Tuning-Math messages 550 - 574

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Message: 550

Date: Tue, 24 Jul 2001 20:11:34

Subject: Re: Hey Carl

From: Paul Erlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
> 
> If you look at the old BP posts I linked yesterday and the recent
> comma-chroma posts, I'm using a generalization that would find the
> simplest 1 or 2D "generators" as identities that fall within a given
> range (i.e., 2:3 for the 1D 7-tone Pythagorean or 4:5:6 for the 2D
> syntonic diatonic).
> 
> This is of course one way to generate periodicity blocks that'll
> accomplish what you've outlined here to two dimensions -- only it 
does
> it by starting with some arbitrary MOS index (i.e., x small steps 
and
> y large steps) and not a set of UVs.
> 
Interesting . . . so what would be the 2D "generator" for Dave 
Keenan's 31-tone planar microtemperament?


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Message: 551

Date: Tue, 24 Jul 2001 20:36:29

Subject: Re: Hey Carl

From: carl@l...

> Hey Carl, just wanted to let you know that my "Hypothesis" is 
> dedicated to you.

Neato!  It's quite an honor to have such a cool hypothesis
dedicated to you.

> You asked what a 2D and higher-dimensional generalization of an
> MOS might be. There was a lot of talk on the Tuning list about
> trivalent scales (scales where each generic interval has exactly
> three specific step sizes) for a while but those seem too rarefied
> to be the "right" answer.

Yeah, "trihill" never seemed right.

> Keenan's 31-tone 11-limit planar microtemperament, where 2 of the 4 
> unison vectors are tempered out.

Is that the pre-Canasta one (Canasta having three unison vectors
tempered out... 'zthat right?  And 31-tet all four?)

-Carl


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Message: 553

Date: Tue, 24 Jul 2001 21:51:43

Subject: Re: Hey Carl

From: Paul Erlich

--- In tuning-math@y..., carl@l... wrote:
> 
> > Keenan's 31-tone 11-limit planar microtemperament, where 2 of the 
4 
> > unison vectors are tempered out.
> 
> Is that the pre-Canasta one (Canasta having three unison vectors
> tempered out... 'zthat right?  And 31-tet all four?)
> 
> -Carl

You got it!


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Message: 554

Date: Tue, 24 Jul 2001 20:54:45

Subject: Re: BP linear temperament

From: David C Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Perhaps my view was too severe, but it definitely seems to contradict 
> the _spirit_ of the near-just approximations to all simple ratios of 
> odd numbers, since the approximations to simple ratios involving even 
> numbers are so poor.

Oh no they're not. For sure they are bad in 13-ED3, but the 7-limit
(1,2,4,5,7) optimum version of the Bohlen-Pierce-Stearns temperament (1:3
period, 442.60 cent generator) has no error greater than 8.4 cents. The
octave itself is only 3.8c narrow.

A chain of 15 generators (16 notes) is required for a complete 7-limit
pentad (e.g. 3:4:5:6:7 or 4:5:6:7:9).

Remember that we have tritave equivalence instead of octave equivalence. So
for example 1:5, 3:5, 5:9 are all "tritave equivalent" so they all have the
same error magnitude in this temperament, and they do not have the same
error as 4:5, 5:6, or 9:10.

This 7-limit MA (max-absolute) optimum generator is a 7:9 widened by 2/15
of the following "BPS-comma".

1647086   2^1 * 7^7
------- = ------- ~= 56.37 c
1594323   3^13

And in fact the following 10-limit ratios come in under 8.4 cents error as
well. 1:10 (9:10), 7:8, 7:10. But not 1:8 (8:9) or 5:8, (11.3c and 12.1c
errors).

The 10-limit (1,2,4,5,7,8,10) MA optimum generator is 442.77 cents with max
error of 8.9c. This is 3/22 BPS-comma.

So we get denominators of convergents (and semiconvergents) for this
temperament being:
4 (5) (9) 13 (17) 30 43 (73) 116
These can be read as MOS cardinalities and the larger ones as possibly
useful ED3s.

--------
Keyboard
--------
I've attached a .gif) of the keyboard mapping (showing cents) for 30 notes
of this temperament, which also makes it clear that the "natural" notation
for it would have only 4 nominals to the tritave.
To see a 3:4:5:6:7 pattern, look at notes marked 0 491 885 1196 1459 (cents).

------
Guitar
------
This 30-note per tritave proper MOS will work incredibly well on a guitar!
Simply tune adjacent open strings one generator (a supermajor third) apart.
The smallest step is 48 cents. This is no worse than a 24-EDO guitar, but
notice that it has 19 steps per octave. 

4:5:6:7:9 barre chords should be playable.

Here's the optimum rotation of the scale for fretting such a guitar. The
zero of the rotation shown on the keyboard map, corresponds to the 16th
fret (1017 cents) below. The keyboard scale is then playable everywhere on
the top string, and as far from the nut as possible on the other strings.

Fret   Step
posn   size  (to the nearest cent) 
-----------
0	48
48	84
132	48
179	84
263	48
311	84
395	48
443	84
526	48
574	48
622	84
706	48
754	84
837	48
885	84
969	48
1017	48
1065	84
1148	48
1196	84
1280	48
1328	84
1411	48
1459	48
1507	84
1591	48
1639	84
1722	48
1770	84
1854	48
1902	

Good work Dan!


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Message: 555

Date: Wed, 25 Jul 2001 22:14:20

Subject: Re: Hey Carl

From: Dave Keenan

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Paul,
> 
> What is Dave Keenan's 31-tone planar microtemperament exactly... I
> either missed it or have forgotten it -- any links?
> 
> --Dan Stearns

It's keenan5.scl in the Scala archive.


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Message: 556

Date: Wed, 25 Jul 2001 22:32:46

Subject: Re: Hey Carl

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Hi Paul,
> > 
> > What is Dave Keenan's 31-tone planar microtemperament exactly... I
> > either missed it or have forgotten it -- any links?
> > 
> > --Dan Stearns
> 
> It's keenan5.scl in the Scala archive.

Dave, where's the tuning list post where you describe it, lattice it, 
and tell us which unison vectors are tempered out and which aren't? I 
was just looking at it a few days ago, but now I can't find it.


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Message: 557

Date: Wed, 25 Jul 2001 22:57:36

Subject: Re: BP linear temperament

From: Paul Erlich

--- In tuning-math@y..., David C Keenan <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > Perhaps my view was too severe, but it definitely seems to 
contradict 
> > the _spirit_ of the near-just approximations to all simple ratios 
of 
> > odd numbers, since the approximations to simple ratios involving 
even 
> > numbers are so poor.
> 
> Oh no they're not.

I meant those producable by a small number of generators. In any 
case, I think it's worth pursuing this from _both_ angles.

How about the triple-BP scale I discovered 
(The Bohlen-Pierce Site: BP Scale Structures *)? Is there a 
particular linear temperament that this cries out for?


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Message: 558

Date: Thu, 26 Jul 2001 00:15:11

Subject: Re: Hey Carl

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> Dave, where's the tuning list post where you describe it, lattice 
it, 
> and tell us which unison vectors are tempered out and which aren't? 
I 
> was just looking at it a few days ago, but now I can't find it.

Yahoo groups: /tuning/message/7202 *
Yahoo groups: /tuning/message/7279 *
Yahoo groups: /tuning/message/7341 *

I don't say anything about commas _not_ tempered out. Those tempered 
out are 224:225 and 384:385.

-- Dave Keenan


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Message: 559

Date: Thu, 26 Jul 2001 00:41:18

Subject: Re: Hey Carl

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > Dave, where's the tuning list post where you describe it, lattice 
> it, 
> > and tell us which unison vectors are tempered out and which 
aren't? 
> I 
> > was just looking at it a few days ago, but now I can't find it.
> 
> Yahoo groups: /tuning/message/7202 *
> Yahoo groups: /tuning/message/7279 *
> Yahoo groups: /tuning/message/7341 *
> 
> I don't say anything about commas _not_ tempered out. Those 
tempered 
> out are 224:225 and 384:385.

I'm almost positive it was a different message I was looking at. 
Dang . . . should have bookmarked it. Anyway, the first two above may 
just confuse people.


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Message: 560

Date: Thu, 26 Jul 2001 00:53:33

Subject: Re: BP linear temperament

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., David C Keenan <D.KEENAN@U...> wrote:
> > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > > Perhaps my view was too severe, but it definitely seems to 
> contradict 
> > > the _spirit_ of the near-just approximations to all simple 
ratios 
> of 
> > > odd numbers, since the approximations to simple ratios involving 
> even 
> > > numbers are so poor.
> > 
> > Oh no they're not.
> 
> I meant those producable by a small number of generators.

I think 7 is a small number. That's how many it needs for the octave 
(and hence the fifth) with 3.8c errors.

Prime  No. Generators
-----  --------------
2       7
3       0
5       2
7      -1

> In any 
> case, I think it's worth pursuing this from _both_ angles.

Sure. If 2s are omitted, the MA optimum generator is 439.82c and the 
max error is 4.8c.

> How about the triple-BP scale I discovered 
> (The Bohlen-Pierce Site: BP Scale Structures *)? Is there a 
> particular linear temperament that this cries out for?

Probably, but I don't have time. A suggested approach: Give the 
simplest frequency ratio you can for each scale degree. Then list the 
step sizes between these, classify them into two sizes, L and s, then 
figure out what the generator is from that.
-- Dave Keenan


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Message: 561

Date: Thu, 26 Jul 2001 01:32:46

Subject: Re: BP linear temperament

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > 
> > I meant those producable by a small number of generators.
> 
> I think 7 is a small number. That's how many it needs for the 
octave 
> (and hence the fifth) with 3.8c errors.
> 
> Prime  No. Generators
> -----  --------------
> 2       7
> 3       0
> 5       2
> 7      -1

By these standards I think 7 is a large number! It's much larger in 
magnitude than 0, 2, or -1. And to get some of the simplest ratios 
involving 2 (such as 4:3, 5:4, and 7:4) you need around 14 
generators. Quite a qualitative difference, in my opinion, compared 
with what you need for the basic BP consonances.


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Message: 563

Date: Thu, 26 Jul 2001 21:07:53

Subject: Re: BP linear temperament

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> 
> Probably, but I don't have time. A suggested approach: Give the 
> simplest frequency ratio you can for each scale degree.

Hmm . . . this would be the Kees van Prooijen definition of a 
periodicity block. Have you studied this page? Manuel, can we do this 
in SCALA?

> Then list the 
> step sizes between these, classify them into two sizes, L and s, 
then 
> figure out what the generator is from that.

The Hypothesis in action!


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Message: 564

Date: Fri, 27 Jul 2001 01:20:01

Subject: 152-tET

From: Paul Erlich

Graham, did you come across anything relating to 152-tET in your 
searches? I notice it's near-just, and also contains 76-tET, which 
contains many linear temperaments (meantone, paultone, double-
negative-7-limit-tetradecatonic-thingy . . .)?


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Message: 565

Date: Fri, 27 Jul 2001 04:58:20

Subject: Re: Hey Carl

From: Dave Keenan

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Dave,
> 
> I don't use Scala, could you either give a link to an old post or 
just
> repost the scale in cents?
> 
> thanks,
> 
> --Dan Stearns

Hi Dan,

As it turns out, the scale that appears as keenan5.scl doesn't have 
only 3 step sizes. It's the second scale I give in the same post, (as 
a "near miss") that has only 3 step-sizes. Although I give a lattice 
for it in that post, I didn't give the cents.
Yahoo groups: /tuning/message/7341 *
So here they are:

0.0	        s
30.40953308	L
79.61111475	m
115.8026469	m
151.994179	L
201.1957607	s
231.6052938	m
267.7968259	L
316.9984076	s
347.4079406	m
383.5994728	L
432.8010544	m
468.9925866	s
499.4021197	L
548.6037013	m
584.7952335	s
615.2047665	m
651.3962987	L
700.5978803	s
731.0074134	m
767.1989456	L
816.4005272	m
852.5920594	s
883.0015924	L
932.2031741	m
968.3947062	s
998.8042393	L
1048.005821	m
1084.197353	m
1120.388885	L
1169.590467	s
1200.0	

What we want to know is:

What are the generators for this particular 31 note hyper-MOS of a 
planar temperament?
How you do you find them?
Are they unique?
What is the mapping from generators to primes?
How do you construct this tuning from the generators?
How do you construct other examples of the same planar temperament 
from them?
How do you find linear temperaments that cover them?
How do you make only those examples having exactly 3 step sizes 
(hyper-MOS)?
Of those, how do you make only strictly proper ones?
Is there a smaller strictly-proper 3-step-size scale in this planar 
temperament?
Are there different 3-step-size scales in this planar temperament, 
having 31 notes?

We can answer all the corresponding questions for linear temperaments.

I think I know the answer to some of these questions for this 
particular planar temperament, but not for planar temperaments in 
general.

Any light you can shed will be much appreciated.
-- Dave Keenan


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Message: 566

Date: Fri, 27 Jul 2001 05:43:01

Subject: Re: Hey Carl

From: Dave Keenan

Here's a set of unison vectors for the tuning in the previous post.

 3       5       7      11
--------------------------
 3	 7	 0	 0
 4	-1	 0	 0
 2	 2	-1	 0
-1	 1	 1	 1

It has a determinant of 31, the first two vectors are chromatic (not 
tempered out), the last two are commatic (tempered out).

 3       7
 4      -1

has a determinant of -31 by itself.

Is it possible that although the scale has 3 step sizes and is 
symmetrical, it is not a hyper-MOS?
-- Dave Keenan


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Message: 567

Date: Fri, 27 Jul 2001 09:48 +0

Subject: Re: 152-tET

From: graham@m...

In-Reply-To: <9jqfk1+ifbe@e...>
Paul wrote:

> Graham, did you come across anything relating to 152-tET in your 
> searches? I notice it's near-just, and also contains 76-tET, which 
> contains many linear temperaments (meantone, paultone, double-
> negative-7-limit-tetradecatonic-thingy . . .)?

No.  I was originally only considering ETs with less than 100 notes.  I've 
expanded that now, but it doesn't seem to be 13-limit consistent, and is 
too high to be considered for the other limits.

I did notice 72 came up a lot in the higher limits.  Here's what you get 
by combining the two:

3/28, 16.2 cent generator

basis:
(0.125, 0.013529015588479165)

mapping by period and generator:
([8, 0], ([13, 19, 23, 28, 30], [-3, -4, -5, -3, -4]))

mapping by steps:
[(152, 72), (241, 114), (353, 167), (427, 202), (526, 249), (562, 266)]

unison vectors:
[[-3, -1, 0, -1, 0, 2], [1, 0, 6, -4, 0, -1], [-7, 1, 0, -3, 4, 0], [1, 
10, 0, -
6, 0, 0]]

highest interval width: 6
complexity measure: 48  (56 for smallest MOS)
highest error: 0.004556  (5.467 cents)

11-limit:

highest interval width: 6
complexity measure: 48  (56 for smallest MOS)
highest error: 0.001072  (1.286 cents)
unique

7-limit:

highest interval width: 5
complexity measure: 40  (48 for smallest MOS)
highest error: 0.001044  (1.253 cents)
unique


                 Graham


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Message: 570

Date: Fri, 27 Jul 2001 19:58:54

Subject: Re: Hey Carl

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> Here's a set of unison vectors for the tuning in the previous post.
> 
>  3       5       7      11
> --------------------------
>  3	 7	 0	 0
>  4	-1	 0	 0
>  2	 2	-1	 0
> -1	 1	 1	 1

FWIW, the Fokker parallelopiped PB corresponding to these four unison 
vectors is entirely within the 3-5 plane:

      cents         numerator    denominator
            0            1            1
       41.059          128          125
       70.672           25           24
       111.73           16           15
       162.85         1125         1024
       203.91            9            8
       223.46          256          225
       274.58           75           64
       315.64            6            5
       335.19         4096         3375
       386.31            5            4
       427.37           32           25
       478.49          675          512
       498.04            4            3
        539.1          512          375
       590.22           45           32
       609.78           64           45
        660.9          375          256
       701.96            3            2
       721.51         1024          675
       772.63           25           16
       813.69            8            5
       864.81         3375         2048
       884.36            5            3
       925.42          128           75
       976.54          225          128
       996.09           16            9
       1037.1         2048         1125
       1088.3           15            8
       1129.3           48           25
       1158.9          125           64

and contains the following step sizes:

       41.059
       29.614
       41.059
        51.12
       41.059
       19.553
        51.12
       41.059
       19.553
        51.12
       41.059
        51.12
       19.553
       41.059
        51.12
       19.553
        51.12
       41.059
       19.553
        51.12
       41.059
        51.12
       19.553
       41.059
        51.12
       19.553
       41.059
        51.12
       41.059
       29.614
       41.059


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Message: 571

Date: Fri, 27 Jul 2001 20:00:20

Subject: Re: 152-tET

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> 
> I did notice 72 came up a lot in the higher limits.  Here's what 
you get 
> by combining the two:

Combining the two what? How?


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Message: 572

Date: Fri, 27 Jul 2001 20:03:15

Subject: Re: Hey Carl

From: Paul Erlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Dave,
> 
> <<Is it possible that although the scale has 3 step sizes and is
> symmetrical, it is not a hyper-MOS?>>
> 
> Right, that's sort of what I just posted, but then again I'm not 
sure
> of exactly what definition of "hyper-MOS" we're going by!

Perhaps Dave is trying to proceed by analogy from, say, the situation 
where a scale like 2 2 1 2 1 2 2 in 12-tET has 2 step sizes and is 
symmetrical, but is not an MOS?


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