Yahoo Tuning Groups Ultimate Backup tuning Another 72-et family

Hi Gene,

It would be good to include some headings to tell us what the numbers
mean.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> I looked at five 7s, three 9s and two fives, and found scales which
are pretty impressive; these being the best of breed. Fifty-eight
11-limit triads!
>
> [0, 7, 14, 21, 30, 37, 42, 51, 56, 63] [7, 7, 7, 9, 7, 5, 9, 5, 7,
9]
>
> 3 6 0 0
> 5 11 2 1
> 7 23 14 3
> 9 27 26 4
> 11 36 58 6
>
>
> [0, 9, 14, 23, 30, 35, 42, 49, 56, 65] [9, 5, 9, 7, 5, 7, 7, 7, 9,
7]
>
> 3 5 0 0
> 5 10 3 0
> 7 21 12 2
> 9 26 24 3
> 11 36 58 6

Do these have a reasonably square boundary in the Byzantine planar
temperament (where 224:225 vanishes)?

🔗genewardsmith <genewardsmith@juno.com>

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Hi Gene,
>
> It would be good to include some headings to tell us what the numbers
> mean.

The odd numbers from 3 to 11 are followed by number of intervals for that limit, number of triads for that limit, and connectivity for that limit.

> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > I looked at five 7s, three 9s and two fives, and found scales which
> are pretty impressive; these being the best of breed. Fifty-eight
> 11-limit triads!
> >
> > [0, 7, 14, 21, 30, 37, 42, 51, 56, 63] [7, 7, 7, 9, 7, 5, 9, 5, 7,
> 9]
> >
> > 3 6 0 0
> > 5 11 2 1
> > 7 23 14 3
> > 9 27 26 4
> > 11 36 58 6
> >
> >
> > [0, 9, 14, 23, 30, 35, 42, 49, 56, 65] [9, 5, 9, 7, 5, 7, 7, 7, 9,
> 7]
> >
> > 3 5 0 0
> > 5 10 3 0
> > 7 21 12 2
> > 9 26 24 3
> > 11 36 58 6

> Do these have a reasonably square boundary in the Byzantine planar
> temperament (where 224:225 vanishes)?

I was thinking of projecting an 11-limit detempering back to the five-limit via the 11-limit JI to 5-limit JI projection defined by 225/224 and 385/384 (which is an 11-limit planar temperament extending Byzantine) and seeing what it looked like.

🔗David C Keenan <d.keenan@uq.net.au>

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > I looked at five 7s, three 9s and two fives, and found scales
which
> are pretty impressive; these being the best of breed. Fifty-eight
> 11-limit triads!
> >
> > [0, 7, 14, 21, 30, 37, 42, 51, 56, 63] [7, 7, 7, 9, 7, 5, 9, 5,
7,
> 9]
> >
> > 3 6 0 0
> > 5 11 2 1
> > 7 23 14 3
> > 9 27 26 4
> > 11 36 58 6
> >
> >
> > [0, 9, 14, 23, 30, 35, 42, 49, 56, 65] [9, 5, 9, 7, 5, 7, 7, 7,
9,
> 7]
> >
> > 3 5 0 0
> > 5 10 3 0
> > 7 21 12 2
> > 9 26 24 3
> > 11 36 58 6
>
> Do these have a reasonably square boundary in the Byzantine planar
> temperament (where 224:225 vanishes)?

Gidday Gene,

By that, I meant: Are they compact on this toroidal rectangular lattice,
which shows the "Byzantine" planar temperamant in 72-ET (whether considered
as 7-limit or 11-limit). The numbers are degrees of 72-ET, fifths are
horizontal, secors are vertical.

57 27 69 39 09 51 21 63 33 03 45 15 57
50 20 62 32 02 44 14 56 26 68 38 08 50
43 13 55 25 67 37 07 49 19 61 31 01 43
36 06 48 18 60 30 00 42 12 54 24 66 36
29 71 41 11 53 23 65 35 05 47 17 59 29
22 64 34 04 46 16 58 28 70 40 10 52 22
15 57 27 69 39 09 51 21 63 33 03 45 15

I ask because I'm not sure they deserve to be considered "another family".

So here are those scales on this lattice:

51 21 63
.. 14 56
37 07 ..
30 00 42

09 .. .. ..
.. .. 14 56
.. .. .. 49
.. 30 00 42
.. 23 65 35

So for the first scale the answer is "Yes, it's very 'Byzantine'". But for
the second, "no", and in fact the location of the "09" leads one to suspect
there has been some mistake.

I thought maybe the second one would make more sense on a neutral-thirds *
secors lattice.

.. .. .. .. .. 49 ..
09 30 .. 00 .. 42 ..
.. 23 .. 65 14 35 56
.. .. .. .. .. ..(49)

But it doesn't really.

There's little doubt that it would be improved harmonically by changing the
09 to an 07.

Byzantine (fifths * secors)
.. 14 56
.. 07 49
30 00 42
23 65 35

neutral-thirds * secors
.. .. 07 .. 49 ..
30 .. 00 .. 42 ..
23 .. 65 14 35 56
.. .. ..(07)..(49)

Rami Vitale's 23 note Byzantine superset scale, when microtempered into
72-ET, consists of the following 19 notes on the Byzantine lattice.

.. .. .. 26 68
37 07 49 19 61
30 00 42 12 54
23 65 35 05 47
16 58 .. .. ..

Which was of course the reason for calling this fifths * secors lattice
"Byzantine" in the first place.

I think this 19 note scale is very likely a "hyper-MOS", with "trihills"
property and is also probably a periodicity block. Gene, I'm still hoping
you will be able to work out (and explain) the mathematics of these
higher-dimension analogs of MOS, on tuning-math. i.e. How does one find
those boundaries within a planar temperament that give scales with these
desirable melodic properties (analogous to properties we are familiar with
for linear temperaments), other than by trial and error?
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗genewardsmith <genewardsmith@juno.com>

--- In tuning@y..., David C Keenan <d.keenan@u...> wrote:

> There's little doubt that it would be improved harmonically by changing the
> 09 to an 07.

That's a heck of a good idea, since seven 7s, two 9s and a 5 are about as close to the 10-note Miracle MOS as one can get without *being* the MOS, so I'm going to look at all of these Keenan-scales.

Gene, I'm still hoping
> you will be able to work out (and explain) the mathematics of these
> higher-dimension analogs of MOS, on tuning-math. i.e. How does one find
> those boundaries within a planar temperament that give scales with these
> desirable melodic properties (analogous to properties we are familiar with
> for linear temperaments), other than by trial and error?

I thought it was pretty clear I was struggling with this. One could simply step through partitions, and then permute them, but there are too many candidates to make this computationally feasible, so I've tried a few systems for getting good partitions of 72 to start out with with some success, but there is obviously much to be done before the problem can be regarded as satisfactorily understood, at least by me. (If anyone else has a clue, please clue us in!)

By the way, do they understand "throw another shrimp on the barbie" jokes in Oz?

🔗dkeenanuqnetau <d.keenan@uq.net.au>

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., David C Keenan <d.keenan@u...> wrote:
>
> > There's little doubt that it would be improved harmonically by
changing the
> > 09 to an 07.
>
> That's a heck of a good idea, since seven 7s, two 9s and a 5 are
about as close to the 10-note Miracle MOS as one can get without
*being* the MOS, so I'm going to look at all of these Keenan-scales.
>
> Gene, I'm still hoping
> > you will be able to work out (and explain) the mathematics of
these
> > higher-dimension analogs of MOS, on tuning-math. i.e. How does one
find
> > those boundaries within a planar temperament that give scales with
these
> > desirable melodic properties (analogous to properties we are
familiar with
> > for linear temperaments), other than by trial and error?
>
> I thought it was pretty clear I was struggling with this.
...
> but there is obviously much to be done before
the problem can be regarded as satisfactorily understood, at least by
me. (If anyone else has a clue, please clue us in!)
>

Good onya. Sorry to hassle you.

> By the way, do they understand "throw another shrimp on the barbie"
jokes in Oz?

I don't think I've ever heard any _jokes_ involving it, but When Paul
Hogan (as Crocodile Dundee) said it in the movie, we thought it was
pretty dopey because Australians call them prawns, not shrimps.

And we're certainly sick of the fact that crocodile-wrestling morons,
and a sycophantic prime minister, are the main things Americans think
of when someone mentions Australia. But that's not your fault.

🔗genewardsmith <genewardsmith@juno.com>

In fact, it isn't that simple--the four best Keenan scales are good scales, but don't knock the other family of scales out of the ring by any means, not managing to get up to 58 triads.

Dave's scale

[0, 7, 14, 21, 30, 37, 44, 51, 60, 67]

3 6 0 0
5 13 4 1
7 25 18 3
9 29 31 4
11 35 53 6

[0, 7, 14, 21, 30, 37, 44, 49, 56, 63]

3 6 0 0
5 15 6 2
7 27 22 5
9 31 36 5
11 35 52 6

[0, 7, 14, 21, 30, 37, 46, 53, 60, 67]

3 4 0 0
5 11 4 0
7 21 12 2
9 27 26 3
11 35 52 6

[0, 7, 14, 21, 30, 37, 44, 53, 60, 67]

3 5 0 0
5 14 5 1
7 25 18 3
9 30 33 4
11 35 51 6

🔗David C Keenan <d.keenan@uq.net.au>

Here's what Partch's 43 note scale from 'Genesis of a Music' looks like on
the 'Byzantine' lattice (with notes named as 72-ET degrees).

.. .. 69 39 09 51 21 63 33 03 .. .. ..
.. .. 62 .. .. 44 14 56 26 .. .. .. ..
.. .. .. 25 67 37 07 49 19 61 31 01 ..
.. .. .. 18 60 30 00 42 12 54 .. .. ..
.. 71 41 11 53 23 65 35 05 47 .. .. ..
.. .. .. .. 46 16 58 28 .. .. 10 .. ..
.. .. ..(69 39 09 51 21 63 33 03).. ..

Notice that it contains Rami Vitale's Byzantine superset scale (at least
when both are tempered), with its 1/1 corresponding to Partch's 4/3 (30deg72).

Here's Partch's earlier(?) 43 from 'Exposition on Monophony'. Four notes
changed. (1, 71, 31, 41) <--> (2, 70, 32, 40).

.. .. 69 39 09 51 21 63 33 03 .. .. ..
.. .. 62 32 02 44 14 56 26 .. .. .. ..
.. .. .. 25 67 37 07 49 19 61 .. .. ..
.. .. .. 18 60 30 00 42 12 54 .. .. ..
.. .. .. 11 53 23 65 35 05 47 .. .. ..
.. .. .. .. 46 16 58 28 70 40 10 .. ..
.. .. ..(69 39 09 51 21 63 33 03).. ..

And here's the full lattice for ease of reference.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗emotionaljourney22 <paul@stretch-music.com>

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

> Gene, I'm still hoping
> > you will be able to work out (and explain) the mathematics of
these
> > higher-dimension analogs of MOS, on tuning-math. i.e. How does
one find
> > those boundaries within a planar temperament that give scales
with these
> > desirable melodic properties (analogous to properties we are
familiar with
> > for linear temperaments), other than by trial and error?
>
> I thought it was pretty clear I was struggling with this. One could
>simply step through partitions, and then permute them, but there are
>too many candidates to make this computationally feasible, so I've
>tried a few systems for getting good partitions of 72 to start out
>with with some success, but there is obviously much to be done
>before the problem can be regarded as satisfactorily understood, at
>least by me. (If anyone else has a clue, please clue us in!)

to dave's question above, doesn't the Hypothesis provide at least one
answer? that is, generate the fokker periodicity block, choosing
unison vectors corresponding to those that vanish in the planar
temperament, and completing the list with two that don't . . .

🔗jonszanto <JSZANTO@ADNC.COM>

🔗genewardsmith <genewardsmith@juno.com>

🔗dkeenanuqnetau <d.keenan@uq.net.au>

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:
> Hello Dave,
>
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > And we're certainly sick of the fact that crocodile-wrestling
> > morons, and a sycophantic prime minister, are the main things
> > Americans think of when someone mentions Australia. But that's not
> > your fault.
>
> Hmm. Not in this order, but when I think Australia I think Uluru
(Ayers Rock), the Sydney Opera House, and vast, vast areas of land
without condominiums.
>

That's wonderful Jon. But I've got some bad news for you and Kraig.
I think Rupert Murdoch took out US citizenship some time ago. :-)

Sorry about this OT stuff folks.

🔗emotionaljourney22 <paul@stretch-music.com>

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > to dave's question above, doesn't the Hypothesis provide at least
one
> > answer? that is, generate the fokker periodicity block, choosing
> > unison vectors corresponding to those that vanish in the planar
> > temperament, and completing the list with two that don't . . .
>
> When I said I've tried a few systems, this is the sort of thing I
>meant; in fact, this is exactly what I did in the most recent round
>of planar temperament examples. However, it *is* only a starting
>point, and I see no guarantee we get all the interesting cases in
>this way.

right -- for example, a hexagonal boundary instead of a quadrilateral
boundary is one alternative.