Approximate consonances of Parch's 43 tone scale

I searched for all instances where one tone of this scale differed from another by an amount within 5 cents of an 11-limit consonance; this led to four commas: 385/384, 8019/8000, 441/440, and 540/539. These commas are linearly independent, and define an equal temperament (as well as a PB, incidentally.) The et they define is
(drum roll please) 72 et.

The linear temperaments obtained by leaving out one of the commas were
<385/384,441/440,540/539>, miracle; <540/539,8019/8000,385/384>,
catakleismic; and two unnamed temperaments with half-octave period:

#5 on my list, <540/539,441/440,8019/8000> with wedgie
[12,34,20,30,26,-2,6,-49,-48,15] and generators a=4.9919/72, b=1/2;

<441/440,8019/8000,385/384> with wedgie
[12,22,-4,-6,7,-40,-51,-71-90,-3] and generators a=10.9910/72, b=1/2.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> I searched for all instances where one tone of this scale differed
from another by an amount within 5 cents of an 11-limit consonance;
this led to four commas: 385/384, 8019/8000, 441/440, and 540/539.
>

This is fascinating.

How far out does the error have to go before another comma appears?
How far in can you come before one of those commas disappears and
which one is it?

> These commas are linearly independent, and define an equal
temperament
(as well as a PB, incidentally.) The et they define is
> (drum roll please) 72 et.

Neato! But that's dependent on your choice of allowable error?

> The linear temperaments obtained by leaving out one of the commas
were
> <385/384,441/440,540/539>, miracle; <540/539,8019/8000,385/384>,
> catakleismic;

What's the generator and period of catakleismic?

> and two unnamed temperaments with half-octave period:
>
> #5 on my list, <540/539,441/440,8019/8000> with wedgie
> [12,34,20,30,26,-2,6,-49,-48,15] and generators a=4.9919/72, b=1/2;

An 83.2 cent generator, half-octave period.

> <441/440,8019/8000,385/384> with wedgie
> [12,22,-4,-6,7,-40,-51,-71-90,-3] and generators a=10.9910/72,
b=1/2.

An 183.2 cent generator, half-octave period

How many notes in contiguous, (equal-length?) chains of generators
does each of these need to encompass Partch's 'Genesis' scale?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > I searched for all instances where one tone of this scale
differed
> from another by an amount within 5 cents of an 11-limit consonance;
> this led to four commas: 385/384, 8019/8000, 441/440, and 540/539.
> >
>
> This is fascinating.

Where was this posted originally?

> How far out does the error have to go before another comma appears?
> How far in can you come before one of those commas disappears and
> which one is it?
>
> > These commas are linearly independent, and define an equal
> temperament
> (as well as a PB, incidentally.) The et they define is
> > (drum roll please) 72 et.
>
> Neato! But that's dependent on your choice of allowable error?

But of course. The commas are, in cents, 4.5026 cents, 4.1068 cents,
3.9302 cents, and 3.209. All about the same -- but you could draw a
line between them, I guess.

> > The linear temperaments obtained by leaving out one of the commas
> were
> > <385/384,441/440,540/539>, miracle; <540/539,8019/8000,385/384>,
> > catakleismic;
>
> What's the generator and period of catakleismic?
>
> > and two unnamed temperaments with half-octave period:
> >
> > #5 on my list, <540/539,441/440,8019/8000> with wedgie
> > [12,34,20,30,26,-2,6,-49,-48,15] and generators a=4.9919/72,
b=1/2;
>
> An 83.2 cent generator, half-octave period.
>
> > <441/440,8019/8000,385/384> with wedgie
> > [12,22,-4,-6,7,-40,-51,-71-90,-3] and generators a=10.9910/72,
> b=1/2.
>
> An 183.2 cent generator, half-octave period
>
> How many notes in contiguous, (equal-length?) chains of generators
> does each of these need to encompass Partch's 'Genesis' scale?

What's the answer for MIRACLE? Wasn't it an non-'Genesis' 43-tone
scale that MIRACLE comprised in 45 consecutive notes in a chain of
generators?

This is really interesting, as it makes one wonder, to what extend
was Secor's original proposal "unique" or "best".

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > What's the generator and period of catakleismic?

From /tuning-math/message/2791

20. Catakleismic

[6, 5, 22, -21, -6, 18, -54, 37, -66, -135]

[225/224, 385/384, 540/539, 4375/4374]

ets 19, 72

[[0, 6, 5, 22, -21], [1, 0, 1, -3, 9]]

[.2639230436, 1]

a = 19.0025/72 = 316.7076522 cents

badness 271.0589693
rms 1.697136764
g 20.98979344

Because of the recent discussion of the 152 et, I thought I would also repost this for your consideration:

3. Octoid

[24, 32, 40, 24, -5, -4, -45, 3, -55, -71]

[540/539, 3025/3024, 4375/4374, 9801/9800]

ets 72, 80, 152, 224, 296

[[0, -3, -4, -5, -3], [8, 16, 23, 28, 31]]

[.1383934690, 1/8]

a = 9.9643/72 = 31.0001/224 = 166.0721626

badness 147.3854996
rms .7687062948
g 23.42160176

> > How many notes in contiguous, (equal-length?) chains of generators
> > does each of these need to encompass Partch's 'Genesis' scale?

As in Genesis of a Music?

> What's the answer for MIRACLE? Wasn't it an non-'Genesis' 43-tone
> scale that MIRACLE comprised in 45 consecutive notes in a chain of
> generators?

I don't know. I got the scale I analyzed from a web search; I didn't know there was more than one 43 tone Partch scale.

> This is really interesting, as it makes one wonder, to what extend
> was Secor's original proposal "unique" or "best".

Miracle does appear here.

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

> 3. Octoid
>
> [24, 32, 40, 24, -5, -4, -45, 3, -55, -71]
>
> [540/539, 3025/3024, 4375/4374, 9801/9800]
>
> ets 72, 80, 152, 224, 296
>
> [[0, -3, -4, -5, -3], [8, 16, 23, 28, 31]]
>
> [.1383934690, 1/8]
>
> a = 9.9643/72

Equivalently, about 1/72 oct. -- right?

> = 31.0001/224 = 166.0721626
>
> badness 147.3854996

This 11-limit badness is not directly comparable to 5-limit badness
for 5-limit temperaments, is it?

> rms .7687062948
> g 23.42160176

Certainly looks like an efficient way of getting 11-limit harmony in
152-tET! Sort of the 152-tET version of MIRACLE, but more accurate.
This implies sort of an adaptive 8-tET scheme, where more and more 16-
cent-apart 8-tET chains are needed the more expansive the JI chords
you want to play. The only JI interval in a single 8-tET chain is the
11/6 (and equivalents). Certainly a major aspect of 152-tET that I
overlooked.

> > This is really interesting, as it makes one wonder, to what
extend
> > was Secor's original proposal "unique" or "best".
>
> Miracle does appear here.

I know, but how do the others compare, in terms of number
of "restored" consonant intervals and chords (hexads on down), and in
terms of minimax 11-limit error?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > [.1383934690, 1/8]
> >
> > a = 9.9643/72
>
> Equivalently, about 1/72 oct. -- right?

Right. It suggests a temperament of 72 in terms of the 224-et, with a generator of 3/224 and another of 1/8.

> > = 31.0001/224 = 166.0721626
> >
> > badness 147.3854996
>
> This 11-limit badness is not directly comparable to 5-limit badness
> for 5-limit temperaments, is it?

Nope; it's similar in a way, because of the flatness condition.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > [.1383934690, 1/8]
> > >
> > > a = 9.9643/72
> >
> > Equivalently, about 1/72 oct. -- right?
>
> Right. It suggests a temperament of 72 in terms of the 224-et, with
a generator of 3/224 and another of 1/8.
>
> > > = 31.0001/224 = 166.0721626
> > >
> > > badness 147.3854996
> >
> > This 11-limit badness is not directly comparable to 5-limit
badness
> > for 5-limit temperaments, is it?
>
> Nope; it's similar in a way, because of the flatness condition.

Right, but is it directly comparable? Is 500 an equally "bad" score
in both frameworks?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > Nope; it's similar in a way, because of the flatness condition.
>
> Right, but is it directly comparable? Is 500 an equally "bad" score
> in both frameworks?

More or less, to the extent the question even makes sense, I suppose.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > Nope; it's similar in a way, because of the flatness condition.
> >
> > Right, but is it directly comparable? Is 500 an equally "bad"
score
> > in both frameworks?
>
> More or less, to the extent the question even makes sense, I
suppose.

Well, what if the question were phrased in terms of the density of
temperaments that pass a "goodness" criterion in the vicinity of a
given g (gens) value?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > > How many notes in contiguous, (equal-length?) chains of
generators
> > > does each of these need to encompass Partch's 'Genesis' scale?
>
> As in Genesis of a Music?
>
> > What's the answer for MIRACLE? Wasn't it an non-'Genesis' 43-tone
> > scale that MIRACLE comprised in 45 consecutive notes in a chain of
> > generators?
>
> I don't know. I got the scale I analyzed from a web search; I didn't
know there was more than one 43 tone Partch scale.

See /tuning/topicId_25575.html#25575