Improved generators for 7-limit linear temperaments

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:

/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. (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