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Message: 3800 - Contents - Hide Contents
Date: Mon, 4 Feb 2002 12:54 +00
Subject: Best Partch keyboard mapping
From: graham@xxxxxxxxxx.xx.xx
The question was raised last week as to what MOS is "best" for the Partch
43 note scale. From what we already know there are three candidates:
Wilson's 11-limit schismic mapping gets all 43 notes into a 41 note MOS.
Of course this can't be unique, but all of the 41 notes have at least one
counterpart in the Partch scale.
Secor's miracle mapping gets the 11-limit diamond into a 45 note generator
chain. Each interval is represented uniquely. Unfortunately, the Genesis
scale requires more than 45 notes, and no scale with a complete diamond
can fit into the 41 note MOS. The next biggest MOS is 72, which is quite
complex.
Any 72 note keyboard can get the scale uniquely, and be tempered to a high
degree of accuracy.
My criteria for "best" mapping are:
All ratios expressed uniquely (no split keys)
All 43 notes fit in a small MOS. This also means the scale is a subset of
a simple periodicity block.
Looking through the scales in <22 26 29 31 41 46 58 72 80 87 89 94 111 113 11... * [with cont.] (Wayb.)> it
looks like the simplest, unique temperament there is h41&h58. This gets
the whole diamond into the 58 note MOS.
I've written a script to check that both versions of the 43 note scale lie
within the 58 note MOS and it runs okay. I've included it below for those
receiving e-mails who don't lose the indentation. The most complex
intervals are 16:15 and 15:8.
So this 58 note mapping looks like the best one by these criteria. You
may be able to do better by not starting with consistent ETs.
import re, temper
h58 = temper.PrimeET(58,temper.primes[:4])
h41 = temper.PrimeET(41,temper.primes[:4])
scale = h41&h58
for scl in ("partch_43.scl", "partch_43a.scl"):
lastStepSize = 0
for n, d in re.findall(r'(?m)^ (\d+)/(\d+)', open(scl).read()):
vector = temper.factorizeRatio(int(n), int(d))[:5]
octs, gens = scale.byMapping(vector)
if abs(gens)>27:
print "%s/%s too big" %(n, d)
stepSize = temper.dotprod(vector, h58.basis)
if stepSize <= lastStepSize:
print "%s/%s not in sequence" %(n,d)
lastStepSize = stepSize
Graham
Message: 3801 - Contents - Hide Contents Date: Mon, 4 Feb 2002 09:08:14 Subject: Re: a notation for Schoenberg's rational implications From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 03, 2002 10:14 PM > Subject: [tuning-math] Re: a notation for Schoenberg's rational implications > > > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>>> it's a chromatic unison vector with respect to a >>> 7-tone PB in an MOS tuning, not with respect to a >>> 12-tone PB in an equal tuning. In the latter it >>> functions as a step vector -- but is a very unlikely >>> choice for one. >>>> It worked, though. >> well clearly the formula wasn't depending on it to any > great extent. no ratios of 11 or multiples of 11 even > made it into the block.yes, i was curious about the 11-limit PB without 7 that Gene derived for Schoenberg way back around Christmas ... if you recall, i posted a question then wondering why there were no 7-limit ratios. so i was very surprised here to see that the PB that i finally derived, as the one which i think represents the first 184 pages of _Harmonielehre_ the best, is only 5-limit! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3802 - Contents - Hide Contents Date: Mon, 04 Feb 2002 05:18:44 Subject: Re: a notation for Schoenberg's rational implications From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> >> so the PB comes _without_ using 441:440? > >> well, no ... when i use 2:1 instead of 441:440,not a good idea to use 2:1 as a unison vector, especially if you're trying to do thing's gene's way.> as i said, i have to use a chromatic unison-vector in > addition to all the commatic ones, in order to get the > full PB. > >>> anyway, how strange to call that a chromatic unison vector. >> are you sure you got that from me? > >> n o ! i didn't get t h a t from you. i got the > necissity of having a chromatic unison-vector from you.this sure isn't an example of where there's such a necessity, or even such a possibility.> i got the idea to use 441:440 as a chromatic unison-vector > when i derived it from the Minkowski-reduced version of the > Schoenberg PB which Gene and you calculated, and you > pointed out to me that it was not a commatic unison-vector > because in fact the pitches separated by it did involve a > change of accidental.it's a chromatic unison vector with respect to a 7-tone PB in an MOS tuning, not with respect to a 12-tone PB in an equal tuning. In the latter it functions as a step vector -- but is a very unlikely choice for one.> so i replaced the 2:1 i had in my matrix with that, and > _voilą_! -- out came the PB!you should clarify with gene what goes into the formula you're using. i expect he'll confirm that what you're putting in is not a unison vector (or in the kernel).> but as you can see from my subsequent post, i believe > there's more validity to Schoenberg's actual theory in > using 45:44 as a chromatic unison-vector instead, and > it does result in a scale which has one different note.of course all these scales are just representing 12-et, so there's no point in 'fretting' over single notes. remember that the parallelogram construction is arbitrary, and differences in it and other constructions are only meaningful when you're not tempering out all the unison vectors. in order to get the intervals to work the way schoenberg intended, you have to temper out all the unison vectors.
Message: 3803 - Contents - Hide Contents Date: Mon, 4 Feb 2002 10:47:19 Subject: interval class From: monz how about a good definition for "interval class"? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3804 - Contents - Hide Contents Date: Mon, 04 Feb 2002 05:22:19 Subject: Re: Gene's notation formula: alternate duodene? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> ok ... i'm s t i l l confused about equivalence interval, > period, and unison-vector, but i thought i had the difference > between commatic and chromatic unison-vectors. > > now, what's the difference between a chromatic unison-vector > and a step-vector?a chromatic unison vector corresponds to zero steps in the PB -- it just takes you from one variant of a note to another.> in the 12-tone scale, of which the duodene is an example, > there is a difference between C and C#, and in this case > that difference is most likely to be 25:24.but 25:24 is a chromatic unison vector with respect to the 7-tone PB that is the diatonic scale, but not in the case of a 12-tone PB.> so why is that a step (which i would equate with a > "diatonic semitone") and not a chromatic unison-vector > (which i would equate with a "chromatic semitone")?clearly in this case, a 12-tone closed system, there is no difference between a diatonic semitone and a chromatic semitone. in a 7-tone PB, step vectors would include 16:15, 10:9, and 9:8.
Message: 3805 - Contents - Hide Contents Date: Mon, 4 Feb 2002 11:10:21 Subject: a pile of new definitions From: monz thanks to Manuel ... Rothenberg efficiency Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) Rothenberg stability Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) Rothenberg redundancy Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) Lumma stability Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) Lumma impropriety Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) as always, i encourage others to submit corrections, criticisms, elaborations, and lots of diagrams! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3806 - Contents - Hide Contents Date: Mon, 4 Feb 2002 12:18:27 Subject: the missing consistencies From: monz ok, i just recently updated the links in my "consistency" definition Definitions of tuning terms: consistent, (c) 1... * [with cont.] (Wayb.) and now i'm getting a "page cannot be found" error when i try to view them where are they hiding now? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3807 - Contents - Hide Contents Date: Mon, 04 Feb 2002 05:31:13 Subject: step vectors and chromatic unison vectors From: paulerlich in gene's notation system just posted to the tuning list, 9:8 is a step vector and all the others are independent chromatic unison vectors. there are no commatic unison vectors because every JI ratio gets a distinct notation -- nothing is tempered out or ignored.
Message: 3808 - Contents - Hide Contents Date: Mon, 4 Feb 2002 12:26:41 Subject: idea for a personal tuning experiment From: monz i thought of a great idea for a tuning experiment. some of you (i can remember Carl) like my _24-eq tune_ Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) i wrote this piece because many tuning theorists give 24-edo a bad rap -- it's inconsistent, etc. -- and i've always liked it, so i wanted to write a piece using it that i thought sounded cool. i didn't approach the task analytically this time. from the webpage:>> Unusually for me, with this piece I have no idea >> what the rational implications of the harmony are >> - I composed it strictly 'by ear', concentrating more >> on the counterpoint than on the harmony. I tried to >> make logical-sounding chord progressions, but I've >> never analyzed what they are. Someday I would like >> to figure them outso here's my idea: can we find several different JI interpretations of this short piece, and then i'll make MIDIs of all the different versions and compare them with the original 24-edo version, and see which rational implications i think are the ones i most likely intended. obviously, this won't have much generality, and i don't claim for it to have validity for anyone else. but it would be interesting for me to clarify what i was doing when i wrote this piece. the one clue i can give about the harmonic implications in this tune: the last measure of the 5-measure phrase uses only even-numbered degrees of the scale in all instruments, so it's 12-edo, which i suppose implies basically a pythagorean tuning for that measure. what about the first four measures? i'll supply any data that's needed, but it should all be easily decipherable from the MIDI-file on the webpage Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) happy hunting -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3809 - Contents - Hide Contents Date: Mon, 04 Feb 2002 05:32:14 Subject: Re: Gene's notation formula: alternate duodene? From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>>> From: paulerlich <paul@s...> >> To: <tuning-math@y...> >> Sent: Sunday, February 03, 2002 9:22 PM >> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene? >> >> > >> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> in the 12-tone scale, of which the duodene is an example, >>> there is a difference between C and C#, and in this case >>> that difference is most likely to be 25:24. >>>> but 25:24 is a chromatic unison vector with respect to the 7-tone PB >> that is the diatonic scale, but not in the case of a 12-tone PB. >>>>> so why is that a step (which i would equate with a >>> "diatonic semitone") and not a chromatic unison-vector >>> (which i would equate with a "chromatic semitone")? >>>> clearly in this case, a 12-tone closed system, there is no difference >> between a diatonic semitone and a chromatic semitone. in a 7-tone PB, >> step vectors would include 16:15, 10:9, and 9:8. > > >> ok ... i think i got it, but it sure is confusing stuff ... > > Paul, i think you might be confusing some of the posts > i sent in today. some are about the Schoenberg 12-tone PB, > which is definitely meant to be 12-edo with all unison-vectors > tempered out, but others are about the duodene, which Ellis > meant to be JI with nothing tempered out.25:24 is still a step vector in the duodede. look! you can find it between distinct pitches in the duodene!
Message: 3810 - Contents - Hide Contents Date: Mon, 04 Feb 2002 20:36:30 Subject: Re: Best Partch keyboard mapping From: paulerlich --- In tuning-math@y..., graham@m... wrote:> The question was raised last week as to what MOS is "best" for the Partch > 43 note scale. From what we already know there are three candidates: > > Wilson's 11-limit schismic mapping gets all 43 notes into a 41 note MOS. > Of course this can't be unique, but all of the 41 notes have at least one > counterpart in the Partch scale. > > Secor's miracle mapping gets the 11-limit diamond into a 45 note generator > chain. Each interval is represented uniquely. Unfortunately, the Genesis > scale requires more than 45 notes, and no scale with a complete diamond > can fit into the 41 note MOS. The next biggest MOS is 72, which is quite > complex. > > Any 72 note keyboard can get the scale uniquely, and be tempered to a high > degree of accuracy. > > > My criteria for "best" mapping are: > > All ratios expressed uniquely (no split keys) > > All 43 notes fit in a small MOS. This also means the scale is a subset of > a simple periodicity block. > > Looking through the scales in <22 26 29 31 41 46 58 72 80 87 89 94 111 113 11... * [with cont.] (Wayb.)> it > looks like the simplest, unique temperament there is h41&h58. This gets > the whole diamond into the 58 note MOS.perhaps you or gene could tell us the errors and unison vectors here.
Message: 3811 - Contents - Hide Contents Date: Mon, 04 Feb 2002 06:04:52 Subject: Re: Gene's PB formula, generalized (was: a notation for Schoenberg's...) From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> i don't really know what to call them, so i'll just make this > do: {hv, hw, hx, hy, hz}. it's the top row of numbers in the > adjoint (or is it a unimodular inverse?) of the kernel.I would call those hv(2), hw(2), hx(2), hy(2) and hz(2), where the h's are vals, which you can equate to column vectors.
Message: 3812 - Contents - Hide Contents Date: Mon, 04 Feb 2002 20:38:35 Subject: Re: idea for a personal tuning experiment From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> the one clue i can give about the harmonic implications > in this tune: the last measure of the 5-measure phrase > uses only even-numbered degrees of the scale in all > instruments, so it's 12-edo, which i suppose implies > basically a pythagorean tuning for that measure.you don't think 12-equal can imply 5-limit? i sure do.
Message: 3813 - Contents - Hide Contents Date: Mon, 4 Feb 2002 13:01:02 Subject: Re: idea for a personal tuning experiment From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, February 04, 2002 12:38 PM > Subject: [tuning-math] Re: idea for a personal tuning experiment > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> the one clue i can give about the harmonic implications >> in this tune: the last measure of the 5-measure phrase >> uses only even-numbered degrees of the scale in all >> instruments, so it's 12-edo, which i suppose implies >> basically a pythagorean tuning for that measure. >> you don't think 12-equal can imply 5-limit? i sure do.yup, i sure do too. in keeping with the overall description of the experiment, i should definitely have mentioned the "double entendre" 3-/5-limit implications of 12-edo. guess that was the theorist getting the best of the musician in me ... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3814 - Contents - Hide Contents Date: Mon, 04 Feb 2002 06:07:38 Subject: Re: Gene's PB formula, generalized (was: a notation for Schoenberg's...) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> i don't really know what to call them, so i'll just make this >> do: {hv, hw, hx, hy, hz}. it's the top row of numbers in the >> adjoint (or is it a unimodular inverse?) of the kernel. >> I would call those hv(2), hw(2), hx(2), hy(2) and hz(2), where theh's are vals, which you can equate to column vectors. So why is there an "extra vector" used in constructing the PB? Is this just an arbitrary vector independent of the others, or is it a step vector, or what?
Message: 3815 - Contents - Hide Contents Date: Mon, 04 Feb 2002 21:29:44 Subject: Re: Best Partch keyboard mapping From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> perhaps you or gene could tell us the errors and unison vectors here.I found all of my so-called "standard" ets up to 100 which represent each note of Genesis, and obtained the following: 58 .3529661000 7.302746897 65 .6150147495 11.35411845 72 .2346001000 3.910001666 73 .5954749001 9.788628497 80 .4116062001 6.174093001 84 .6814441795 9.734916846 87 .4564030700 6.295214756 89 .4717246800 6.360332766 91 .6610206501 8.716755827 94 .3698782499 4.721849998 95 .8571248997 10.82684084 The second column is maximum relative error, and the third maximum 11-limit consonance error in cents. I'm not sure what you want unisons for--do you want to find an 11-limit block containing Genesis?
Message: 3816 - Contents - Hide Contents Date: Mon, 04 Feb 2002 06:07:50 Subject: Re: a notation for Schoenberg's rational implications From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> it's a chromatic unison vector with respect to a 7-tone PB in an MOS > tuning, not with respect to a 12-tone PB in an equal tuning. In the > latter it functions as a step vector -- but is a very unlikely choice > for one.It worked, though.
Message: 3817 - Contents - Hide Contents Date: Mon, 04 Feb 2002 21:46:17 Subject: Re: Best Partch keyboard mapping From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> perhaps you or gene could tell us the errors and unison vectors here. >> I found all of my so-called "standard" ets up to 100 whichrepresent each note of Genesis, and obtained the following:> > 58 .3529661000 7.302746897 > 65 .6150147495 11.35411845 > 72 .2346001000 3.910001666 > 73 .5954749001 9.788628497 > 80 .4116062001 6.174093001 > 84 .6814441795 9.734916846 > 87 .4564030700 6.295214756 > 89 .4717246800 6.360332766 > 91 .6610206501 8.716755827 > 94 .3698782499 4.721849998 > 95 .8571248997 10.82684084 > > The second column is maximum relative error, and the third maximum > 11-limit consonance error in cents.gene -- graham did not specify an ET for the scale -- he specified a linear temperament. so this doesn't answer my question. i'd also like to know the generator, etc.
Message: 3818 - Contents - Hide Contents Date: Tue, 05 Feb 2002 09:16:17 Subject: 35 linear temperaments compatible with Genesis From: genewardsmith I took the list of 319 11-limit linear temperaments which result from wedging three superparticulars together, and wedged each of them with the 43 tones of Partch's Genesis scale. Since the wedge product is distributive, two such wedge products will differ if and only if the associated linear product distiguishes the two tones in question. Hence I rather quickly found that there are 35 temperaments from this list which distinguish all 43 tones of Genesis. Since this is Partch, I sorted them by rms error (maximum error would have been better, but I haven't written the code for that) instead of badness. Here they are: Hemiennealimmal wedgie [36, 54, 36, 18, 2, -44, -96, -68, -145, -74] map [[0, 2, 3, 2, 1], [18, 12, 17, 34, 54]] bad 78.02777025 rms .1987978555 g 36. generators [.4591217956, 1/18] 550.9461547 cents 200/3 cents Octoid wedgie [24, 32, 40, 24, -5, -4, -45, 3, -55, -71] map [[0, -3, -4, -5, -3], [8, 19, 27, 33, 34]] bad 147.3855058 rms .7687063272 g 23.42160176 generators [.2633934691, 1/8] 316.0721629 cents 150 cents wedgie [24, 20, 16, -12, -24, -42, -102, -19, -97, -89] map [[0, -6, -5, -4, 3], [4, 12, 14, 15, 11]] bad 308.3069734 rms 1.086898632 g 29.62624319 generators [.2359353963, 1/4] 283.1224756 cents 300 cents Undecimal augmented fifth wedgie [12, 22, -4, -6, 7, -40, -51, -71, -90, -3] map [[0, -6, -11, 2, 3], [2, 5, 8, 5, 6]] bad 169.9769336 rms 1.249417068 g 19.06380265 generators [.1526523192, 1/2] 183.1827830 cents 600 cents Ennealammal wedgie [18, 27, 18, -27, 1, -22, -105, -34, -156, -138] map [[0, -2, -3, -2, 3], [9, 19, 28, 30, 24]] bad 432.2032364 rms 1.297305750 g 32.62777081 generators [.2635689420, 1/9] 316.2827304 cents 400/3 cents wedgie [12, 34, 20, 30, 26, -2, 6, -49, -48, 15] map [[0, -6, -17, -10, -15], [2, 4, 7, 7, 9]] bad 179.9856041 rms 1.462301383 g 17.95231779 generators [.6933142420e-1, 1/2] 83.19770904 cents 600 cents wedgie [12, -2, 20, -6, -31, -2, -51, 52, -7, -86] map [[0, -6, 1, -10, 3], [2, 7, 4, 12, 5]] bad 195.0280472 rms 1.584514409 g 17.95231779 generators [.3193466476, 1/2] 383.2159771 cents 600 cents Ennealimmal wedgie [18, 27, 18, 45, 1, -22, 9, -34, 11, 64] map [[0, -2, -3, -2, -5], [9, 19, 28, 30, 43]] bad 290.9447340 rms 1.614770432 g 22.56419413 generators [.2635543717, 1/9] 316.2652460 cents 400/3 cents Catakleismic wedgie [6, 5, 22, -21, -6, 18, -54, 37, -66, -135] map [[0, 6, 5, 22, -21], [1, 0, 1, -3, 9]] bad 271.0589693 rms 1.697136764 g 20.98979344 generators [.2639230436, 1] 316.7076523 cents 1200 cents wedgie [12, -14, -4, -42, -50, -40, -108, 30, -49, -104] map [[0, -6, 7, 2, 21], [2, 8, -1, 4, -10]] bad 602.7223540 rms 1.755131812 g 33.22649545 generators [.4027409219, 1/2] 483.2891063 cents 600 cents wedgie [12, 10, 44, 30, -12, 36, 6, 74, 35, -68] map [[0, 6, 5, 22, 15], [2, 0, 2, -6, -1]] bad 336.3466116 rms 1.811029208 g 22.97825059 generators [.2639312147, 1/2] 316.7174576 cents 600 cents wedgie [0, 0, 0, 72, 0, 0, 114, 0, 167, 202] map [[0, 0, 0, 0, 1], [72, 114, 167, 202, 216]] bad 965.8716892 rms 1.828087456 g 43.02822994 generators [.4575975148, 1/72] 549.1170178 cents 50/3 cents wedgie [6, 17, -26, 15, 13, -58, 3, -108, -24, 132] map [[0, 6, 17, -26, 15], [1, -1, -5, 14, -3]] bad 353.9145617 rms 1.860428748 g 23.31155446 generators [.4305698169, 1] 516.6837803 cents 1200 cents wedgie [0, 24, -24, 0, 38, -38, 0, -123, -83, 83] map [[0, 0, 1, -1, 0], [24, 38, 48, 75, 83]] bad 327.4119970 rms 1.864367444 g 22.21968240 generators [.3197865867, 1/24] 383.7439040 cents 50 cents wedgie [0, 12, 24, 36, 19, 38, 57, 22, 42, 18] map [[0, 0, 1, 2, 3], [12, 19, 24, 26, 30]] bad 331.3058163 rms 1.886539844 g 22.21968240 generators [.3193329987, 1/12] 383.1995984 cents 100 cents wedgie [6, 5, 22, 51, -6, 18, 60, 37, 101, 67] map [[0, 6, 5, 22, 51], [1, 0, 1, -3, -10]] bad 471.0957653 rms 1.895128508 g 27.37047418 generators [.2638837183, 1] 316.6604620 cents 1200 cents Miracle wedgie [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] map [[0, 6, -7, -2, 15], [1, 1, 3, 3, 2]] bad 125.5016755 rms 1.901465778 g 12.35198075 generators [.9722688696e-1, 1] 116.6722644 cents 1200 cents wedgie [26, 11, 12, 34, -43, -54, -36, -3, 41, 54] map [[0, 26, 11, 12, 34], [1, -11, -3, -3, -13]] bad 564.0012695 rms 2.379622379 g 26.59887216 generators [.4840763552, 1] 580.8916262 cents 1200 cents wedgie [17, 6, 15, 27, -30, -24, -16, 18, 42, 24] map [[0, 17, 6, 15, 27], [1, -5, 0, -3, -7]] bad 297.3513487 rms 2.412281097 g 17.96822592 generators [.3874002607, 1] 464.8803128 cents 1200 cents wedgie [8, -16, -6, -38, -44, -32, -88, 31, -33, -86] map [[0, -4, 8, 3, 19], [2, 4, 3, 5, 3]] bad 665.4908578 rms 2.577579336 g 28. generators [.1032506025, 1/2] 123.9007230 cents 600 cents wedgie [2, -4, 30, 22, -11, 42, 28, 81, 65, -42] map [[0, -1, 2, -15, -11], [2, 4, 3, 18, 16]] bad 409.8176894 rms 2.781040890 g 20. generators [.4126371803, 1/2] 495.1646164 cents 600 cents wedgie [18, 27, 18, 72, 1, -22, 52, -34, 74, 140] map [[0, -2, -3, -2, -8], [9, 23, 34, 34, 66]] bad 1096.758484 rms 2.869219268 g 35.43303543 generators [.4841701986, 1/9] 581.0042383 cents 400/3 cents wedgie [6, -12, 10, -14, -33, -1, -43, 57, 9, -74] map [[0, -3, 6, -5, 7], [2, 4, 3, 7, 5]] bad 310.1807132 rms 2.986631473 g 16.21286967 generators [.1376269090, 1/2] 165.1522908 cents 600 cents wedgie [11, 18, 5, 41, 3, -23, 27, -39, 33, 98] map [[0, -11, -18, -5, -41], [1, 2, 3, 3, 5]] bad 504.1807894 rms 3.109758191 g 21.17950492 generators [.3753460343e-1, 1] 45.04152412 cents 1200 cents wedgie [16, 26, 46, 40, 4, 28, 8, 34, 3, -47] map [[0, 8, 13, 23, 20], [2, -4, -7, -15, -11]] bad 632.2967355 rms 3.130234693 g 24.16609194 generators [.4480823356, 1/2] 537.6988027 cents 600 cents wedgie [2, -4, -16, -24, -11, -31, -45, -26, -42, -12] map [[0, -1, 2, 8, 12], [2, 4, 3, -1, -3]] bad 336.4543492 rms 3.182069339 g 16.38814903 generators [.4135137199, 1/2] 496.2164639 cents 600 cents Nonkleismic wedgie [10, 9, 7, 25, -9, -17, 5, -9, 27, 46] map [[0, 10, 9, 7, 25], [1, -1, 0, 1, -3]] bad 240.3020098 rms 3.316530343 g 13.06303399 generators [.2584558980, 1] 310.1470776 cents 1200 cents wedgie [16, 14, 10, -8, -15, -29, -68, -16, -67, -57] map [[0, -8, -7, -5, 4], [2, 7, 8, 8, 5]] bad 544.2372567 rms 3.755921842 g 19.79898987 generators [.2394815594, 1/2] 287.3778713 cents 600 cents wedgie [8, 1, -13, 20, -17, -43, 4, -33, 43, 101] map [[0, 8, 1, -13, 20], [1, -1, 2, 7, -3]] bad 441.7730342 rms 3.771188024 g 17.42740043 generators [.3228611737, 1] 387.4334084 cents 1200 cents wedgie [18, 22, 30, 16, -7, -3, -37, 8, -39, -59] map [[0, -9, -11, -15, -8], [2, 10, 13, 17, 13]] bad 477.7638530 rms 4.000177346 g 17.63114128 generators [.3794325393, 1/2] 455.3190472 cents 600 cents wedgie [20, 18, 14, -8, -18, -34, -82, -18, -81, -71] map [[0, 10, 9, 7, -4], [2, -2, 0, 2, 9]] bad 817.4342779 rms 4.100337156 g 23.97617865 generators [.2586851394, 1/2] 310.4221673 cents 600 cents wedgie [4, -8, 26, 10, -22, 30, 2, 83, 51, -62] map [[0, 2, -4, 13, 5], [2, 2, 7, -2, 4]] bad 458.2298456 rms 4.200154135 g 16.69901622 generators [.2931210457, 1/2] 351.7452548 cents 600 cents wedgie [7, 9, 13, -15, -2, 1, -48, 5, -66, -87] map [[0, 7, 9, 13, -15], [1, -1, -1, -2, 9]] bad 475.7351256 rms 4.791027076 g 15.78199698 generators [.3694108067, 1] 443.2929680 cents 1200 cents wedgie [7, 9, 13, 31, -2, 1, 25, 5, 41, 42] map [[0, 7, 9, 13, 31], [1, -1, -1, -2, -8]] bad 461.2544269 rms 4.975617526 g 15.14454169 generators [.3697081632, 1] 443.6497958 cents 1200 cents wedgie [14, 18, 26, 35, -4, 2, 7, 10, 19, 8] map [[0, 14, 18, 26, 35], [1, -1, -1, -2, -3]] bad 705.4531354 rms 5.860579055 g 17.71399769 generators [.1846380967, 1] 221.5657160 cents 1200 cents
Message: 3819 - Contents - Hide Contents Date: Tue, 5 Feb 2002 10:58 +00 Subject: Re: Best Partch keyboard mapping From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a3mvf9+db4f@xxxxxxx.xxx> Gene:>> I found all of my so-called "standard" ets up to 100 which> represent each note of Genesis, and obtained the following: >>>> 58 .3529661000 7.302746897 >> 65 .6150147495 11.35411845 >> 72 .2346001000 3.910001666 >> 73 .5954749001 9.788628497 >> 80 .4116062001 6.174093001 >> 84 .6814441795 9.734916846 >> 87 .4564030700 6.295214756 >> 89 .4717246800 6.360332766 >> 91 .6610206501 8.716755827 >> 94 .3698782499 4.721849998 >> 95 .8571248997 10.82684084 >> >> The second column is maximum relative error, and the third maximum >> 11-limit consonance error in cents.Thanks for that. I had checked the prime-mappings below 58 to verify that there isn't a simpler constant structure lurking. Checking they don't uniquely represent each note in the right order. I also checked that 53 couldn't work with any 11-limit mapping so long as you preserved its good 5-limit mapping. There may be other possible inconsistent mappings. I don't know if you'd want to use them. Paul:> gene -- graham did not specify an ET for the scale -- he specified a > linear temperament. so this doesn't answer my question. i'd also like > to know the generator, etc.The first step is to find an ET or constant structure that uniquely represents each note in sequence. Only then can you start looking for a linear temperament with an MOS that doesn't take you beyond that number of notes. Showing there aren't any valid constant structures below 58 is sufficient proof that there are no better mappings by my criteria, although some other mapping of 58 may be equally good. Here's the temperament listing. You could have got it from the .key list I linked to, or the CGI for ETs to linear temperaments. 29/99, 351.5 cent generator basis: (1.0, 0.29293920388170502) mapping by period and generator: [(1, 0), (1, 2), (-5, 25), (-1, 13), (2, 5)] mapping by steps: [(58, 41), (92, 65), (135, 95), (163, 115), (201, 142)] highest interval width: 25 complexity measure: 25 (41 for smallest MOS) highest error: 0.005264 (6.317 cents) unique 128:99 =~ 9:7 14:11 =~ 81:64 consistent with: 41, 58 For commatic unison vectors, I get 243:242, 540:539 and 896:891 Graham
Message: 3820 - Contents - Hide Contents Date: Tue, 5 Feb 2002 16:23:19 Subject: Re: the missing consistencies From: manuel.op.de.coul@xxxxxxxxxxx.xxx They're not hiding, you mistook the .html extensions for .txt. Manuel [This message contained attachments]
Message: 3821 - Contents - Hide Contents Date: Tue, 5 Feb 2002 08:07:29 Subject: Re: the missing consistencies From: monz> From: <manuel.op.de.coul@xxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, February 05, 2002 7:23 AM > Subject: Re: [tuning-math] the missing consistencies > > > They're not hiding, you mistook the .html extensions > for .txt.thanks, Manuel, it's fixed now. Definitions of tuning terms: consistent, (c) 1... * [with cont.] (Wayb.) paul, you had asked me to change something about my graphics here and in the "unique" definition ... what was it? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3822 - Contents - Hide Contents Date: Tue, 05 Feb 2002 21:48:01 Subject: Re: Best Partch keyboard mapping From: paulerlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <a3mvf9+db4f@e...> > Gene: >>>> I found all of my so-called "standard" ets up to 100 which>> represent each note of Genesis, and obtained the following: >>>>>> 58 .3529661000 7.302746897 >>> 65 .6150147495 11.35411845 >>> 72 .2346001000 3.910001666 >>> 73 .5954749001 9.788628497 >>> 80 .4116062001 6.174093001 >>> 84 .6814441795 9.734916846 >>> 87 .4564030700 6.295214756 >>> 89 .4717246800 6.360332766 >>> 91 .6610206501 8.716755827 >>> 94 .3698782499 4.721849998 >>> 95 .8571248997 10.82684084 >>> >>> The second column is maximum relative error, and the third maximum >>> 11-limit consonance error in cents. >> Thanks for that. I had checked the prime-mappings below 58 to verify that > there isn't a simpler constant structure lurking. Checking they don't > uniquely represent each note in the right order. I also checked that 53 > couldn't work with any 11-limit mapping so long as you preserved its good > 5-limit mapping. There may be other possible inconsistent mappings. I > don't know if you'd want to use them. > > Paul:>> gene -- graham did not specify an ET for the scale -- he specified a >> linear temperament. so this doesn't answer my question. i'd also like >> to know the generator, etc. >> The first step is to find an ET or constant structure that uniquely > represents each note in sequence. Only then can you start looking for a > linear temperament with an MOS that doesn't take you beyond that number of > notes. Showing there aren't any valid constant structures below 58 is > sufficient proof that there are no better mappings by my criteria, > although some other mapping of 58 may be equally good. > > Here's the temperament listing. You could have got it fromthe .key list> I linked to, or the CGI for ETs to linear temperaments. > > > 29/99, 351.5 cent generator > > basis: > (1.0, 0.29293920388170502) > > mapping by period and generator: > [(1, 0), (1, 2), (-5, 25), (-1, 13), (2, 5)] > > mapping by steps: > [(58, 41), (92, 65), (135, 95), (163, 115), (201, 142)] > > highest interval width: 25 > complexity measure: 25 (41 for smallest MOS) > highest error: 0.005264 (6.317 cents) > unique > > 128:99 =~ 9:7 > 14:11 =~ 81:64 > > consistent with: 41, 58 > > > For commatic unison vectors, I get > > 243:242, 540:539 and 896:891 > > > Graham thanks graham!you and gene between you have made enough valuable discoveries to fill several issues of xenharmonikon. i encourage both of you to begin working on publishable papers as soon as possible!
Message: 3823 - Contents - Hide Contents Date: Tue, 05 Feb 2002 21:53:01 Subject: Re: the missing consistencies From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> From: <manuel.op.de.coul@e...> >> To: <tuning-math@y...> >> Sent: Tuesday, February 05, 2002 7:23 AM >> Subject: Re: [tuning-math] the missing consistencies >> >> >> They're not hiding, you mistook the .html extensions >> for .txt. > >> thanks, Manuel, it's fixed now. > Definitions of tuning terms: consistent, (c) 1... * [with cont.] (Wayb.) > > > paul, you had asked me to change something about my graphics > here and in the "unique" definition ... what was it?well, before i remind you of that, which data exactly are you plotting here? i would have expected only odd number limits, since you're restricing yourself to equal temperaments with an integer number of notes per octave . . .
Message: 3824 - Contents - Hide Contents Date: Tue, 05 Feb 2002 22:02:58 Subject: question: partch scale as 41-tone periodicity block From: paulerlich in Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.), i found that partch's scale, seen as a 41-tone scale with wilson's two auxillaries, is quite similar to the fokker (parallelepiped) periodicity block with unison vectors 896/891, 441/440, 245/243, and 100/99 (the latter being the deviation of the auxillaries, hence a sort of 'chromatic' unison vector). questions: 1. is there a better choice, in terms of capturing more of Partch's choices correctly? 2. (a) if the answer to (1) is no, what is the linear temperament defined by the unison vectors 896/891, 441/440 and 245/243? (b) if the answer to (1) is yes, then . . . 3. is there a different convex shape, rather than a parallelepiped, that can be specified in a general way (perhaps using a few extra, *dependent* unison vectors) which captures Partch's choices even better than any parallelepiped could? 4. peter piper picked a peck of pickled peppered parellelepipeds. how much wood could a woodchuck chuck if a woodchuck could chuck wood?
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