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Message: 8853 - Contents - Hide Contents

Date: Tue, 23 Dec 2003 04:24:13

Subject: Re: Chord mapping

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Just a note, chord lookup table with common-tone matching is the > basis of my adaptive tuning algorithm... > > MIDI-based adaptive tuning by common-tone matc... * [with cont.] (Wayb.)
So when do you code it?
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Message: 8854 - Contents - Hide Contents

Date: Tue, 23 Dec 2003 22:38:33

Subject: plot of 5-limit commas

From: Paul Erlich

I plotted all 5-limit commas for which

tenney complexity [TC] < 30
and
tenney-based heuristic error [(n-d)/TC/log(TC)] < 0.001
and
the comma is not a higher power of some other comma

Yahoo groups: /tuning-math/files/Paul/waterfal... * [with cont.] 

who can explain the "woof" and "warp", or curved 'grid', apparent in 
this plot?

here is the plot with the ratios of the commas written in:

Yahoo groups: /tuning-math/files/Paul/waterfal... * [with cont.] 

the size of the font used for each comma was 16^(1-epimericity)
where the epimerity is tenney based and equal to log(n-d)/log(n*d).

one can easily imagine the epimericity 'contours' running across this 
plot, though i wasn't able to show them as easily as i thought i 
would . . .

anyway, i hope herman miller, at least, takes a look at this!


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Message: 8856 - Contents - Hide Contents

Date: Tue, 23 Dec 2003 23:39:44

Subject: Re: epimorphism

From: Manuel Op de Coul

>Manuel, you are wrong. This is indeed a torsional block. The four >determinants are 20, 32, 46, and 56 -- obviously these are all >multiples of 2, so we have torsion!
Drag, you're right. Why is it that when you know there's a bug in the code you can spot it immediately, when otherwise it remains unnoticed. Manuel
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Message: 8857 - Contents - Hide Contents

Date: Tue, 23 Dec 2003 22:42:14

Subject: Re: epimorphism

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" 
<manuel.op.de.coul@e...> wrote:
>
>> Manuel, you are wrong. This is indeed a torsional block. The four >> determinants are 20, 32, 46, and 56 -- obviously these are all >> multiples of 2, so we have torsion! >
> Drag, you're right. Why is it that when you know there's a bug in > the code you can spot it immediately, when otherwise it remains unnoticed. > > Manuel
so now can you find a *real* counterexample?
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Message: 8858 - Contents - Hide Contents

Date: Tue, 23 Dec 2003 10:21:32

Subject: Magical mystery meanpop wreckage

From: Gene Ward Smith

If we take the duodene, which is the genus 3^i 5^j where i runs from
-1 to 2 and j from -1 to 1, we get something which mapped by meantone
gives us i+4j, which runs from -1-4=-5 to 2+4=6, the meantone 12-note
MOS. If we take instead the major/minor thirds parallogram, which is
the same as the duodene but with 5/3 in place of 3, we do not get a 
MOS. Instead, we get 3i+4j, which gives us the values 
-7,-4,-3,-1,0,1,2,3,4,6,7,10. In septimal meantone, this gives us the 
magic chords of the augmented triad, based on (5/4)^2(9/7)/2 =225/224,
and the diminished seventh, based on (6/5)^3(7/6)/2 = 126/125. We 
don't need 81/80 to find this temperament; the existence of the above 
two magic chords determines it. Beyond the 7-limit, septimal meantone 
divides into what I call "meantone" and "meanpop", depending on how 
11 maps, with the dividing line being 31-et for which the two are the 
same. Meanpop territory is where the fifths are just a tad flatter, 
as for example 50-et or Wilson meantone.

Further magic awaits us in meanpop land. We have (9/7)/(6/5)/(20/13) =
351/350 (the ratwolf triad) and (5/4)(7/6)/(16/11) = 385/384 (no name 
known to me.) If we ask for all four of these magic chords at once, 
we get 13-limit meanpop, for which 50-et is a good choice.

Consider what happens to the "thirds" scale discussed above when the 
temperament is meanpop. The 5-limit JI scale is

! thirds.scl
!
Major and minor thirds 
paralleogram                                           
 12
!
25/24
10/9
6/5
5/4
4/3
25/18
3/2
8/5
5/3
125/72
48/25
2

If we take the 50-et version of this, we get

! wreckpop.scl
!
"Wreckmeister" 13-limit meanpop (50 et) tempered 
thirds.scl                                         
 12
!
72.000000
192.000000
312.000000
384.000000
504.000000
576.000000
696.000000
816.000000
888.000000
960.000000
1128.000000
1200.000000

The 0-4-7 triads for this goes

[432, 312, 744]
[384, 312, 696]
[384, 264, 648]
[432, 312, 744]
[384, 312, 696]
[384, 312, 696]
[432, 264, 696]
[384, 312, 696]
[384, 312, 696]
[432, 312, 744]
[384, 264, 648]
[384, 312, 696]

We have six meantone major triads, [384, 312, 696], but we also have
three ratwolf triads [432, 312, 744], and two as yet unnamed magical
[384, 264, 648] triads. Finally, we have a nice, old-fashioned 
supermajor triad, [432, 264, 696]. Each scale step has a triad, magic 
or mundane, to go with it (plus the minor versions of course.)

This is, of course, related to my Wreckmeister A scale, which however 
I only tempered via 126/125 planar. Looking at the matter from the 
point of view of meanpop seems like a much better way to go about 
Wreckmeistering.



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Message: 8860 - Contents - Hide Contents

Date: Wed, 24 Dec 2003 21:21:32

Subject: Re: epimorphism

From: Manuel Op de Coul

Paul wrote:
>so now can you find a *real* counterexample?
I guess not... I've uploaded a new release with the torsion and epimorphism bugs fixed, updated Sagittal symbols, improved periodicity block dialog, and smaller improvements. http://www.huygens-fokker.org/software/Scala_Setup.exe - Type Ok * [with cont.] (Wayb.) Manuel
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Message: 8861 - Contents - Hide Contents

Date: Wed, 24 Dec 2003 21:19:23

Subject: Re: plot of 5-limit commas

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >
>> who can explain the "woof" and "warp", or curved 'grid', apparent > in >> this plot? >
> I'd start by taking ratios of commas along lines, and see if there is > a single high-quality comma behind the line.
No; the lines are essentially single numerators in one direction, and single denominators in the other.
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Message: 8862 - Contents - Hide Contents

Date: Wed, 24 Dec 2003 14:42:57

Subject: definitions

From: Carl Lumma

Gene,

If you could collect all your definitions, "scale", "tuning",
"notation", etc. and put them on xenharmony.org or Wikipedia
I would be enthralled.

-Carl



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Message: 8863 - Contents - Hide Contents

Date: Wed, 24 Dec 2003 08:52:13

Subject: Re: plot of 5-limit commas

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> who can explain the "woof" and "warp", or curved 'grid', apparent in > this plot?
I'd start by taking ratios of commas along lines, and see if there is a single high-quality comma behind the line. ________________________________________________________________________ ________________________________________________________________________ To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
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Message: 8864 - Contents - Hide Contents

Date: Thu, 25 Dec 2003 23:59:12

Subject: Re: Transitive groups of degree 12 and low order containing a 12-cycle

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Gene, > > If it's not too much trouble, could you run S(4) X C(4) and also > S(4) X S(3)? And give the generators and polynomials, if its not too > much trouble
S(4)XC(3) order 72 Generators a := [[12, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]] e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] B := [[1, 10], [2, 5], [6, 9]] Polynomial x^12+x^11+3*x^10+8*x^9+14*x^8+19*x^7+24*x^6+19*x^5+14*x^4+8*x^3+3*x^2+x+1 S(4)XS(3) order 144 Generators a := [[12, 4, 8], [1, 5, 9], [2, 6, 10], [3, 7, 11]] b := [[1, 5], [2, 10], [4, 8], [7, 11]] e := [[12, 3, 6, 9], [1, 4, 7, 10], [2, 5, 8, 11]] B := [[1, 10], [2, 5], [6, 9]] Polynomial x^12+x^11+3*x^10+6*x^9+11*x^8+13*x^7+17*x^6+13*x^5+11*x^4+6*x^3+3*x^2+x+1 Merry Christmas!
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Message: 8866 - Contents - Hide Contents

Date: Fri, 26 Dec 2003 00:44:15

Subject: Re: Attention Gene

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> Yahoo groups: /tuning-math/message/8269 * [with cont.] > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" > <gwsmith@s...> >> wrote:
>>> After fixing my program, here is what I am getting for Prooijen > and
>>> geometric 11-limit reductions: >>> >>> ! red72_11pro.scl >>> Prooijen 11-limit reduced scale >>> 72 >>> ! >>> 81/80 >>> 64/63 >>
>> Gene -- why isn't this 45/44?
I guess I'm using the wrong definition of Prooijen complexity.
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Message: 8868 - Contents - Hide Contents

Date: Fri, 26 Dec 2003 06:16:03

Subject: Re: Attention Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >> Yahoo groups: /tuning-math/message/8269 * [with cont.] >>
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote:
>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" >> >> wrote:
>>>> After fixing my program, here is what I am getting for Prooijen >> and
>>>> geometric 11-limit reductions: >>>> >>>> ! red72_11pro.scl >>>> Prooijen 11-limit reduced scale >>>> 72 >>>> ! >>>> 81/80 >>>> 64/63 >>>
>>> Gene -- why isn't this 45/44? >
> I guess I'm using the wrong definition of Prooijen complexity.
It's called 'expressibility', and it's simply (the log of) the "ratio of" (or, imprecisely speaking, "odd-limit") measure of the ratio. Definitions of tuning terms: ratio of, (c) 200... * [with cont.] (Wayb.) Searching Small Intervals * [with cont.] (Wayb.) Since log(45)<log(63), you must indeed have the wrong definition.
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Message: 8871 - Contents - Hide Contents

Date: Fri, 26 Dec 2003 06:23:06

Subject: Re: Attention Gene

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> Since log(45)<log(63), you must indeed have the wrong definition.
log(11*45)>log(63), which is what I think I used.
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Message: 8872 - Contents - Hide Contents

Date: Fri, 26 Dec 2003 08:32:43

Subject: Re: Attention Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >
>> Since log(45)<log(63), you must indeed have the wrong definition. >
> log(11*45)>log(63), which is what I think I used.
I thought I had already straightened you out on that particular misunderstanding. ________________________________________________________________________ ________________________________________________________________________ To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx ------------------------------------------------------------------------ Yahoo! Groups Links To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
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Message: 8873 - Contents - Hide Contents

Date: Sat, 27 Dec 2003 20:52:20

Subject: Re: 5-limit, 12-note Fokker blocks

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> The Duodene is well-known as a 5-limit, 12-note Fokker block; > I > decided to check on whether Scala knows of four other examples, and > got this.
Scala has many others, including Marpurg's Monochord #1, Ramos/Ramis, etc . . . see the Gentle Introduction to Fokker Periodicity Blocks.
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Message: 8874 - Contents - Hide Contents

Date: Sat, 27 Dec 2003 00:21:53

Subject: 5-limit, 12-note Fokker blocks

From: Gene Ward Smith

The Duodene is well-known as a 5-limit, 12-note Fokker block; I 
decided to check on whether Scala knows of four other examples, and 
got this. (The scales can be obtained by letting i range from 1 to 
12.)


(16/15)^i (2048/2025)^(-5i/12) (81/80)^(-2i/12)
No Scala name


(16/15)^i (81/80)^(8i/12) (648/625)^(-5i/12)
No Scala name


(16/15)^i  (128/125)^(-8i/12) (648/625)^(3i/12)
No Scala name, a mode of what I've called the thirds scale


(16/15)^i (2048/2025)^(-3i/12) (128/125)^(-2i/12)
Scala identifies it as lumma5r.scl

! lumma5r.scl
!
Carl Lumma's scale, 5-limit just version, TL 19-2-
99                            
 12
!
 16/15
 9/8
 75/64
 5/4
 4/3
 45/32
 3/2
 8/5
 5/3
 225/128
 15/8
 2/1


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