[5 2]

> >[7 3]

> >[7 4]

> >

> >a = 3.026421762 / 26; b = 1/2

> >measure 8510

>

> This and the above look suspiciously like

> the decatonic and double-diatonic systems.

> But they're not, are they?

One is the 9/7 generator thing of 22-et, and the other doesn't look diatonic to me. Aside from 26-et, it could be tried on h34+v7.

>> Not on monz's chart. What's "g"?

>

>An average number of generator steps to get to the consonances.

Aha! The most important measure of all! Is this the mean?

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> >> Not on monz's chart. What's "g"?

> >

> >An average number of generator steps to get to the consonances.

>

> Aha! The most important measure of all! Is this the mean?

gene was using rms, i believe.