Planar temperaments -- 1029/1024

The basic idea of this one is (8/7)^3 ~ 3/2. We could use 2,5,7 as a
basis, but I'll use 2,5/4,8/7:

35/32 -- 5/4 -- 10/7 -- 80/49 -- 15/8
| | | | |
7/4 -- 1 -- 8/7 -- 21/16 -- 3/2
| | | | |
7/4 -- 8/5 --64/35 -- 21/20 -- 6/5

For the 7-limit least squares, the fifth is 1.62 cents flat, the
major third 1.3 cents flat and the 7 2.27 cents flat.

--- In tuning@y..., genewardsmith@j... wrote:
> The basic idea of this one is (8/7)^3 ~ 3/2.

Does anyone know how 1029/1024 got the name "gamelan residue"? It makes me wonder if there may be more to this temperament than I reported.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:
> > The basic idea of this one is (8/7)^3 ~ 3/2.
>
> Does anyone know how 1029/1024 got the name "gamelan residue"? It
>makes me wonder if there may be more to this temperament than I
>reported.

I'll bet that some JI enthusiast noted that Slendro divides the
perfect fifth into three roughly equal parts, decided that each part
sounded like 8/7, and then noticed that (8/7)^3 falls short of 3/2 by
1029/1024.

Since there's no 5 in the ratio, it's sort of a "free dimension" in a
full 7-limit planar temperament based on 1029/1024.

But a _linear temperament_ based on only the primes 3 and 7 (and 2
for octaves) and the 1029/1024 should be quite interesting . . . in
fact I was going to suggest that we additionally investigate not only
this set of primes, but also the BP case of primes 5 and 7 (and _3_
for equivalence) for our linear temperament paper.

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

in
> fact I was going to suggest that we additionally investigate not only
> this set of primes, but also the BP case of primes 5 and 7 (and _3_
> for equivalence) for our linear temperament paper.

I was thinking on similar lines--as well as a 5-limit comma search, which I just completed, I could do a 2,3,7 and 2,5,7 comma search; to that we could add 3,5,7 if you like.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> in
> > fact I was going to suggest that we additionally investigate not
only
> > this set of primes, but also the BP case of primes 5 and 7 (and
_3_
> > for equivalence) for our linear temperament paper.
>
> I was thinking on similar lines--as well as a 5-limit comma search,
>which I just completed, I could do a 2,3,7 and 2,5,7 comma search;
>to that we could add 3,5,7 if you like.

Glad we agree! I do want to use a *weighted* gens measure,
though . . . OK, this belongs on tuning-math.

Gene:
> > Does anyone know how 1029/1024 got the name "gamelan residue"? It
> >makes me wonder if there may be more to this temperament than I
> >reported.

Paul:
> I'll bet that some JI enthusiast noted that Slendro divides the
> perfect fifth into three roughly equal parts, decided that each part
> sounded like 8/7, and then noticed that (8/7)^3 falls short of 3/2 by
> 1029/1024.

I thought last time this came up, Manuel said the name came from Fokker. Another
way of looking at it is that four steps of 8:7 and one of 7:6 almost add up to an
octave.

> But a _linear temperament_ based on only the primes 3 and 7 (and 2
> for octaves) and the 1029/1024 should be quite interesting . . . in
> fact I was going to suggest that we additionally investigate not only
> this set of primes, but also the BP case of primes 5 and 7 (and _3_
> for equivalence) for our linear temperament paper.

And I also thought that Margo had experimented with this temperament. Haven't I
already done that BP search? I've had a look at the 3.7.9 diamond, and these are
the unison vectors for the top 10 temperaments:

[-11, -9, 0, 9] 40353607:40310784
[1, 10, 0, -6] 118098:117649
[-4, -1, 0, 2] 49:48
[-10, 1, 0, 3] 1029:1024
[25, -14, 0, -1] 33554432:33480783
[-11, -9, 0, 9] 40353607:40310784
[-11, -9, 0, 9] 40353607:40310784
[-10, 1, 0, 3] 1029:1024
[-35, 15, 0, 4] 34451725707L:34359738368L
[6, -2, 0, -1] 64:63

There's some repetitition, so some of the temperaments will have torsion. [25,
-14, 0, -1] or 33554432:33480783 is the 2-3-7 schisma. The top one, [-11, -9, 0,
9] or 40353607:40310784 gives two 9-equals 35 cents apart. It's accurate to around
0.2 cents.

The second one has mapping [(2, 0), (4, -3), (7, -5)] and a 166 cent generator, two
otonalities in the 8-note MOS and is accurate to a cent or so. Consistent with 14-
and 22-equal.

The third one is mapping [(1, 0), (2, -2), (3, -1)], consistent with 4, 5 and
9-equal. It's 12 cents out, but you already get 2 otonalities in the 4 note MOS.

Graham

--- In tuning@y..., graham@m... wrote:

> > But a _linear temperament_ based on only the primes 3 and 7 (and
2
> > for octaves) and the 1029/1024 should be quite interesting . . .
in
> > fact I was going to suggest that we additionally investigate not
only
> > this set of primes, but also the BP case of primes 5 and 7 (and
_3_
> > for equivalence) for our linear temperament paper.
>
> And I also thought that Margo had experimented with this
temperament.

In a JI form, yes (chain of 8:7s)

>I've had a look at the 3.7.9 diamond, and these are
> the unison vectors for the top 10 temperaments:
>
> [-11, -9, 0, 9] 40353607:40310784
> [1, 10, 0, -6] 118098:117649
> [-4, -1, 0, 2] 49:48
> [-10, 1, 0, 3] 1029:1024
> [25, -14, 0, -1] 33554432:33480783
> [-11, -9, 0, 9] 40353607:40310784
> [-11, -9, 0, 9] 40353607:40310784
> [-10, 1, 0, 3] 1029:1024
> [-35, 15, 0, 4] 34451725707L:34359738368L
> [6, -2, 0, -1] 64:63
>
> There's some repetitition, so some of the temperaments will have
torsion. [25,
> -14, 0, -1] or 33554432:33480783 is the 2-3-7 schisma. The top
one, [-11, -9, 0,
> 9] or 40353607:40310784 gives two 9-equals 35 cents apart.

That's the one I suggested for Margo on MakeMicroMusic, isn't it?

I am supplimenting my collection of 72-et scales based on 225/224~1 with ones starting from the "gamelan residue", 1029/1024~1. This has no five in it, and is a 36-et comma also, so it is not surprising I found some 72-et scales which were also 36-et scales. One nice one is the following pentatonic:

[0 14 28 42 56]
[14 14 14 14 16]

This has seven intervals and a connectivity of two in the 7-limit, and is a version of something we discussed in connection with my question about the "gamelan residue". It is four tempered 8/7's and a tempered 7/6; the above scale in this mode can be approximated by
1--8/7--21/16--3/2--12/7, but all the modes are of equal interest. While it could be treated as a sort of slendro, it does have some very precisely tuned intervals, and slendro is hardly the only way to treat it.

Here's a bit of the previous conversation:

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

> I'll bet that some JI enthusiast noted that Slendro divides the
> perfect fifth into three roughly equal parts, decided that each part
> sounded like 8/7, and then noticed that (8/7)^3 falls short of 3/2 by
> 1029/1024.
>
> Since there's no 5 in the ratio, it's sort of a "free dimension" in a
> full 7-limit planar temperament based on 1029/1024.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> I am supplimenting my collection of 72-et scales based on 225/224~1
with ones starting from the "gamelan residue", 1029/1024~1. This has
no five in it, and is a 36-et comma also, so it is not surprising I
found some 72-et scales which were also 36-et scales. One nice one is
the following pentatonic:
>
> [0 14 28 42 56]
> [14 14 14 14 16]
>
> This has seven intervals and a connectivity of two in the 7-limit,
and is a version of something we discussed in connection with my
question about the "gamelan residue". It is four tempered 8/7's and a
tempered 7/6; the above scale in this mode can be approximated by
> 1--8/7--21/16--3/2--12/7, but all the modes are of equal interest.
While it could be treated as a sort of slendro, it does have some
very precisely tuned intervals, and slendro is hardly the only way to
treat it.

Hi Gene . . . nice that you're rediscovered this. This Slendro is
every other note of the Decimal scale, and thus quite prevalent in
Blackjack. It was discussed quite a bit on this list in the year 2001.

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

> Hi Gene . . . nice that you're rediscovered this. This Slendro is
> every other note of the Decimal scale, and thus quite prevalent in
> Blackjack. It was discussed quite a bit on this list in the year 2001.

Decimal came up in the gamelan residue search also, by the way.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Hi Gene . . . nice that you're rediscovered this. This Slendro is
> > every other note of the Decimal scale, and thus quite prevalent
in
> > Blackjack. It was discussed quite a bit on this list in the year
2001.
>
> Decimal came up in the gamelan residue search also, by the way.

Tuning-math seems to be down now. Why not do a search that takes all
the commas of 72-tET into account?

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

> Tuning-math seems to be down now. Why not do a search that takes all
> the commas of 72-tET into account?

There are an infinite number of them is one reason not to. I could go on to 2401/2400 and 4375/4374, and also to the 11-limit; I've been thinking of both. Then there is the question of two-step-size scales, three-step-size scales or even four-step-size to consider. I'm beating up on the 72-et partly because there is preeixsting interest, and partly because I want to find out the best way of doing this sort of thing, and what we might run across, and concentrating on one or a few ets seems like a good way to do that. I don't know if there is much interest in the 46-et, or what else people might like to look at.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Tuning-math seems to be down now. Why not do a search that takes
all
> > the commas of 72-tET into account?
>
> There are an infinite number of them is one reason not to.

Well, the whole thing is just a hyper-torus (in the 7-limit, anyway).
Can't you apply your graph theory to that?

>I could go on to 2401/2400 and 4375/4374,

See above.

>and also to the 11-limit

A higher-dimensional, but still finite, structure.

>I've been thinking of both. Then there is the question of two-step-
>size scales, three-step-size scales or even four-step-size to
>consider. I'm beating up on the 72-et partly because there is
>preeixsting interest, and partly because I want to find out the best
>way of doing this sort of thing, and what we might run across, and
>concentrating on one or a few ets seems like a good way to do that.
>I don't know if there is much interest in the 46-et, or what else
>people might like to look at.

22, 31.

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

/tuning/topicId_30085.html#32339

> I am supplimenting my collection of 72-et scales based on 225/224~1
with ones starting from the "gamelan residue", 1029/1024~1.

Thanks so much for this addition, Gene. (or however you did it,
probably involved more than "addition...")

My thought is that once all these new 72-tET materials are assembled
they should be forwarded to the 72-tET enthusiasts in Boston, as well
as, possibly, being published in Xenharmonicon or some such. Very
useful...

best,

JP

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

/tuning/topicId_30085.html#32349

> Hi Gene . . . nice that you're rediscovered this. This Slendro is
> every other note of the Decimal scale, and thus quite prevalent in
> Blackjack. It was discussed quite a bit on this list in the year
2001.

Maybe Gene would be interested in the post that Dave Keenan made that
showed all the "traditional" scales associated with Blackjack...

Unfortunatly, I don't have the post number at the moment...

JP

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

/tuning/topicId_30085.html#32369

> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Tuning-math seems to be down now. Why not do a search that takes
all
> > the commas of 72-tET into account?
>
> There are an infinite number of them is one reason not to. I could
go on to 2401/2400 and 4375/4374, and also to the 11-limit; I've been
thinking of both. Then there is the question of two-step-size scales,
three-step-size scales or even four-step-size to consider. I'm
beating up on the 72-et partly because there is preeixsting interest,
and partly because I want to find out the best way of doing this sort
of thing, and what we might run across, and concentrating on one or a
few ets seems like a good way to do that. I don't know if there is
much interest in the 46-et, or what else people might like to look at.

Please keep at the 72, Gene! The more we know, the merrier...

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> /tuning/topicId_30085.html#32339
>
>
> > I am supplimenting my collection of 72-et scales based on
225/224~1
> with ones starting from the "gamelan residue", 1029/1024~1.
>
>
> Thanks so much for this addition, Gene. (or however you did it,
> probably involved more than "addition...")

Joseph, you've already played with this scale, right? Remember,
"every fourth note"?

> My thought is that once all these new 72-tET materials are
assembled
> they should be forwarded to the 72-tET enthusiasts in Boston,

They'll toss it. They don't give two $#!&@%$ about ratios, or
anything like that. Julia Werntz composes at a 12-tET piano, and
"imagines" the "inflections".

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

/tuning/topicId_30085.html#32403

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> >
> > /tuning/topicId_30085.html#32339
> >
> >
> > > I am supplimenting my collection of 72-et scales based on
> 225/224~1
> > with ones starting from the "gamelan residue", 1029/1024~1.
> >
> >
> > Thanks so much for this addition, Gene. (or however you did it,
> > probably involved more than "addition...")
>
> Joseph, you've already played with this scale, right? Remember,
> "every fourth note"?
>

Thank, Paul, for reminding me... Yes, I remember these scales now and
forgot to delete that message...

JP

I wrote,

> Julia Werntz composes at a 12-tET piano, and
> "imagines" the "inflections".

My apologies to all and to Julie especially. Julie does not compose
at a piano, she sings.