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Message: 1725 - Contents - Hide Contents Date: Tue, 02 Oct 2001 21:41:47 Subject: Re: Digest Number 124 From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Better yet, can we avoid bringing 12 into this at all? I see no >> reason it should be brought in. >> Since we aren't tempering, leaving it out is the thing to do.Well . . . 81:80 being a commatic unison vector sort of implies tempering . . . but in no way selects 12 over other meantones like 19, 31, 43, 55, or 74.
Message: 1726 - Contents - Hide Contents Date: Tue, 02 Oct 2001 21:57:52 Subject: Re: Digest Number 124 From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Well . . . 81:80 being a commatic unison vector sort of implies > tempering . . . but in no way selects 12 over other meantones like > 19, 31, 43, 55, or 74.Those are all of the form n h7 + m h5, which is what I was talking about.
Message: 1727 - Contents - Hide Contents Date: Tue, 02 Oct 2001 21:51:47 Subject: Re: 41, 46 and 58 From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote:> And "less than or equal to l" should be "less than or equal to w". I was > wondering how a prime could be less than 1.No, I mean that for those values of w, cons(w, n) is less than 1.> It looks like h(q_i) is the number of steps in the ET for the ratio q_i. > So h(2) is the number of steps to the octave. So cons(w, n) assumes > consistency which -- warning! -- doesn't hold for 46-equal in the > 15-limit, because there's one ambiguous interval.Consistency always holds automatically the way I define things, since I calculate everything based on the generators--that is, "h" is defined not just by h(2) but also h(3), h(5) etc.> The n^(1/d) means you're scaling according to the size of the octave and > the number of prime dimensions. Right. > That could be re-written > > n^(1/d) * n * max(|tempered_pitch - just_pitch|) > > where pitches are in octaves. Which can be simplified to Right again. > n^(1+1/d) * max(|tempered_pitch - just_pitch|) > > Is that right? I think I almost understand it! So why the 1+1/d?If you are doing a simultaneous Diophantine approximation of d different numbers, at least one of them irrational, by numbers with denominator n, then we can find an infinite number of n such that all of the numbers are within 1/n^(1+1/d) of the correct value. Choosing a p-limit scale division involves the simultaneous approximation of d values log_2(3), ..., log_2(p) by numbers of the form m/n. In this case we add the condition that we also want to approximate log_2(5/3), etc. but to get the dimensions right I still scale by n^(1+1/d), since only d of these are independent.
Message: 1728 - Contents - Hide Contents
Date: Wed, 3 Oct 2001 16:05 +01
Subject: Re: How consistent are cents?
From: graham@xxxxxxxxxx.xx.xx
In-Reply-To: <OF6575791C.83BBE7D8-ONC1256ADA.004B4ECC@xxx.xx>
I agree with Manuel: 1200-equal is 11-limit consistent.
Graham
Message: 1729 - Contents - Hide Contents Date: Wed, 03 Oct 2001 18:40:48 Subject: Re: Digest Number 124 From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Well . . . 81:80 being a commatic unison vector sort of implies >> tempering . . . but in no way selects 12 over other meantones like >> 19, 31, 43, 55, or 74. >> Those are all of the form n h7 + m h5, which is what I was talking > about.Gene, I really liked your n h7 + m h5 discussion!
Message: 1730 - Contents - Hide Contents Date: Wed, 03 Oct 2001 18:48:05 Subject: Re: How consistent are cents? From: Paul Erlich --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:> > Paul wrote:>> Again, I don't know what you mean by consistent here. 1200-tET is >> consistent through the 9-limit, but not the 11-limit. >> Are you sure? According to my calculation, 1200-tET is 12-limit > consistent (hey, funny). This is the integer limit, Paul > prefers to use the odd limit which is one less for octave > divisions. The 12-limit region is 1199.97488 - 1200.01221 tET. > The margin is very small, 11/7 is almost halfway between two > steps: 782.49 cents. > > ManuelHi Manuel, you are right. 1200-tET is 11-limit consistent, but not 13- limit consistent. I typed "1" instead of "11" in my program and got an error, so I mistakenly concluded that 11 didn't work. It does.
Message: 1731 - Contents - Hide Contents Date: Wed, 03 Oct 2001 19:59:41 Subject: Re: How consistent are cents? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Hi Manuel, you are right. 1200-tET is 11-limit consistent, but not 13- > limit consistent. I typed "1" instead of "11" in my program and got > an error, so I mistakenly concluded that 11 didn't work. It does.In any case, what I gave was not a measure of consistency, but a consistent measure. I enforce consistency, and then simply give a measure of how far out of tune the result is in a relative sense.
Message: 1732 - Contents - Hide Contents
Date: Wed, 03 Oct 2001 06:47:58
Subject: Re: blackjack "problem" solved (for Gene)
From: Pierre Lamothe
Gene,
I post here only a part of my answer to your questions. I
will continue as soon as I can. You wrote:
<< "Multilinear" as in tensor or wedge products, or what
exactly do you mean? >>
As you can see here
<http://www.aei.ca/~plamothe/sys72/re1215.gif - Type Ok * [with cont.] (Wayb.)>
the ib1215 set is convex in the lattice <3 5 7 11> z^4 and
that in the current sense (i.e. any segment between nodes
has no hole, the intermediate nodes are occupied). Multilinear
refer simply to the fact it's a 4D convex body (differ in R^4).
*** I note for anyone else that <3 5 7 11> == [42 23 58 33]
yellow == 3, red == 5, green == 7, blue == 11
so it's very easy to recognize the axis and to know the
JI corresponding values. Nodes have color of the minimal
N-limit.
You wrote:
<< It seems to me that a sequence defined by one generator
is trivially convex. >>
Sure. What is in question is the apparently obvious choice of
a convex segment centered on 0 (of a such linear sequence) to
obtain a pertinent object in musical sense.
That follows by simple imitation of the pure fifth generator.
There is no problem with the powers of the unique prime 3 for
there exist only two JI steps and a unique unison vector. In
each turn of the octave a new unison vector appears and the
precedent one replace a precedent step.
When a microgenerator like the Secor 7/72 is chosen for its
quality of very good approximation of the intervals in the
11-limit, if there is pretention to obtain more than a space
where possible tempered harmonies in 11-limit are very good, it
appears to me that one have to pay attention at what differs
from the one dimensional fifth generator.
Obviously, if someone believes JI reference is useless to build
scales in the melodic sense, I would say "Continue to imitate a
well-known simple schema and build long scales". I believe, for
my part, that if JI reference is important for harmony, it remains
important for melodic structuration since I discovered that the
two "logic" are inextricably related in algebraic structures.
The gammier theory indicates that the longest non-degenerated
structures (definition forward) within both the odd 21-limit and
the prime 7-limit or 11-limit (i.e. using lattices <2 3 5 7> Z^4
or <2 3 5 7 11> Z^5) have 10 degrees and there exist only two
such structures:
ib1183 in 7-limit
ib1215 in 11-limit.
Comparing the 41 intervals of ib1215 with the 41 intervals of
the convex linear Miracle sequence, it is easy to show there is
few differences in the following like-keyboard representation
<http://www.aei.ca/~plamothe/sys72/cl1215.gif - Type Ok * [with cont.] (Wayb.)>
Which of these two has more sense for melodical structuration?
First, I argue against the Miracle sequence that as long as one
see within it a reflect of an underlying decatonic structure one
have to admit that this sequence contains unison vectors, namely
2, 4, 68, 70. (What ask to distinguish tuning and structure).
For sure, imitating the fifth generator, one could consider that
sequence as a bivalent 41-tone scale 0 2 4 5 7 9 11 12 ... having
1 and 2 as steps and also 1 (whoops!) as unison vector (i.e.
step difference). Yet one could consider it as a bunch of bivalent
21-tone scales, having 2 and 5 as steps and also (whoops!) 3 as
unison vector.
68 70 72
61 63 65 67
54 56 58 60
47 49 51 53
40 42 44 46
33 35 37 39
26 28 30 32
19 21 23 25
12 14 16 18
5 7 9 11
0 2 4
Besides their so much logic quality one could ask Joseph Perhson
to verify their melodic practicability.
-----
Looking now at ib1215 as a simple set, it is easy to find first
it contains 6 atoms (i.e. intervals that can't be factorized
into lesser intervals in the set), namely
<22/21 21/20 16/15 15/14 12/11 10/9>
If ib1215 is coherent (as set), there exist a unique epimorphism
H : ib1215 --> Z
such that
H(22/21) = H(21/20) = ... = H(10/9) = 1
what is the case and then
H(2/1) = 10.
As any gammier structure ib1215 contains more than one periodicity
block whose intersection is the unison.
I add that the corresponding tempered values in 72-tET are
5 == <22/21> == <21/20>
7 == <16/15> == <15/14>
9 == <12/11>
11 == <10/9>
and that the only cases of ambiguity in detempering are 5, 7 (and
octave reverse).
PROBLEM
(a) Try to make that with the Miracle set whose atoms
have to correspond to 2 and 5.
(b) Where's the evidence that that may correspond to a
decatonic structure?
(c) What would be the atoms removing 2 and 4? and which
periodicity could be then possible?
-----
... to continue
Pierre
Message: 1733 - Contents - Hide Contents
Date: Wed, 3 Oct 2001 14:32 +01
Subject: Temperament finder update
From: graham@xxxxxxxxxx.xx.xx
I noticed a mistake in my temperament finder program. For alternative
equivalence intervals and timbres, I wasn't using these to select the
consistent ETs. So, although all the temperaments found are still valid,
some would previously have been missing.
Schismic temperament is now included in the standard 5-limit list. I
don't think that's because of anything I changed this time. It probably
came in when I increased the number of ETs considered from 20 to 21, so
that schismic would appear as a 5-limit microtemperament. To make this
appear less arbitrary, I've made the number of ETs 25 now. It doesn't
make much difference.
This looks like a good temperament from the "9-limit" nonoctave tubulong
list:
1/5, 46.9 cent generator (really 70.4 cents)
basis:
(0.16666666666666666, 0.03910923891316731)
mapping by period and generator:
[(6, 0), (10, -1), (13, -2), (15, -1)]
mapping by steps:
[(24, 6), (39, 10), (50, 13), (59, 15)]
unison vectors:
[[2, -1, 1, -1], [1, 8, -2, -4]]
highest interval width: 2
complexity measure: 12 (18 for smallest MOS)
highest error: 0.003877 (4.653 cents) (really 7.0 cents)
The period is 1/4 of an octave, and you need three notes within that
period. So it works out at 12 notes per octave.
The scripts have been updated at
<# Temperament finding library -- definitions * [with cont.] (Wayb.)>
<#!/usr/local/bin/python * [with cont.] (Wayb.)>
<import temper, string * [with cont.] (Wayb.)>
<import temper, string * [with cont.] (Wayb.)>
and the output files that have changed significantly
<2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ... * [with cont.] (Wayb.)>
<2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ... * [with cont.] (Wayb.)>
<2 10 12 14 16 18 20 22 24 26 30 34 38 44 46 50... * [with cont.] (Wayb.)>
<2 4 6 10 14 16 18 20 22 26 30 34 36 38 44 46 5... * [with cont.] (Wayb.)>
<2 4 6 10 14 16 18 20 22 26 30 34 36 38 44 46 5... * [with cont.] (Wayb.)>
<2 4 6 16 18 20 22 36 38 44 52 54 60 66 68 76 7... * [with cont.] (Wayb.)>
<2 4 6 16 20 38 48 52 54 68 76 78 82 92 96 98 1... * [with cont.] (Wayb.)>
<2 4 6 16 20 38 48 52 54 68 76 78 82 92 96 98 1... * [with cont.] (Wayb.)>
<1 2 3 5 6 8 9 10 11 13 14 15 16 17 19 20 21 23... * [with cont.] (Wayb.)>
<404 Not Found * [with cont.] Search for http://www.microtonal.co.uk/limit7.tubulong.nonoctave in Wayback Machine>
<5 6 8 11 13 16 19 24 30 32 35 37 43 46 54 59 6... * [with cont.] (Wayb.)>
<6 8 13 16 19 24 30 35 37 43 59 72 78 88 102 12... * [with cont.] (Wayb.)>
<13 16 19 30 35 37 72 102 136 142 147 152 160 1... * [with cont.] (Wayb.)>
<13 16 19 35 102 136 147 160 166 179 182 195 20... * [with cont.] (Wayb.)>
Graham
Message: 1734 - Contents - Hide Contents Date: Wed, 3 Oct 2001 15:40:56 Subject: Re: Tetrachordality and Scala (was Re: semi-periodic scales) From: manuel.op.de.coul@xxxxxxxxxxx.xxx [me] Perhaps an average value for transpositions by a fifth starting on all keys of the scale. [Carl] Can you illustrate with an example? Let's forget it, on second thought it was a bad idea.>This is exactly what we want. Except, instead of comparing >the scale and its rotation to a degree near a 3:2, we compare >it with its transposition by an exact 3:2, simply taking the >lowest value for all rotations of the transposed version, >using a simple ordered pairing of intervals in each case.Can _you_ illustrate with an example? Manuel
Message: 1735 - Contents - Hide Contents Date: Wed, 3 Oct 2001 15:48:32 Subject: Re: How consistent are cents? From: manuel.op.de.coul@xxxxxxxxxxx.xxx Paul wrote:>Again, I don't know what you mean by consistent here. 1200-tET is >consistent through the 9-limit, but not the 11-limit.Are you sure? According to my calculation, 1200-tET is 12-limit consistent (hey, funny). This is the integer limit, Paul prefers to use the odd limit which is one less for octave divisions. The 12-limit region is 1199.97488 - 1200.01221 tET. The margin is very small, 11/7 is almost halfway between two steps: 782.49 cents. Manuel
Message: 1736 - Contents - Hide Contents Date: Thu, 04 Oct 2001 17:47:31 Subject: Re: 72 owns the 11-limit From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> I just finished an interesting calculation, where I took the nine > smallest superparticular ratios belonging to the 11-limit, namely > 225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024, > 4375/4374 and 9801/9800. I then checked all 126 4-element subsets, > finding 69 of rank 4. Astonisingly, all 69 had the 72 et as the > generator of its null space; in 59 of those cases the determinant of > the 5x5 matrix with 5 indeterminates for the first row was +-h72.What was it in the other 10 cases? I did some similar calculations for the 7-limit a couple of years ago; I found that, for the smallest of Fokker's unison vectors, 22 and 171 were the "co-owners".> > At the moment, at least until my > mother has her brain surgery,My prayers for a successful operation and a full recovery.
Message: 1737 - Contents - Hide Contents Date: Thu, 04 Oct 2001 19:01:25 Subject: 3rd-best 11-limit temperament From: Paul Erlich While the top two temperaments in Graham's 11-limit list are essentially 31-out-of-72 and 46-out-of-72, the third one has the lowest complexity measure of all in this list. Can anyone discuss this, in terms of unison vectors, etc.? 12/53, 271.1 cent generator basis: (1.0, 0.22594789337911292) mapping by period and generator: [(1, 0), (0, 7), (3, -3), (1, 8), (3, 2)] mapping by steps: [(31, 22), (49, 35), (72, 51), (87, 62), (107, 76)] unison vectors: [[7, 8, 0, -7, 0], [-21, 3, 7, 0, 0], [-21, -2, 0, 0, 7]] highest interval width: 17 complexity measure: 17 (22 for smallest MOS) highest error: 0.007764 (9.317 cents) Why is this better than an ME 22-out-of-46, which has a maximum error of 8.6 cents in the 11-limit, probably reducable further in a non-ET setting?
Message: 1738 - Contents - Hide Contents Date: Thu, 04 Oct 2001 04:23:33 Subject: 72 owns the 11-limit From: genewardsmith@xxxx.xxx I just finished an interesting calculation, where I took the nine smallest superparticular ratios belonging to the 11-limit, namely 225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024, 4375/4374 and 9801/9800. I then checked all 126 4-element subsets, finding 69 of rank 4. Astonisingly, all 69 had the 72 et as the generator of its null space; in 59 of those cases the determinant of the 5x5 matrix with 5 indeterminates for the first row was +-h72. It seems to me I had better check some other prime limits, and it would be helpful if someone could post some of the Xenharmonicon data so I don't need to recalculate it. At the moment, at least until my mother has her brain surgery, I don't feel I am able to leave the house for long and do this myself.
Message: 1739 - Contents - Hide Contents Date: Thu, 4 Oct 2001 00:48:26 Subject: OFFLIST Re: 72 owns the 11-limit From: monz Hi Gene, 1st) Just nitpicking here: Technically, the name of the journal published in the 1970s and 80s by John Chalmers, continued by Daniel Wolf, and then picked up again by Dr. Chalmers, is "Xenharmonikôn". Most email posters don't worry about the "ô", but everyone uses the "k". 2nd) My prayers are with your mother for her surgery. Hope all goes well. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" ----- Original Message ----- From: <genewardsmith@xxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Wednesday, October 03, 2001 9:23 PM Subject: [tuning-math] 72 owns the 11-limit> I just finished an interesting calculation, where I took the nine > smallest superparticular ratios belonging to the 11-limit, namely > 225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024, > 4375/4374 and 9801/9800. I then checked all 126 4-element subsets, > finding 69 of rank 4. Astonisingly, all 69 had the 72 et as the > generator of its null space; in 59 of those cases the determinant of > the 5x5 matrix with 5 indeterminates for the first row was +-h72. > > It seems to me I had better check some other prime limits, and it > would be helpful if someone could post some of the Xenharmonicon data > so I don't need to recalculate it. At the moment, at least until my > mother has her brain surgery, I don't feel I am able to leave the > house for long and do this myself. > > > > 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.) > _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 1740 - Contents - Hide Contents Date: Thu, 4 Oct 2001 00:58:09 Subject: OFFLIST Re: 72 owns the 11-limit From: monz Oops... my bad. It was supposed to be OFFLIST. Sorry, Gene. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 1741 - Contents - Hide Contents Date: Thu, 04 Oct 2001 10:34:12 Subject: Re: blackjack "problem" solved (for Gene) From: Pierre Lamothe Gene, I'll wait later to continue my answer. Hope all goes well for your mother. Pierre
Message: 1742 - Contents - Hide Contents Date: Fri, 05 Oct 2001 18:46:38 Subject: Re: 3rd-best 11-limit temperament From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:>> Why is this better than an ME 22-out-of-46, which has a maximum error >> of 8.6 cents in the 11-limit, probably reducable further in a non- ET >> setting? >> Presumably, you mean >>>>> temper.Temperament(46, 22, temper.primes[:4]) >> 3/34, 52.2 cent generator > > basis: > (0.5, 0.043499613319368802) > > mapping by period and generator: > [(2, 0), (3, 2), (5, -4), (5, 7), (7, -1)] > > mapping by steps: > [(46, 22), (73, 35), (107, 51), (129, 62), (159, 76)] > > unison vectors: > [[-3, -1, -1, 0, 2], [4, 0, -2, -1, 1], [0, -5, 1, 2, 0]] > > highest interval width: 11 > complexity measure: 22 (24 for smallest MOS)24? What about the 22-tone MOS?> highest error: 0.007005 (8.406 cents) > > The increase in complexity is greater than the improved accuracy. OK.
Message: 1743 - Contents - Hide Contents Date: Fri, 05 Oct 2001 11:51:26 Subject: Re: Tetrachordality and Scala From: Carl Lumma>> >his is exactly what we want. Except, instead of comparing >> the scale and its rotation to a degree near a 3:2, we compare >> it with its transposition by an exact 3:2, simply taking the >> lowest value for all rotations of the transposed version, >> using a simple ordered pairing of intervals in each case. >>Can _you_ illustrate with an example?Maybe I shouldn't have said exactly. Since we're dealing with transposition by a perfect 3:2 we can't compare 2nds because they're the same. However, since I can losslessly convert a list of 2nds into a list of "pitches (intervals measured from a 1/1) and back again, I assume I can plug a list of "pitches" (1/1 9/8 5/4 etc.) into the formula and preserve the equivalence to the list of intervals in the set of all subsets of the scale that we agreed on... Pentatonic scale in 12-tET: scale- 0 200 500 700 900, sum of squares= 1,590,000 *(3:2), mod (2:1)- 702 902 2 202 402, corr= 0.430566 mode 2- 902 2 202 402 702, corr= 0.638113 mode 3- 2 202 402 702 902, corr= 0.971446 mode 4- 202 402 702 902 2, corr= 0.669559 mode 5- 402 702 902 2 202, corr= 0.487169 tetrachordality index= 0.971446 ...Using pitches may goof up the formula, though, since lists of 2nds must sum to an octave while lists of "pitches" do not. I don't know enough about how the correlation formula works statistically to tell if this is a problem. I roughly remember using a "correlation" in Physics class to measure the fit of a linear function to a set of data... finding the best fit... but I don't remember the math. I am sure the first pitch = zero causes a problem... 6-tET: scale- 0 200 400 600 800 1000, sum of squares= 2,200,000 *(3:2), mod (2:1)- 702 902 1102 102 302 502, corr= 0.648181 mode 2- 902 1102 102 302 502 702, corr= mode 3- 1102 102 302 502 702 902, corr= mode 4- 102 302 502 702 902 1102, corr= 1.139090 mode 5- 302 502 702 902 1102 102, corr= 0.866363 -- d'oh! mode 6- 502 702 902 1102 102 302, corr= ...Here mode 5 is closer to 1 than mode 4. There's probably a way around this, but maybe we don't need correlation at all -- just the mean difference between the pitches in the best-fit mode of the transposed scale from the pitches in the original scale. It might also be argued that we don't want to enforce order of the pitches during comparison. Rather than testing each transposed mode for the best fit, we could just line up the new pitches with their nearest partners in the original scale, in any order. I suspect this would break our agreed-upon equivalence with the intervals of all subets. But do we care about an interval pattern or pitches, as far as the perceptual basis for this measure? You there, Paul? -Carl
Message: 1744 - Contents - Hide Contents Date: Fri, 05 Oct 2001 05:03:58 Subject: Re: 3rd-best 11-limit temperament From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> While the top two temperaments in Graham's 11-limit list are > essentially 31-out-of-72 and 46-out-of-72, the third one has the > lowest complexity measure of all in this list. Can anyone discuss > this, in terms of unison vectors, etc.?One thing you might note about it is that it is the 22-31 linear temperament, with tunings calculated by Graham's minimax condition. If we set B = 2^(0.2259478...), then his basis is 2^r B^s; we could also have a basis U = 2^5 B^(-22), V = 2^(-7) B^31 and in this basis we approximate rational numbers q in the 11-limit by U^h31(q) V^h22(q).
Message: 1745 - Contents - Hide Contents Date: Fri, 05 Oct 2001 18:51:23 Subject: Re: 3rd-best 11-limit temperament From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., genewardsmith@j... wrote: >>> One thing you might note about it is that it is the 22-31 linear >> temperament, with tunings calculated by Graham's minimax condition. >> If we set B = 2^(0.2259478...), then his basis is 2^r B^s; we could >> also have a basis U = 2^5 B^(-22), V = 2^(-7) B^31 and in this > basis>> we approximate rational numbers q in the 11-limit by >> U^h31(q) V^h22(q). >> Another thing to note is how easy it is to find the generators and > MOS scales if you don't proceed all the way down to the PB. In the > 22-31 case, we have a continued fraction [1,2,2,4] for 31/22; the > penultimate convergent is 7/5, and our generator is U^7 V^5. For the > 41-31 temperament, we have a continued fraction [1, 3, 10], which > leads to a penultimate continued fraction 4/3, and a generator > W^4 X^3 which is the miracle generator; here W and X are to be > selected by some optimality condition, such as minimax or least > squares on a tonality diamond (if I have the jargon right.)Well, that's Graham's way of putting it . . . Anyway, the rest of this stuff looks wonderful, but needs to be presented in a much more user-friendly way for musicians. I think you could do a great service by putting all these ideas in a "for dummies" kind of document, with familiar examples and such. If you wrote such a paper in the next couple of months, you could probably get it published in Xenharmonikon 18 . . .
Message: 1746 - Contents - Hide Contents Date: Fri, 05 Oct 2001 06:18:49 Subject: Re: 3rd-best 11-limit temperament From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> One thing you might note about it is that it is the 22-31 linear > temperament, with tunings calculated by Graham's minimax condition. > If we set B = 2^(0.2259478...), then his basis is 2^r B^s; we could > also have a basis U = 2^5 B^(-22), V = 2^(-7) B^31 and in this basis > we approximate rational numbers q in the 11-limit by > U^h31(q) V^h22(q).Another thing to note is how easy it is to find the generators and MOS scales if you don't proceed all the way down to the PB. In the 22-31 case, we have a continued fraction [1,2,2,4] for 31/22; the penultimate convergent is 7/5, and our generator is U^7 V^5. For the 41-31 temperament, we have a continued fraction [1, 3, 10], which leads to a penultimate continued fraction 4/3, and a generator W^4 X^3 which is the miracle generator; here W and X are to be selected by some optimality condition, such as minimax or least squares on a tonality diamond (if I have the jargon right.) This is easy enough that I've been meaning to suggest that Manuel consider putting into Scala a routine to calculate Gen(m, n, p) and Mos(n,m,p) for two ets m and n and a prime limit p; in case m and n are not relatively prime this needs to be adjusted by working inside of the interval of repetition. Of course one can also think of this in terms of the ets generated by linear combinations of hm and hn, as for instance h53 = h22 + h31 and h72 = h31 + h41.
Message: 1747 - Contents - Hide Contents Date: Fri, 05 Oct 2001 18:54:11 Subject: Re: Tetrachordality and Scala From: Paul Erlich --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> But do we care about an > interval pattern or pitches, as far as the perceptual basis for > this measure? You there, Paul? >I'm afraid I've lost you on this. I defined omnitetrachordality on the tuning list, and Gene said he was going to come up with some theorems about it . . .
Message: 1748 - Contents - Hide Contents Date: Fri, 05 Oct 2001 03:50:57 Subject: Convexity lemma From: Pierre Lamothe --------------- Convexity lemma --------------- Let L be a lattice of rank k in which is defined an epimorphism L --> Z having a periodicity p in the octave, and within it, a set S of intervals modulo 2 . Any convex segment of S, containing more than 2p-1 elements, and having an element in each class, contains a unison vector. ----- Indeed, any linear sequence in L/<2>Z has a periodicity equal to 1 or p or any factor of p. A line may contain an infinite convex sequence of elements without any element of class 0 if its periodicity is less than p. A convex segment containing an element of each class has one of it in class 0 and has the periodicity p. Since it contains more than 2p-1 elements, it contains forcely two elements of class 0 due to the periodical return for each of its classes. One of these elements may be the unison, the other is forcely a unison vector. -------------------- Plunging the lattice L in an Euclidean space to obtain a linear temperament with an orthogonal projection on a line, the longest convex sequence possible, without a hole corresponding to a unison vector, and containing a mode (so an element in each class), is then equal to 2p-1, for a good temperament don't permit that an element of another class, in the same vicinity, might be projected at the same place than the unison vector. Pierre
Message: 1749 - Contents - Hide Contents Date: Fri, 05 Oct 2001 19:16:14 Subject: Re: 3rd-best 11-limit temperament From: Paul Erlich --- In tuning-math@y..., graham@m... wrote: