Tuning-Math Digests messages 5354 - 5378

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Message: 5354

Date: Sat, 19 Oct 2002 01:47:30

Subject: Re: for monzoni: bloated list of 5-limit linear temperaments

From: monz

thanks, paul!  i'll add it to my "linear temperaments"
definition when i get a chance.

because of the tunings used in some of my favorites
of Herman Miller's _Pavane for a warped princess_,
there's a family of equal-temperaments which i've become
interested in lately, which all temper out the apotome,
{2,3}-vector [-11 7],  ratio 2187:2048, ~114 cents:
14-, 21-, and 28-edo.

i noticed that these EDOs all have cardinalities which
are multiples of the exponent of 3 of the "vanishing comma".

looking at the lattices on my "bingo-card-lattice" definition
Yahoo groups: /monz/files/dict/bingo.htm *
i can see it works the same way for 10-, 15-, and 20-edo,
which all temper out the _limma_, {2,3}-vector [8 -5] = ~90 cents.


so apparently, at least in these few cases (but my guess
is that it happens in many more), there is some relationship
between the logarithmic division of 2 which creates the
EDO and the exponent of 3 of a comma that's tempered out.

has anyone noted this before?  any further comments on it?
is it possible that for these two "commas" it's just
a coincidence?

-monz



----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Friday, October 18, 2002 6:40 PM
Subject: [tuning-math] for monzoni: bloated list of 5-limit linear
temperaments


> monzieurs,
>
> someone let me know if anything is wrong or missing . . .
>
> 25/24 ("neutral thirds"?)
> generators [1200., 350.9775007]
> ets 3 4 7 10 11 13 17
>
> 81/80 (3)^4/(2)^4/(5) meantone
> generators [1200., 696.164845]
> ets 5 7 12 19 31 50
>
> 128/125 (2)^7/(5)^3 augmented
> generators [400.0000000, 91.20185550]
> ets 3 9 12 15 27 39 66
>
> 135/128 (3)^3*(5)/(2)^7 pelogic
> generators [1200., 677.137655]
> ets 7 9 16 23
>
> 250/243 (2)*(5)^3/(3)^5 porcupine
> generators [1200., 162.9960265]
> ets 7 8 15 22 37
>
> 256/243 (2)^8/(3)^5 quintal (blackwood?)
> generators [240.0000000, 84.66378778]
> ets 5 10 15 25
>
> 648/625 (2)^3*(3)^4/(5)^4 diminished
> generators [300.0000000, 94.13435693]
> ets 4 8 12 16 20 28 32 40 52 64
>
> 2048/2025 (2)^11/(3)^4/(5)^2 diaschismic
> generators [600.0000000, 105.4465315]
> ets 10 12 34 46 80
>
> 3125/3072 (5)^5/(2)^10/(3) magic
> generators [1200., 379.9679493]
> ets 3 13 16 19 22 25
>
> 15625/15552 (5)^6/(2)^6/(3)^5 kleismic
> generators [1200., 317.0796753]
> ets 4 11 15 19 34 53 87
>
> 16875/16384 negri
> generators [1200., 126.2382718]
> ets 9 10 19 28 29 47 48 66 67 85 86
>
> 20000/19683 (2)^5*(5)^4/(3)^9 quadrafifths
> generators [1200., 176.2822703]
> ets 7 13 20 27 34 41 48 61 75 95
>
> 32805/32768 (3)^8*(5)/(2)^15 shismic
> generators [1200., 701.727514]
> ets 12 17 29 41 53 65
>
> 78732/78125 (2)^2*(3)^9/(5)^7 hemisixths
> generators [1200., 442.9792975]
> ets 8 11 19 27 46 65 84
>
> 393216/390625 (2)^17*(3)/(5)^8 wuerschmidt
> generators [1200., 387.8196733]
> ets 3 28 31 34 37 40
>
> 531441/524288 (3)^12/(2)^19 pythagoric (NOT pythagorean)/aristoxenean?
> generators [100.0000000, 14.66378756]
> ets 12 48 60 72 84 96
>
> 1600000/1594323 (2)^9*(5)^5/(3)^13 amt
> generators [1200., 339.5088256]
> ets 7 11 18 25 32
>
> 2109375/2097152 (3)^3*(5)^7/(2)^21 orwell
> generators [1200., 271.5895996]
> ets 9 13 22 31 53 84
>
> 4294967296/4271484375 (2)^32/(3)^7/(5)^9 septathirds
> generators [1200., 55.27549315]
> ets 22 43 65 87


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Message: 5355

Date: Sat, 19 Oct 2002 02:00:42

Subject: Re: for monzoni: bloated list of 5-limit linear temperaments

From: monz

oh, and of course, your list already shows that this
also happens with the "Pythagoric" temperaments, which
all temper out the Pythagorean comma, {2,3}-vector [-19 12],
and which all have cardinalities which are multiples of 12.

so it seems that any EDO which tempers out a 3-limit
"comma" has a cardinality (= logarithmic division of 2)
which is a multiple of the exponent of 3 in that "comma".

interesting.  looks to me like there's some kind of
"bridge between incommensurable primes" going on here.


-monz


----- Original Message -----
From: "monz" <monz@xxxxxxxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Saturday, October 19, 2002 1:47 AM
Subject: Re: [tuning-math] for monzoni: bloated list of 5-limit linear
temperaments


> thanks, paul!  i'll add it to my "linear temperaments"
> definition when i get a chance.
>
> because of the tunings used in some of my favorites
> of Herman Miller's _Pavane for a warped princess_,
> there's a family of equal-temperaments which i've become
> interested in lately, which all temper out the apotome,
> {2,3}-vector [-11 7],  ratio 2187:2048, ~114 cents:
> 14-, 21-, and 28-edo.
>
> i noticed that these EDOs all have cardinalities which
> are multiples of the exponent of 3 of the "vanishing comma".
>
> looking at the lattices on my "bingo-card-lattice" definition
> Yahoo groups: /monz/files/dict/bingo.htm *
> i can see it works the same way for 10-, 15-, and 20-edo,
> which all temper out the _limma_, {2,3}-vector [8 -5] = ~90 cents.
>
>
> so apparently, at least in these few cases (but my guess
> is that it happens in many more), there is some relationship
> between the logarithmic division of 2 which creates the
> EDO and the exponent of 3 of a comma that's tempered out.
>
> has anyone noted this before?  any further comments on it?
> is it possible that for these two "commas" it's just
> a coincidence?
>
> -monz


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Message: 5359

Date: Sun, 20 Oct 2002 12:31:16

Subject: Re: A common notation for JI and ETs

From: monz

> From: "David C Keenan" <d.keenan@xx.xxx.xx>
> To: "George Secor" <gdsecor@xxxxx.xxx>
> Cc: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, September 18, 2002 6:12 PM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> At 06:19 PM 17/09/2002 -0700, George Secor wrote:
> > From:  George Secor (9/17/02, #4626)
> >
> > Neither 306 nor 318 are 7-limit consistent, so I don't see much point
> > in doing these, other than they may have presented an interesting
> > challenge.
>
> Good point. Forget 318-ET, but 306-ET is of interest for being strictly
> Pythagorean. The fifth is so close to 2:3 that even god can barely tell
the
> difference. ;-)


what an interesting coincidence!  i just noticed this bit because
Dave quoted it in his latest post.

just yesterday, i "discovered" for myself that 306edo is a great
approximation of Pythagorean tuning, and that one degree of it
designates "Mercator's comma" (2^84 * 3^53), which i think makes
it particularly useful to those who are really interested in
exploring Pythagorean tuning.

see my latest additions to:
Yahoo groups: /monz/files/dict/pythag.htm *



-monz
"all roads lead to n^0"


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Message: 5360

Date: Sun, 20 Oct 2002 12:41:48

Subject: Re: MUSIC OF THE SPHERES

From: monz

hello Bill,


> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, October 20, 2002 9:07 AM
> Subject: [tuning-math] Re: MUSIC OF THE SPHERES


> --- In tuning-math@y..., Bill Arnold <billarnoldfla@y...> wrote:
> 
> > In conclusion: if someone KNOWS of a message board which WELCOMES a
> > discussion of MUSIC OF THE SPHERES and the MATH and PHYSICS thereof,
> > let me know.  I will take my question THERE.
> 
> I thought someone had made a list for that very topic.


Yahoo groups: /celestial-tuning/ *


i know you already subscribe, and i also know that 
your questions have gone unanswered by members of the
celestial-tuning list, but if you're going to lurk on
tuning and tuning-math, i encourage you to keep posting
there on celestial-tunings as your work is entirely
relevant to the subject matter of that group.

(and this time *i'll* aplogize for the cross-post)


-monz


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Message: 5371

Date: Mon, 21 Oct 2002 12:44:22

Subject: Re: Epimorphic

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Gene wrote:

>Great! It seems to me it would be better to say "JI-epimorphic" or
>"RI-epimorphic", leaving open the possibility of also implementing
>"meantone-epimorphic" or "starling-epimorphic" some fine day.

It turns out the question was moot since Pierre showed that it's
equivalent to CS. Anyway I don't need to throw the new code straight
away if I use it to print out the characterising val. I'll call that 
epimorphic prime-degree mapping.

Isn't "meantone-epimorphic" covered by Myhill's property?

Manuel


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Message: 5373

Date: Mon, 21 Oct 2002 19:40:45

Subject: Re: Digest Number 497

From: John Chalmers

Gene asked:

>Has anyone paid attention to scales which have a number of steps a
>multiple of a MOS? They inherit structure from the MOS, and using a 2MOS
>or a 3MOS seems like a good way to fill in those annoying gaps.

I think most of Messiaien's "Modes of Limited Transposition" in 12-tet
are multiple MOS's of 3, 4 and 6-tet. I don't have a list handy on this 
computer to check, unfortunately. IIRC, William Lyman Young (in his
"Report 
to the Swedish Royal Academy of Music" etc.) proposed a decatonic scale
in 
24-tet which was two 5-tone MOS's of 12 (2322323223) and a 14-tone scale 
of 2 sections of the 7-tone diatonic sequence as 22122212212221 in
24-tet.
He considered these as generated from cycles of half-fourths or
half-fiths.

I suspect that some of Wyschnegradski's scales might be multiple MOS's
too,
but I don't have a list either.

--John


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