Notating the Carlos Beta scale

I recently posted a Sagittal notation for Wendy Carlos' Alpha scale to the tuning list. It turned out to be pretty easy to find a good notation. I went on to try Beta, but that isn't going as well. Here's what I've got so far (see if anyone has any ideas to fill in the gaps):

A perfect fifth is 11 steps of beta, so I only need to notate 5 pairs of notes between D and A (or C and G, or whatever). The major and minor thirds fall into place. One step of beta is near 28/27, so 4 steps can be notated as 81/70 and 7 steps as 35/27.

That leaves 2, 3, 8, and 9. But there don't seem to be many options. 2 steps is close to 14/13 if you want to include the 13 limit, but there doesn't seem to be a very good way to notate 14/13. 9 steps could be notated as 25/18. 14 steps could be 42/25.

I guess what I'm looking for is a de-tempered Beta scale, something that fits the scale reasonably well with intervals that aren't too complex, and is symmetrical within the 3/2 interval. This does seem like a pretty good match, so maybe I'll just use this.

0: 1/1 0.000 unison, perfect prime
1: 28/27 62.961 Archytas' 1/3-tone
2: 27/25 133.238 large limma, BP small semitone
3: 28/25 196.198 middle second
4: 81/70 252.680 Al-Hwarizmi's lute middle finger
5: 6/5 315.641 minor third
6: 5/4 386.314 major third
7: 35/27 449.275 9/4-tone, septimal semi-diminished fourth
8: 75/56 505.757
9: 25/18 568.717 classic augmented fourth
10: 81/56 638.994
11: 3/2 701.955 perfect fifth

D D|\\ E)!!( E~! E/|) F/| F||\ G\!) G~| G)||( A!// A

Hi Herman,

Without looking at this too closely... I'm sure you know
that beta is very close to 19-ET. It has 1212-cent octaves.
The 7-limit TOP stretch for 19-ET octaves is already 1204
cents. The patent val

< 19 30 44 53 |

works with a 2,3 map of

< 1 0 -4 -13 |
< 0 1 4 10 |

So what's wrong with standard meantone notation?

-Carl

At 07:16 PM 6/10/2008, you wrote:
>I recently posted a Sagittal notation for Wendy Carlos' Alpha scale to
>the tuning list. It turned out to be pretty easy to find a good
>notation. I went on to try Beta, but that isn't going as well. Here's
>what I've got so far (see if anyone has any ideas to fill in the gaps):
>
>A perfect fifth is 11 steps of beta, so I only need to notate 5 pairs of
>notes between D and A (or C and G, or whatever). The major and minor
>thirds fall into place. One step of beta is near 28/27, so 4 steps can
>be notated as 81/70 and 7 steps as 35/27.
>
>That leaves 2, 3, 8, and 9. But there don't seem to be many options. 2
>steps is close to 14/13 if you want to include the 13 limit, but there
>doesn't seem to be a very good way to notate 14/13. 9 steps could be
>notated as 25/18. 14 steps could be 42/25.

Carl Lumma wrote:
> Hi Herman,
> > Without looking at this too closely... I'm sure you know
> that beta is very close to 19-ET. It has 1212-cent octaves.
> The 7-limit TOP stretch for 19-ET octaves is already 1204
> cents. The patent val
> > < 19 30 44 53 |
> > works with a 2,3 map of
> > < 1 0 -4 -13 |
> < 0 1 4 10 |
> > So what's wrong with standard meantone notation?
> > -Carl

Hmm, that could be one way to notate it. It would be like using 12-ET notation for diminished, a practical notation for the sake of simplicity. But I want to see if I can find a notation that fits the specific tuning of Beta and emphasizes the non-purity of the octaves.

I thought it would be useful to look at notation for sycamore temperament, which could be seen as Beta with octaves. 5-limit sycamore has a limited choice of accidentals, so I found a 7-limit version that works out reasonably well.

[<1, 1, 2, 1], <0, 11, 6, 34]>
(-1, +19) ~|( [0, -5, 1, 2>
(-2, +38) /| [-4, 4, -1>
(-3, +57) ~|) [-4, -1, 0, 2>
(+1, -18) /|) [2, 2, -1, -1>
(+0, +1) |\\ [2, -3, 0, 1>
(-1, +20) ~~|| [-7, 8, 0, -2>
(-2, +39) ||\ [-7, 3, 1>
(-4, +77) /||\ [-11, 7>

The ~|( is one of two 7-limit alternatives -- the other is [1, 2, -3, 1> --, but this is the best fit of the two and the flag arithmetic works with it (but not the other). Besides, [1, 2, -3, 1> maps to the same pitch as /| [-4, 4, -1> . So there's not much of a chance anyone would pick ~|( to stand for (-2, +38), with /| being such an obvious choice.

This fits my original suggestion for Beta with the exception of ~|( replacing ~| , and ~~|| in place of )||( .

D D|\\ E~~!! E~!( E/|) F/| F||\ G\!) G~~|| A!// A

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Carl Lumma wrote:
> > Hi Herman,
> >
> > Without looking at this too closely... I'm sure you know
> > that beta is very close to 19-ET. It has 1212-cent octaves.
> > The 7-limit TOP stretch for 19-ET octaves is already 1204
> > cents. The patent val
> >
> > < 19 30 44 53 |
> >
> > works with a 2,3 map of
> >
> > < 1 0 -4 -13 |
> > < 0 1 4 10 |
> >
> > So what's wrong with standard meantone notation?
> >
> > -Carl
>
> Hmm, that could be one way to notate it. It would be like using 12-
ET
> notation for diminished, a practical notation for the sake of
> simplicity. But I want to see if I can find a notation that fits
the
> specific tuning of Beta and emphasizes the non-purity of the
octaves.
>
> I thought it would be useful to look at notation for sycamore
> temperament, which could be seen as Beta with octaves. 5-limit
sycamore
> has a limited choice of accidentals, so I found a 7-limit version
that
> works out reasonably well.
>
> [<1, 1, 2, 1], <0, 11, 6, 34]>
> (-1, +19) ~|( [0, -5, 1, 2>
> (-2, +38) /| [-4, 4, -1>
> (-3, +57) ~|) [-4, -1, 0, 2>
> (+1, -18) /|) [2, 2, -1, -1>
> (+0, +1) |\\ [2, -3, 0, 1>
> (-1, +20) ~~|| [-7, 8, 0, -2>
> (-2, +39) ||\ [-7, 3, 1>
> (-4, +77) /||\ [-11, 7>
>
> The ~|( is one of two 7-limit alternatives -- the other is [1, 2, -
3, 1>
> --, but this is the best fit of the two and the flag arithmetic
works
> with it (but not the other). Besides, [1, 2, -3, 1> maps to the
same
> pitch as /| [-4, 4, -1> . So there's not much of a chance anyone
would
> pick ~|( to stand for (-2, +38), with /| being such an obvious
choice.
>
> This fits my original suggestion for Beta with the exception of ~|(
> replacing ~| , and ~~|| in place of )||( .
>
> D D|\\ E~~!! E~!( E/|) F/| F||\ G\!) G~~|| A!// A

Could you please show how you came up with the generator mapping
[<1, 1, 2, 1], <0, 11, 6, 34]> (which generators?)

Thanks

PGH

Paul G Hjelmstad wrote:

> Could you please show how you came up with the generator mapping
> [<1, 1, 2, 1], <0, 11, 6, 34]> (which generators?)
> > Thanks
> > PGH

I ran the Beta generator (63.8 cents) through a program that finds the closest approximation to the primes. Since the Sagittal notation system repeats at the octaves, I used 1200.0 cents for the other generator (the period). Here's a summary of the results (ignoring anything off by more than 20 cents):

Mappings of 3
(2, -8) : 1889.60c (-12.36c)
(1, 11) : 1901.80c (-0.16c)
Mappings of 5
(2, 6) : 2782.80c (-3.51c)
(7, -88) : 2785.60c (-0.71c)
Mappings of 7
(2, 15) : 3357.00c (-11.83c)
(1, 34) : 3369.20c (+0.37c)
Mappings of 11
(4, -10) : 4162.00c (+10.68c)
(5, -29) : 4149.80c (-1.52c)
Mappings of 13
(3, 13) : 4429.40c (-11.13c)
(2, 32) : 4441.60c (+1.07c)
Mappings of 17
(5, -17) : 4915.40c (+10.44c)
(6, -36) : 4903.20c (-1.76c)
Mappings of 19
(5, -14) : 5106.80c (+9.29c)
(6, -33) : 5094.60c (-2.91c)
Mappings of 23
(5, -9) : 5425.80c (-2.47c)

The mappings of 3 and 5 need to be (1, 11) and (2, 6) to match the description of Beta (6 steps to the major third, 11 steps to the perfect fifth). That leaves (1, 34) as the best mapping of 7. (2, 15) is another option, but it's not as good a match.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Paul G Hjelmstad wrote:
>
> > Could you please show how you came up with the generator mapping
> > [<1, 1, 2, 1], <0, 11, 6, 34]> (which generators?)
> >
> > Thanks
> >
> > PGH
>
> I ran the Beta generator (63.8 cents) through a program that finds
the
> closest approximation to the primes. Since the Sagittal notation
system
> repeats at the octaves, I used 1200.0 cents for the other generator
(the
> period). Here's a summary of the results (ignoring anything off by
more
> than 20 cents):
>
> Mappings of 3
> (2, -8) : 1889.60c (-12.36c)
> (1, 11) : 1901.80c (-0.16c)
> Mappings of 5
> (2, 6) : 2782.80c (-3.51c)
> (7, -88) : 2785.60c (-0.71c)
> Mappings of 7
> (2, 15) : 3357.00c (-11.83c)
> (1, 34) : 3369.20c (+0.37c)
> Mappings of 11
> (4, -10) : 4162.00c (+10.68c)
> (5, -29) : 4149.80c (-1.52c)
> Mappings of 13
> (3, 13) : 4429.40c (-11.13c)
> (2, 32) : 4441.60c (+1.07c)
> Mappings of 17
> (5, -17) : 4915.40c (+10.44c)
> (6, -36) : 4903.20c (-1.76c)
> Mappings of 19
> (5, -14) : 5106.80c (+9.29c)
> (6, -33) : 5094.60c (-2.91c)
> Mappings of 23
> (5, -9) : 5425.80c (-2.47c)
>
> The mappings of 3 and 5 need to be (1, 11) and (2, 6) to match the
> description of Beta (6 steps to the major third, 11 steps to the
perfect
> fifth). That leaves (1, 34) as the best mapping of 7. (2, 15) is
another
> option, but it's not as good a match.

Thanks. I thought that you started with a matrix
1 0 0 0; Gen1; Gen2; Beta Comma, which was inverted to give
[<1, 1, 2, 1], <0, 11, 6, 34]>, the generator mapping,

Obviously, this is more "direct", I take it, you use this
information for your "tuning map" then too.

PGH

Paul G Hjelmstad wrote:

> Thanks. I thought that you started with a matrix
> 1 0 0 0; Gen1; Gen2; Beta Comma, which was inverted to give
> [<1, 1, 2, 1], <0, 11, 6, 34]>, the generator mapping,

That's a good method if you know the comma you want to temper out, but Beta isn't so much defined by its commas. There was a recent thread in the tuning list on Petr Pařízek's planar (rank 3) temperaments, where I used matrix inversion to find a generator mapping, but I didn't go into details since the math is beyond the scope of that list.