Since 9-odd-limit and 7-odd-limit involve the same set of primes, we can justify using either. If we regard them both as important, we can justify using anything between the ranges of the poptimal generators even when these are disjoint. To be sure, when 3 has high complexity we might be interested in 7 but not 9, but I think this is pretty much of a quibble. I calculated a 9-limit poptimal, (the second listed below) and then found the best which can be had by putting 7 and 9 together. The "univeral" generators listed below are where all the p values give the same result; since this didn't happen for both 7 and 9 I still got an et.

At this point I would say things are generally looking good. We now have 72 for Miracle, 84 for Orwell, 12 for Diminished and Dominant Seventh, 22 for Porcupine, 31 for Semififths and Supermajor Seconds,

41 for Superkleismic and 46 for Semisixths. The 112-et for Meantone is still a little over the top, but is at least worth considering; the tendency for Meantone to want a generator in this very small interval is rather striking.

pops:=[

"Decimal",

[4, 2, 2, -1, 8, -6], [[2, 0, 3, 4], [0, 2, 1, 1]],

[1/2 l2(4/3)), "universal"], [1/2, 5/24], [1/2, 5/24],

"Dominant seventh",

[1, 4, -2, -16, 6, 4], [[1, 0, -4, 6], [0, 1, 4, -2]],

[1, 24/41], [1, 7/12], [1, 7/12],

"Diminished",

[4, 4, 4, -2, 5, -3], [[4, 0, 3, 5], [0, 1, 1, 1]],

[1/4, 1/16], [1/4, l2(3 sqrt(2)/4), "universal"], [1/4, 1/12],

"Blackwood",

[0, 5, 0, -14, 0, 8], [[5, 8, 0, 14], [0, 0, 1, 0]],

[1/5, 3/25], [1/5, 1/15], [1/5, 1/15],

"Augmented",

[3, 0, 6, 14, -1, -7], [[3, 0, 7, -1], [0, 1, 0, 2]],

[1/3, 8/33], [1/3, 11/45], [1/3, 8/33],

"Pajara",

[2, -4, -4, 2, 12, -11], [[2, 0, 11, 12], [0, 1, -2, -2]],

[1/2, 5/56], [1/2, 5/56], [1/2, 5/56],

"Hexadecimal",

[1, -3, 5, 20, -5, -7], [[1, 0, 7, -5], [0, 1, -3, 5]],

[1, 23/41], [1, 41/73], [1, 23/41],

"Tertiathirds",

[4, -3, 2, 13, 8, -14], [[1, 2, 2, 3], [0, 4, -3, 2]],

[1, 5/48], [1, 5/48], [1, 5/48],

"Kleismic",

[6, 5, 3, -7, 12, -6], [[1, 0, 1, 2], [0, 6, 5, 3]],

[1, 14/53], [1, 14/53], [1, 14/53],

"Tripletone",

[3, 0, -6, -14, 18, -7], [[3, 0, 7, 18], [0, 1, 0, -2]],

[1/3, 2/27], [1/3, 3/33], [1/3, 1/12],

"Hemifourths",

[2, 8, 1, -20, 4, 8], [[1, 0, -4, 2], [0, 2, 8, 1]],

[1, 4/19], [1, 4/19], [1, 4/19],

"Meantone",

[1, 4, 10, 12, -13, 4], [[1, 0, -4, -13], [0, 1, 4, 10]],

[1, 119/205], [1, 65/112], [1, 65/112],

"Injera",

[2, 8, 8, -4, -7, 8], [[2, 0, -8, -7], [0, 1, 4, 4]],

[1/2, 2/26], [1/2, 2/26], [1/2, 2/26],

"Double wide",

[8, 6, 6, -3, 13, -9], [[2, 1, 3, 4], [0, 4, 3, 3]],

[1/2, 7/26], [1/2, 13/48], [1/2, 7/26],

"Porcupine",

[3, 5, -6, -28, 18, 1], [[1, 2, 3, 2], [0, 3, 5, -6]],

[1, 8/59], [1, 3/22], [1, 3/22],

"Superpythagorean",

[1, 9, -2, -30, 6, 12], [[1, 0, -12, 6], [0, 1, 9, -2]],

[1, 45/76], [1, 29/49], [1, 29/49],

"Muggles",

[5, 1, -7, -19, 25, -10], [[1, 0, 2, 5], [0, 5, 1, -7]],

[1, 62/197], [1, 53/168], [1, 17/54],

"Beatles",

[2, -9, -4, 16, 12, -19], [[1, 1, 5, 4], [0, 2, -9, -4]],

[1, 19/64], [1, 8/27], [1, 8/27],

"Flattone",

[1, 4, -9, -32, 17, 4], [[1, 0, -4, 17], [0, 1, 4, -9]],

[1, 63/109], [1, 37/64], [1, 37/64],

"Magic",

[5, 1, 12, 25, -5, -10], [[1, 0, 2, -1], [0, 5, 1, 12]],

[1, 13/41], [1, 71/224], [1, 13/41],

"Nonkleismic",

[10, 9, 7, -9, 17, -9], [[1, 9, 9, 8], [0, 10, 9, 7]],

[1, 23/89], [1, 38/147], [1, 23/89],

"Semisixths",

[7, 9, 13, 5, -1, -2], [[1, 6, 8, 11], [0, 7, 9, 13]],

[1, 44/119], [1, 17/46], [1, 17/46],

"Orwell",

[7, -3, 8, 27, 7, -21], [[1, 0, 3, 1], [0, 7, -3, 8]],

[1, 26/115], [1, 43/190], [1, 19/84],

"Miracle",

[6, -7, -2, 15, 20, -25], [[1, 1, 3, 3], [0, 6, -7, -2]],

[1, 17/175], [1, 32/329], [1, 7/72],

"Quartaminorthirds",

[9, 5, -3, -21, 30, -13], [[1, 1, 2, 3], [0, 9, 5, -3]],

[1, 9/139], [1, 17/262], [1, 7/108],

"Supermajor seconds",

[3, 12, -1, -36, 10, 12], [[1, 1, 0, 3], [0, 3, 12, -1]],

[1, 53/274], [1, 6/31], [1, 6/31],

"Schismic",

[1, -8, -14, -10, 25, -15], [[1, 0, 15, 25], [0, 1, -8, -14]],

[1, 55/94], [1, 55/94], [1, 55/94],

"Superkleismic",

[9, 10, -3, -35, 30, -5], [[1, 4, 5, 2], [0, 9, 10, -3]],

[1, 37/138], [1, 106/395], [1, 11/41],

"Squares",

[4, 16, 9, -24, -3, 16], [[1, 3, 8, 6], [0, -4, -16, -9]],

[1, 93/262], [1, 104/293], [1, 93/262],

"Semififths",

[2, 8, -11, -48, 23, 8], [[1, 1, 0, 6], [0, 2, 8, -11]],

[1, 119/410], [1, 9/31], [1, 9/31],

"Diaschismic",

[2, -4, -16, -26, 31, -11], [[2, 0, 11, 31], [0, 1, -2, -8]],

[1/2, 9/104], [1/2, 9/104], [1/2, 9/104],

"Octafifths",

[8, 18, 11, -25, 5, 10], [[1, 1, 1, 2], [0, 8, 18, 11]],

[1, 13/177], [1, 8/109], [1, 8/109],

"Tritonic",

[5, -11, -12, 3, 33, -29], [[1, 4, -3, -3], [0, 5, -11, -12]],

[1, 163/337], [1, 191/395], [1, 59/122],

"Supersupermajor",

[3, 17, -1, -50, 10, 20], [[1, 1, -1, 3], [0, 3, 17, -1]],

[1, 25/128], [1, 25/128], [1, 25/128],

"Shrutar",

[4, -8, 14, 55, -11, -22], [[2, 1, 9, -2], [0, 2, -4, 7]],

[1/2, 8/182], [1/2, 9/205], [1/2, 8/182],

"Catakleismic",

[6, 5, 22, 37, -18, -6], [[1, 0, 1, -3], [0, 6, 5, 22]],

[1, 52/197], [1, 52/197], [1, 52/197],

"Hemiwuerschmidt",

[16, 2, 5, 6, 37, -34], [[1, 15, 4, 7], [0, 16, 2, 5]],

[1, 37/229], [1, 37/229], [1, 37/229],

"Hemikleismic",

[12, 10, -9, -49, 48, -12], [[1, 0, 1, 4], [0, 12, 10, -9]],

[1, 25/189], [1, 62/469], [1, 16/121],

"Hemithirds",

[15, -2, -5, -6, 50, -38], [[1, 4, 2, 2], [0, -15, 2, 5]],

[1, 24/149], [1, 43/267], [1, 24/149],

"Wizard",

[12, -2, 20, 52, 2, -31], [[2, 1, 5, 2], [0, 6, -1, 10]],

[1/2, 23/72], [1/2, 99/310], [1/2, 23/72],

"Duodecimal",

[0, 12, 24, 22, -38, 19], [[12, 19, 0, -22], [0, 0, 1, 2]],

[1/12, 3/228], [1/12, 3/228], [1/12, 3/228],

"Slender",

[13, -10, 6, 42, 27, -46], [[1, 2, 2, 3], [0, 13, -10, 6]],

[1, 5/156], [1, 4/125], [1, 4/125],

"Amity",

[5, 13, -17, -76, 41, 9], [[1, 3, 6, -2], [0, 5, 13, -17]],

[1, 155/548], [1, 99/350], [1, 99/350],

"Hemififths",

[2, 25, 13, -40, -15, 35], [[1, 1, -5, -1], [0, 2, 25, 13]],

[1, 70/239], [1, 70/239], [1, 70/239],

"Ennealimmal",

[18, 27, 18, -34, 22, 1], [[9, 1, 1, 12], [0, 2, 3, 2]],

[1/9, 43/612]], [1/9, 227/3231], [1/9, 43/612]]:

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> Since 9-odd-limit and 7-odd-limit involve the same set of primes,

we can justify using either. If we regard them both as important, we

can justify using anything between the ranges of the poptimal

generators even when these are disjoint. To be sure, when 3 has high

complexity we might be interested in 7 but not 9, but I think this is

pretty much of a quibble. I calculated a 9-limit poptimal, (the

second listed below) and then found the best which can be had by

putting 7 and 9 together. The "univeral" generators listed below are

where all the p values give the same result; since this didn't happen

for both 7 and 9 I still got an et.

>

> At this point I would say things are generally looking good. We now

have 72 for Miracle, 84 for Orwell, 12 for Diminished and Dominant

Seventh, 22 for Porcupine, 31 for Semififths and Supermajor Seconds,

> 41 for Superkleismic and 46 for Semisixths. The 112-et for Meantone

is still a little over the top, but is at least worth considering;

the tendency for Meantone to want a generator in this very small

interval is rather striking.

>

> ...

> "Kleismic",

> [6, 5, 3, -7, 12, -6], [[1, 0, 1, 2], [0, 6, 5, 3]],

>

> [1, 14/53], [1, 14/53], [1, 14/53],

I'm a little puzzled how 53-ET popped up (instead of 72-ET) for this

one. In this regard, please refer to:

There I evaluated by minimax only, but I thought that it seemed

pretty clear that 53 was not our choice to notate kleismic. (I

haven't waded through all of the details of your explanation, because

I'm just interested in the results, and this one puzzles me.)

--George

--- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:

> I'm a little puzzled how 53-ET popped up (instead of 72-ET) for this

> one. In this regard, please refer to:

>

> /tuning-math/message/5253

>

> There I evaluated by minimax only, but I thought that it seemed

> pretty clear that 53 was not our choice to notate kleismic.

In a nutshell, 53 is smaller than 72; it is very close to the minimax value while 72 is very close to the rms value, but either will work.

We have:

Least squares 316.664

Least cubes 316.466

Least fourth powers 316.566

Minimax 316.993

Both the 53-et value of 316.981 cents and the 72-et value of 316.667 cents are between the rms and minimax optimal generators.

How differently would the notation look in 53 as opposed to 72?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>"

<gdsecor@y...> wrote:

>

> > I'm a little puzzled how 53-ET popped up (instead of 72-ET) for

this

> > one. In this regard, please refer to:

> >

> > /tuning-math/message/5253

> >

> > There I evaluated by minimax only, but I thought that it seemed

> > pretty clear that 53 was not our choice to notate kleismic.

>

> In a nutshell, 53 is smaller than 72; it is very close to the

minimax value while 72 is very close to the rms value, but either

will work.

>

> We have:

>

> Least squares 316.664

> Least cubes 316.466

> Least fourth powers 316.566

> Minimax 316.993

>

>

> Both the 53-et value of 316.981 cents and the 72-et value of

316.667 cents are between the rms and minimax optimal generators.

>

> How differently would the notation look in 53 as opposed to 72?

To illustrate, I will repeat something from my earlier posting

(#5253):

<< Something else I noticed about the choice of 72 as the notation

for both the Miracle and kleismic temperaments: the progression of

sagittal symbols for a 72-ET panchromatic scale (one passing through

all the tones) is the same as that for a sequence of tones differing

by the generating interval in both temperaments. To illustrate:

72-ET: C C\! C!) C\!/ B|) B/| B

Miracle: C B\! Bb!) A\!/ G|) F#/| F

kleismic: C A\! F#!) Eb\!/ B|) G#/| F

The pattern then repeats. I believe that this is a useful property

that provides a further justification for basing the kleismic

notation on 72. >>

In 53-ET sagittal notation the last line of the above table would be:

kleismic: C A\! F#\! Eb\!/ B/| G# F

or Ab/|

The Ab/| respelling in 53 is the same in 72 (since G#=Ab in 72). The

essential differences between the 53 and 72 symbols are:

1) The 7 comma symbol |) occurs only in the 72 standard set; it is

the same number of degrees as the 5 comma /| in 53; and

2) The apotome minus the 11 diesis is the same number of degrees as

the 11 diesis in 72 -- so A\!/ equals G#/|\ -- but in 53 it is

different, so A\!/ -- A lowered by ~32:33, the 11 diesis -- would be

respelled with a different symbol, G#(|) -- G# raised by ~704:729,

the 11' diesis.

--George