Tuning-Math Digests messages 5000 - 5024

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Message: 5000

Date: Tue, 11 Jun 2002 19:48:51

Subject: bye

From: Carl Lumma

I'm singing off of yahoo groups.  Tuning-math and harmonic-entropy
tomorrow, and tuning when anything I'm involved in has died down.
As always, I reserve the right to change my mind at any point.  :)

I'll keep up the tuning-math list at freelists.org for a time, if
anybody's interested in it.  If folks want to switch, I'm happy to
do the grunt work, or anyone else who'd like to do it is welcome
to what I have so far.  The list is very configurable, so there's
all sorts of things to vote on, though the current config should be
at least as good as anything we've had so far.  It's possible to
subscribe many people in one go, such as everybody on tuning-math
here.  Maybe Robert Walker knows how to snag archives.  Freelists'
web interface and archive search seem quite good.

As always, feel free to mail me at carl-lumma.org, where the - is
to be replaced by an @.  In particular, if Gene ever gets interested
in harmonic entropy, or if Paul ever runs the validation exercise,
if the new notation is released, or the search of planar temperaments
turns up any really good 5-10 tone generalized-diatonics... and if
any of you create music; I always love to listen, so drop me a note!

-Carl


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Message: 5001

Date: Tue, 11 Jun 2002 13:48 +0

Subject: Re: Help requested

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <ae4erq+a8rm@xxxxxxx.xxx>
kalleaho wrote:

> What should I read in the Web and in the Lists to get a good 
> understanding of the notation and terminology used in tuning-math? 

As we haven't written up the processes yet, the best place is still the 
list archives.

> I understand what linear temperaments are but the notation used is 
> not self-evident to me. I also have a basic understanding of 
> periodicity blocks but hmm... wedges? commatic/chromatic unison 
> vectors? 

Wedge products are explained at 
<http://mathworld.wolfram.com/wedgeproduct.html *>.  The importance here is 
that the wedge product of the commas defining a linear temperament family 
is the complement of the wedge product of two equal temperaments belonging 
to the same family.

The chromatic unison vector is the one you don't temper out to get a 
linear temperament.  So for 7 note meantone, this is the chromatic 
semitone 25:24.


                      Graham


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Message: 5003

Date: Tue, 11 Jun 2002 19:49:22

Subject: A twelve-note, 11-limit scale

From: genewardsmith

This results from tempering a variety of Fokker blocks using the planar
temperament defined by 126/125~176/175~1. I've used the 120-et for the
results; since I already called the 108-et the crazy uncle of the
family, I don't know where to place 120.

Scale in 120-et
[0, 8, 23, 31, 39, 50, 62, 70, 81, 89, 101, 112]

Interval and triad count
5: 23, 12
7: 36, 36
9: 42, 58
11: 49, 82

Connectivities: 2   5   5   8

Fokker blocks which temper to this scale

1, 21/20, 8/7, 6/5, 5/4, 168/125, 10/7, 3/2, 8/5, 42/25, 25/14, 48/25

1, 21/20, 8/7, 25/21, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3, 25/14, 40/21

1, 25/24, 144/125, 6/5, 5/4, 4/3, 36/25, 3/2, 8/5, 5/3, 9/5, 48/25

1, 21/20, 8/7, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3, 9/5, 40/21

1, 22/21, 63/55, 6/5, 5/4, 4/3, 63/44, 3/2, 8/5, 5/3, 9/5, 21/11


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Message: 5005

Date: Tue, 11 Jun 2002 21:22:41

Subject: Re: A twelve-note, 11-limit scale

From: genewardsmith

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> i mention it in my paper, it's a pajara temperament.

You mentioned the temperament, or 120-et? I don't see what either has to do with pajara, since 50/49 is 4 120-et steps and 64/63 is 3.


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Message: 5007

Date: Wed, 12 Jun 2002 22:39:34

Subject: Two 9-note scales in the temperamentt

From: Gene W Smith

Commas {49/48, 21/20, 99/98, 121/120}

Block [1, 12/11, 7/6, 14/11, 4/3, 3/2, 11/7, 12/7, 11/6]

22-et version

[0, 3, 5, 8, 9, 13, 14, 17, 19]

5: 8, 0
7: 19, 11
9: 29, 41
11: 32, 56

31-et version

[0, 4, 7, 11, 13, 18, 20, 24, 27]

5: 5, 0
7: 15, 6
9: 22, 17
11: 32, 56

53-et version

[0, 7, 12, 19, 22, 31, 34, 41, 46]

5: 5, 0
7: 15, 6
9: 22, 17
11: 32, 56

Commas {128/125, 36/35, 99/98, 121/120}

Block [1, 35/33, 33/28, 5/4, 175/132, 264/175, 8/5, 56/33, 66/35]

22-et version

[0, 2, 5, 7, 9, 13, 15, 17, 20]

5: 12, 4
7: 22, 17
9: 27, 32
11: 30, 45

31-et version

[0, 3, 7, 10, 13, 18, 21, 24, 28]

5: 12, 4
7: 21, 14
9: 25, 26
11: 30, 45

53-et version

[0, 5, 12, 17, 22, 31, 36, 41, 48]

5: 12, 4
7: 21, 14
9: 25, 26
11: 30, 45


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Message: 5010

Date: Thu, 13 Jun 2002 05:03:16

Subject: Re: A twelve-note, 11-limit scale

From: genewardsmith

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> that depends on your mapping! if you use pajara with a 710-cent 
> generator, you're in 120-equal!

In this case, the mapping was defined by the fact that it had to temper out 126/125 and 176/175.


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Message: 5011

Date: Thu, 13 Jun 2002 15:28:26

Subject: A 9-note scale in the planar temperamentt

From: Gene W Smith

I looked at a number of these, and this was the best I found:

Block
[1, 11/10, 8/7, 5/4, 11/8, 16/11, 8/5, 7/4, 20/11]

22-et version [0,3,4,7,10,12,15,18,19]

3:   2   0
5:   11   4
7:   19   12
9:   29   39
11: 33   63
11-limit connectivity 7

31-et version [0,4,6,10,14,17,21,25,27]

3:   2   0
5:   11   4
7:   19   12
9:   22   15
11: 32   58
11-limit connectivity 6

46-et version [0,6,9,15,21,25,31,37,40]

3:   2   0
5:   11   4
7:   19   12
9:   19   12
11: 32   58
11-limit connectivity 6


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Message: 5015

Date: Fri, 14 Jun 2002 18:47:34

Subject: Re: Finding linear temperaments

From: Gene W Smith

Of course, people studying the Riemann Zeta function may in effect have
used computers to find ets before anyone, without knowing it. On the last
page of Titchmarsh, "The Theory of the Riemann Zeta-Function" (Oxford,
1951) he mentions the 140-et without knowing it, and one of the first
things to get worked over when computers came along was Zeta(s).


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Message: 5016

Date: Sat, 15 Jun 2002 19:32:03

Subject: Re: figurate number expansions as scales

From: genewardsmith

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Some interesting expansions and scales can be derived from figurate
> numbers.

Numbers of the form n/(n-1) where n is figurate show up a lot; you could look at my discussion of "jacks", for instance. The fact that
triangle and square demomenators lead to other triangle and square denomenators allows us to create series of scales.


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Message: 5018

Date: Sat, 15 Jun 2002 11:30:02

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> I would therefore recommend going back to the rational 
> complementation system and doing the ET's that way as well.

Agreed. Provided we _always_ use rational complements, whether this 
results in matching half-apotomes or not.

> Or, if you like, we could do them both ways and then decide.

No need.

> I would be agreeable to doing all of the ET's (with the rational 
> complementation scheme) using the symbols that we agreed on in 
> message #4443.

OK.

I will respond to your suggestions for the remaining ones of 6 or less 
steps per apotome when I get more time. Then move on to

7 steps per apotome
42,49,56,63,70,77,84,91,98,105

8 steps per apotome
54,61,68,75,82,89,96,103,110,117

9 steps per apotome
59,66,73,80,87,94,101,108,115,122,129

10 steps per apotome
71,78,85,92,99,106,113,120,127,134,141

etc.

I think we can do some with 23 steps per apotome, maybe even 25.


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Message: 5021

Date: Mon, 17 Jun 2002 03:12:55

Subject: Seven and eleven limit comma lists

From: genewardsmith

Here are comma lists for the 7 and 11 limits. Each comma is less than fifty cents, and each has the property that if the comma is p/q>1
in reduced form, then ln(p-q)/ln(q) < .5 in the 7-limit, and < .3 in
the 11-limit. I've found this weaking of the superparticularity
condition useful in the past, and it occurred to me it would be one
way of getting a finite list of temperaments a la Dave--we could
simply require it to have a basis of commas passing such a condition.
The lists below may be complete; at least, I haven't been able to add
to them.


Seven limit list, ln(p-q)/ln(q)<1/2, cents < 50

[1029/1000, 250/243, 36/35, 525/512, 128/125, 49/48, 50/49, 
3125/3072, 686/675, 64/63, 875/864, 81/80, 3125/3087, 2430/2401,
2048/2025, 245/243, 126/125, 4000/3969, 1728/1715, 1029/1024,
15625/15552, 225/224, 19683/19600, 16875/16807, 10976/10935,
3136/3125, 5120/5103, 6144/6125, 65625/65536, 32805/32768,
703125/702464, 420175/419904, 2401/2400, 4375/4374, 
250047/250000, 78125000/78121827]

Eleven limit list ln(p-q)/ln(q) < .3, cents < 50

[36/35, 77/75, 128/125, 45/44, 49/48, 50/49, 55/54, 56/55, 64/63, 
81/80, 245/242, 99/98, 100/99, 121/120, 245/243, 126/125, 1331/1323,
176/175, 896/891, 1029/1024, 225/224, 243/242, 3136/3125, 385/384,
441/440, 1375/1372, 6250/6237, 540/539, 4000/3993, 5632/5625,
43923/43904, 2401/2400, 3025/3024, 4375/4374, 9801/9800,
151263/151250, 3294225/3294172]


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