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Message: 5300 - Contents - Hide Contents
Date: Sat, 12 Oct 2002 00:31:47
Subject: Historical well-temeraments, 612, and 412
From: Gene Ward Smith
It seems that Werckmeister III is not the only well-temperament to be
nailed by 612. Here are some others, using data taken from Manual's
list of scales:
! young.scl
!
Thomas Young well temperament (1807), also Luigi Malerbi nr.2 (1794)
12
!
256/243
196.09000
32/27
392.18000
4/3
1024/729
698.04500
128/81
894.13500
16/9
1090.22500
2/1
The 612-et version of this is again perfection itself:
[0, 46, 100, 150, 200, 254, 300, 356, 404, 456, 508, 556]
Note that all the steps are even, so 306 also works.
! young2.scl
!
Thomas Young well temperament no.2, ca. 1800
12
!
94.13500
196.09000
298.04500
392.18000
500.00000
592.18000
698.04500
796.09000
894.13500
1000.00000
1092.18000
2/1
Again, the 612-et version is insanely accurate:
[0, 48, 100, 152, 200, 255, 302, 356, 406, 456, 510, 557]
Here is one by Marpurg:
! marpurg2.scl
!
Marpurg 2. Neue Methode (1790)
12
!
109.775 cents
9/8
313.685 cents
81/64
4/3
607.820 cents
3/2
811.730 cents
27/16
1015.640 cents
1105.865 cents
2/1
Once again, 306 would work also:
[0, 56, 104, 160, 208, 254, 310, 358, 414, 462, 518, 564]
Finally, here is an example where 612 does not work, but 412 works
excellently:
! marpurg.scl
!
Marpurg, Versuch ueber die musikalische Temperatur (1776), p. 153
12
!
101.955 cents
200.978 cents
300.000 cents
401.955 cents
500.978 cents
600.000 cents
3/2
800.978 cents
900.000 cents
1001.955 cents
1100.978 cents
2/1
In terms of the 412-et:
[0, 35, 69, 103, 138, 172, 206, 241, 275, 309, 344, 378]
(The 1200-et isn't bad here either.)
Message: 5301 - Contents - Hide Contents Date: Sat, 12 Oct 2002 00:44:25 Subject: Re: Historical well-temeraments, 612, and 412 From: Gene Ward Smith --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> Finally, here is an example where 612 does not work, but 412 works > excellently: > > ! marpurg.scl > ! > Marpurg, Versuch ueber die musikalische Temperatur (1776), p. 153However, 1224 works very, very well, so we still have a nice version of this using 612 as a basic measure: [0,52,102.5,153,205,255.5,306,358,408.5,459,511,561.5]
Message: 5302 - Contents - Hide Contents Date: Sat, 12 Oct 2002 07:41:19 Subject: Re: Historical well-temeraments, 612, and 412 From: Gene Ward Smith --- In tuning-math@y..., "monz" <monz@a...> wrote:> my guess is that the reason 612 works so well has something > to do with the fact that these temperaments temper out the > Pythagorean comma. wanna look into that more?My assumption is that the fact that the Pythagorean comma and 3 are both well represeted by 612 has something to do with it, but that's not the whole story or 665 would dominate.
Message: 5303 - Contents - Hide Contents Date: Sat, 12 Oct 2002 08:37:05 Subject: Re: Historical well-temeraments, 612, and 412 From: Gene Ward Smith --- In tuning-math@y..., "monz" <monz@a...> wrote:> wow, Gene, thanks for these!!! > they'll eventually all become Tuning Dictionary webpages.Great. Here are a couple more historical temperaments which can be nicely expressed in terms of shismas: ! kirnberger1.scl ! Kirnberger's temperament 1 (1766) 12 ! 256/243 9/8 32/27 5/4 4/3 45/32 3/2 128/81 895.11200 16/9 15/8 2/1 [0, 46, 104, 150, 197, 254, 301, 358, 404, 456.5, 508, 555] ! kirnberger2.scl ! Kirnberger 2: 1/2 synt. comma. "Die Kunst des reinen Satzes" (1774) 12 ! 135/128 9/8 32/27 5/4 4/3 45/32 3/2 405/256 895.11186 16/9 15/8 [0, 47, 104, 150, 197, 254, 301, 358, 405, 456.5, 508, 555] Just for kicks, here is the Ellis Duodene: [0, 57, 104, 161, 197, 254, 301, 358, 415, 451, 519, 555]
Message: 5304 - Contents - Hide Contents Date: Sun, 13 Oct 2002 21:48:54 Subject: Re: scales and periodicity blocks (from tuning-math2) From: Carl Lumma>> >ot epimorphic, eh? From monz's tuning dictionary, I get that >> a scale is epimorphic when there's a single val that can map >> all its degrees to integers. I'm missing some details. Can >> you (Gene) show how one of these scales is not "epimorphic"? >>For a p-limit JI scale of n steps, you get n-1 linear equations >in a number of unknowns equal to the number of primes up to p >which must have a solution for the scale to be epimorphic.Primes -- what about the 9-limit? Wouldn't we need an unknown for the 9-axis?>Take, for intance, recta3c1: >[1,15/14,7/6,6/5,5/4,9/7,7/5,3/2,8/5,12/7,7/4,15/8]. > >If h=[a,b,c,d] is our val, then h(15/14)=1 for example gives us >the equation -a+b+c-d-1=0.You expect the taxicab distance from the unison to a scale member to equal its position in the scale?>Taking the equations for 15/14, 7/6, 6/5, and 5/4 we find there >is no solution,...and you expect a single set of values for the unknowns will work for all scale members? The only difference between equations will be the signs and the term representing the scale position in question?>hence no epimorphic scale can start out 1--15/14--7/6--6/5--5/4...You mean no epimorphic scale _with p unknowns_, right?>> "epimorphic/CS" -- forgive me if we've been over this (I >> can't find anything in the archives), but does epimorphic >> equal constant structures? >>Epimorphic ==> CS, clearly.Hopefully it will be clear to me soon...>For the other way I need to be sure I really understand >exactly what is CS.I wish I knew the language you're using, I could tell you! Gene, your tools are so cool, it would be a shame if you remain the only person around here who knows how to use them... have you thought of writing a 'Gene's tools for dummies' paper, that explains why these tools work in terms of lattice geometry (how most musician folks around here think of scales and blocks, I reckon)? Maybe it would help if you explained how you came to think epimorphic was a Good Thing (since things like CS and Rothenberg propriety apparently weren't part of your reasoning). -Carl
Message: 5305 - Contents - Hide Contents Date: Sun, 13 Oct 2002 17:01:34 Subject: Re: scales and periodicity blocks (from tuning-math2) From: monz> From: "Carl Lumma" <clumma@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, October 13, 2002 2:48 PM > Subject: [tuning-math] Re: scales and periodicity blocks (from tuning-math2) > > > Gene, your tools are so cool, it would be a shame if you > remain the only person around here who knows how to use > them... have you thought of writing a 'Gene's tools for > dummies' paper, that explains why these tools work in terms > of lattice geometry (how most musician folks around here > think of scales and blocks, I reckon)? > > Maybe it would help if you explained how you came to think > epimorphic was a Good Thing (since things like CS and > Rothenberg propriety apparently weren't part of your > reasoning).i made an initial attempt here: Yahoo groups: /monz/files/dict/genemath.htm * [with cont.] but Gene, Carl's right ... i'm another who would really appreciate nice long-winded explanations, using lots of English words to translate the cryptic mathmatical explanations you give of your work. -monz
Message: 5307 - Contents - Hide Contents Date: Sun, 13 Oct 2002 01:30:20 Subject: scales and periodicity blocks (from tuning-math2) From: Carl Lumma>Periodicity blocks? They [glumma scales] aren't epimorphic/CS.Well, I thought we decided that block = epimorphic + convex. It looks to me like the glumma family is convex. (Paul, how do you feel about the convexity condition in light of its exclusion of the melodic minor?) Not epimorphic, eh? From monz's tuning dictionary, I get that a scale is epimorphic when there's a single val that can map all its degrees to integers. I'm missing some details. Can you (Gene) show how one of these scales is not "epimorphic"? "epimorphic/CS" -- forgive me if we've been over this (I can't find anything in the archives), but does epimorphic equal constant structures?>They really aren't very regular, which I presume is why you >don't like them.That's right, and I think Paul had decided that blocks whose smallest 2nd was larger than their smallest unison vector would be... something. Proper? CS? I forget. -Carl
Message: 5308 - Contents - Hide Contents Date: Sun, 13 Oct 2002 05:20:40 Subject: Quartaminorthirds From: Gene Ward Smith Quartaminorthirds is the 7-limit linear temperament with wedgie [9, 5, -3, -21, 30, -13] and period matrix [[1, 1, 2, 3], [0, 9, 5, -3]]. It tempers out 1029/1024, 6144/6125 and their product, 126/125. It is covered by 31, 46, 77 and 108 among others, but 139 gives almost exactly the rms optimal values with the generator 9/139. It has MOS of size 15 and 16, which in 139 terms are [(9)*14, 13] and [(9)*15, 4]. The first is more regular, but the second I think is more interesting because like Blackjack it can be described in terms of a linear array of tetrads. The complexity measure most useful if we are interested in complete tetrads is Graham complexity, and here we have a value of 12, as opposed to 13 for miracle. The signature is of unital type, [-7,1,17]. From a complexity of 12 we conclude we have four major and four minor tetrads, and these can be identified with the tetrads from [0,0,0] to [0,0,7], which give us the steps defined by the generator running from -3 to 12.
Message: 5309 - Contents - Hide Contents Date: Sun, 13 Oct 2002 05:39:33 Subject: Re: scales and periodicity blocks (from tuning-math2) From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> Not epimorphic, eh? From monz's tuning dictionary, I get that > a scale is epimorphic when there's a single val that can map > all its degrees to integers. I'm missing some details. Can > you (Gene) show how one of these scales is not "epimorphic"?For a p-limit JI scale of n steps, you get n-1 linear equations in a number of unknowns equal to the number of primes up to p which must have a solution for the scale to be epimorphic. Take, for intance, recta3c1: [1,15/14,7/6,6/5,5/4,9/7,7/5,3/2,8/5,12/7,7/4,15/8]. If h=[a,b,c,d] is our val, then h(15/14)=1 for example gives us the equation -a+b+c-d-1=0. Taking the equations for 15/14, 7/6, 6/5, and 5/4 we find there is no solution, and hence no epimorphic scale can start out 1--15/14--7/6--6/5--5/4...> "epimorphic/CS" -- forgive me if we've been over this (I > can't find anything in the archives), but does epimorphic > equal constant structures?Epimorphic ==> CS, clearly. For the other way I need to be sure I really understand exactly what is CS.
Message: 5310 - Contents - Hide Contents Date: Sun, 13 Oct 2002 05:42:17 Subject: Re: Quartaminorthirds From: Gene Ward Smith --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: From a complexity of 12 we conclude we have four major and> four minor tetrads, and these can be identified with the tetrads from > [0,0,0] to [0,0,7], which give us the steps defined by the generator > running from -3 to 12.This should be the tetrads from [0,0,0] to [0,7,0], of course.
Message: 5311 - Contents - Hide Contents Date: Mon, 14 Oct 2002 13:38:24 Subject: Re: EDO superset containing approximation of Werckmeister III? From: manuel.op.de.coul@xxxxxxxxxxx.xxx Gene wrote:>Nice! Is there a way to go beyond the stop point, and *not* show alldivisions? I notice 612 popping up a lot >with temperaments I hadn't looked at yet, but the stop point is set too low to easily see it. Not at the moment I'm afraid. I'll add another qualifier to the next version to make this possible. Manuel
Message: 5312 - Contents - Hide Contents Date: Mon, 14 Oct 2002 11:02:29 Subject: Re: scales and periodicity blocks (from tuning-math2) From: monz> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, October 14, 2002 1:11 AM > Subject: [tuning-math] Re: scales and periodicity blocks (from tuning-math2) > > > --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote: >>>> I was never quite sure what CS meant,that's all. Epimorphic has >>> a clear definition. >>>> How'd you come up with it? >> It seemed like a clearly important property; after defining it, > I noticed that it seemed to be very close to, if not identical with, > CS. I'd go look up the definition right now and start the > long-overdue analysis of the two but I can't seem to find > Monzo's dictionary anymore.for now, it's here: Yahoo groups: /monz/files/dict/index.htm * [with cont.] -monz
Message: 5314 - Contents - Hide Contents Date: Mon, 14 Oct 2002 00:17:58 Subject: Re: scales and periodicity blocks (from tuning-math2) From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:> Primes -- what about the 9-limit? Wouldn't we need an unknown > for the 9-axis?Odd limits aren't relevant here.>> Take, for intance, recta3c1: >> [1,15/14,7/6,6/5,5/4,9/7,7/5,3/2,8/5,12/7,7/4,15/8]. >> >> If h=[a,b,c,d] is our val, then h(15/14)=1 for example gives us >> the equation -a+b+c-d-1=0. >> You expect the taxicab distance from the unison to a scale member > to equal its position in the scale?No; the idea is that there is a val h such that if qn is the nth scale step, starting at 0, then h(qn)=n. If the val does not exist, the scale is not epimorphic.>> Taking the equations for 15/14, 7/6, 6/5, and 5/4 we find there >> is no solution, >> ...and you expect a single set of values for the unknowns will > work for all scale members?The definition is, there *is* a val. The only difference between> equations will be the signs and the term representing the scale > position in question? >>> hence no epimorphic scale can start out 1--15/14--7/6--6/5--5/4... >> You mean no epimorphic scale _with p unknowns_, right?No 7-limit epimorphic scale, I should have said.> Maybe it would help if you explained how you came to think > epimorphic was a Good Thing (since things like CS and > Rothenberg propriety apparently weren't part of your > reasoning).I was never quite sure what CS meant,that's all. Epimorphic has a clear definition.
Message: 5315 - Contents - Hide Contents Date: Mon, 14 Oct 2002 00:30:15 Subject: Re: scales and periodicity blocks (from tuning-math2) From: Carl Lumma>>> >f h=[a,b,c,d] is our val, then h(15/14)=1 for example gives us >>> the equation -a+b+c-d-1=0. >>>> You expect the taxicab distance from the unison to a scale member >> to equal its position in the scale? >>No; the idea is that there is a val h such that if qn is the nth >scale step, starting at 0, then h(qn)=n. If the val does not exist, >the scale is not epimorphic.That's what the tuning dictionary says. But from the signs on the equation above, it looks like you expect a to be an exponent for 2, b for 3, c for 5, d for 7, in the factorization of 15/14.>> ...and you expect a single set of values for the unknowns will >> work for all scale members? >> The definition is, there *is* a val. Right. Why?>> You mean no epimorphic scale _with p unknowns_, right? >>No 7-limit epimorphic scale, I should have said. Ok. >I was never quite sure what CS meant,that's all. Epimorphic has >a clear definition.How'd you come up with it? -Carl
Message: 5316 - Contents - Hide Contents Date: Mon, 14 Oct 2002 08:11:27 Subject: Re: scales and periodicity blocks (from tuning-math2) From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> I was never quite sure what CS meant,that's all. Epimorphic has >> a clear definition. >> How'd you come up with it?It seemed like a clearly important property; after defining it, I noticed that it seemed to be very close to, if not identical with, CS. I'd go look up the definition right now and start the long-overdue analysis of the two but I can't seem to find Monzo's dictionary anymore.
Message: 5317 - Contents - Hide Contents Date: Tue, 15 Oct 2002 16:55:42 Subject: Re: scales and periodicity blocks (from tuning-math2) From: Gene Ward Smith --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:>> It seemed like a clearly important property; after defining it, I >> noticed that it seemed to be very close to, if not identical with, >> CS. I'd go look up the definition right now and start the long- >> overdue analysis of the two but I can't seem to find Monzo's >> dictionary anymore. >> Hopefully, you'll see his note os to its current (temporary) > location. The definition at the top is what you want: > > "A tuning system where each interval occurs always subtended by the > same number of steps."I don't see any reference to JI in this definition, so I don't think it means the same as epimorphic; certainly, however, any epimorphic scale will have this property.
Message: 5319 - Contents - Hide Contents Date: Tue, 15 Oct 2002 23:07:43 Subject: Re: mathematical model of torsion-block symmetry? From: wallyesterpaulrus --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:> BTW, I think the definition of torsion can be made simpler. You do not need > the condition that some power of the interval is in the unison vector group, > because this is always the case (at least when the periodicityblock is finite).> Do I see this correctly? > > Hans Straubi'm not sure. torsion is a pathological condition, especially from the point of view of generating temperaments. it's certainly not true that all finite periodicity blocks exhibit torsion. so what exactly are you saying?
Message: 5320 - Contents - Hide Contents Date: Tue, 15 Oct 2002 05:43:15 Subject: Re: scales and periodicity blocks (from tuning-math2) From: Carl Lumma>It seemed like a clearly important property; after defining it, I >noticed that it seemed to be very close to, if not identical with, >CS. I'd go look up the definition right now and start the long- >overdue analysis of the two but I can't seem to find Monzo's >dictionary anymore.Hopefully, you'll see his note os to its current (temporary) location. The definition at the top is what you want: "A tuning system where each interval occurs always subtended by the same number of steps." -Carl
Message: 5321 - Contents - Hide Contents Date: Tue, 15 Oct 2002 23:17:20 Subject: Re: Historical well-temeraments, 612, and 412 From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "monz" <monz@a...> wrote: >>> my guess is that the reason 612 works so well has something >> to do with the fact that these temperaments temper out the >> Pythagorean comma. wanna look into that more? >> My assumption is that the fact that the Pythagorean comma and 3 are >both well represeted by 612 has something to do with it, but that's >not the whole story or 665 would dominate. guys:it's because these tunings distrubute the pythagorean comma in various ways, typically chopping it into thirds, quarters, sixths, or twelfths. clearly the solution itself will have to be a multiple of 12 (since 12-equal forms the "baseline" where the pythagorean comma is tempered out), and because of the above, it also has to express the pythagorean comma as a multiple of 12. in 612, the pythagorean comma is 12, so 612 is the simplest solution. where the pythagorean comms is chopped into *eighths*, we need to go to 1224.
Message: 5322 - Contents - Hide Contents Date: Tue, 15 Oct 2002 05:46:09 Subject: Re: scales and periodicity blocks (from tuning-math2) From: Carl Lumma [monz wrote...] You did a good job of collecting the key thoughts from that very ConfuSing thread. The only gotcha is that the last quote on the page is by Paul, not me (the message number is correct). -Carl
Message: 5324 - Contents - Hide Contents Date: Wed, 16 Oct 2002 21:26:53 Subject: for those not following the tuning list . . . From: wallyesterpaulrus these . . . Yahoo groups: /tuning/message/39667 * [with cont.] Yahoo groups: /tuning/message/39695 * [with cont.] may be of interest . . .
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