This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 98000 8050 8100 8150 8200 8250 8300 8350 8400 8450 8500 8550 8600 8650 8700 8750 8800 8850 8900 8950
8750 - 8775 -
Message: 8776 - Contents - Hide Contents Date: Thu, 11 Dec 2003 22:39:50 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> ... > I hope George, if it's not too late, will consider using these ratios > for his 72-equal keyboard diagram -- the unevenness is probably not > important for him since he told me he intended the ratios to show how > *intervals* look on the keyboard, not as a representation of the > *pitches*The ratios on the keys illustrate pitches, but my intention was that interval vectors may be deduced from these.> (those of you who have been reading my posts for years know > what i mean -- intervals I notate as a:b, while pitches I notate > a/b) . . .Yes, I am in total agreement with that, except that I agree with Dave Keenan that (unless context dictates otherwise) the small number should precede the colon to reflect the practice of building intervals (and chords) from the bottom up, and also to be consistent with the manner in which we indicate ratios for chords, e.g., 4:5:6. I believe that Helmholtz and Ellis (and most other pre-20th-century) writers followed this practice. Could it be that putting the larger number first in a ratio is an *American* convention?> George, you already have some 3-digit numbers, so the below shouldn't > be a problem, should it?These are ones I have I had to squeeze into a limited amount of space by redoing the characters pixel-by-pixel, as was also the case for ratios having 2 digits in both numerator and denominator. It was very time-consuming and I'm sorry, but there just isn't enough time now to change this many ratios. (For example, why are 64/63 and 63/32 being replaced by 45/44 and 88/45?) Besides, something that I wanted to illustrate with the ratios that I have is that the keyboard does not limit a JI tuning to 72 pitches per octave.> If this isn't acceptable, maybe a 17-limit version of same? > > Or feel free to ignore me.I'm not trying to ignore you. If I've been slow to respond, it's because there has been so much to do lately (including microtonal projects) that I have hardly had time to read the postings. Several days ago I had nearly a week's worth of digests unread, and I had to get the oldest ones out of the way by just scanning the tables of contents, searching for occurrences of my last name, and then deleting them. (I now notice that I probably would have missed replying to this one if I had not been reading the latest ones more carefully (once I noticed that the presence of a new member on the main list caused quite a bit of controversy.) --George
Message: 8777 - Contents - Hide Contents Date: Thu, 11 Dec 2003 09:36:27 Subject: Re: Enumerating pitch class sets algebraically From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:>> i think the main reason harmonic analysis would be >> characterised as an "art" is precisely *because* of >> the ambiguity available to a composer like Brahms, >> whether his intended tuning is 12edo or a meantone >> (the only two likely possibilities for Brahms IMO). >> >> my point: that *temperament* allows composers to play >> the kinds of games ("punning") that aren't possible >> in JI. >> Irrelevant -- Dante and I were talking about 'conventional' tonal > harmonic analysis, which never distinguishes any 81:80s anyway.OK, my bad. i didn't follow the thread from the beginning and probably should have just stayed out of it. ... in fact, my eyes have glazed over with nearly every post i've seen here over the last week, except for this one.>> and of course JI is the tuning which offers >> the straightforward "scientific" approach to harmonic >> analysis. > > BS.i guess i didn't express myself clearly enough. you and i both already know each other's viewpoints on this. anyway, since i did miss so much here in the last week, it's probably not worth it for me to try to clarify now ... -monz
Message: 8779 - Contents - Hide Contents Date: Thu, 11 Dec 2003 00:24:42 Subject: Re: Digest Number 862 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:> When I said that some > physicists would agree with me and some with you, I was talking > about agreement on the *real* issue at hand which is the > _mathematical _verses _the _musical _definition _of _the > _harmonic _series, which - I'm sorry - does not involve > acoustics!So you think some physicists would agree with you and say what, exactly? I honestly want to understand you better.> > Sheesh, this has become a royal waste of time... > > Aaron
Message: 8780 - Contents - Hide Contents Date: Thu, 11 Dec 2003 01:58:02 Subject: Re: An 11-limit linear temperament top 100 list From: Paul Erlich Dave, I have a different interpretation of what's going on here than you. Gene has gone about the process of finding 11-limit linear temperaments in several ways. Proceeding directly from a prescribed set of commas was never one that anyone thought would capture _the_ top 100, or top N, according to any badness function (of complexity and error). That's why Gene wrote "An" and not "The" in the title. Proceeding directly from commas is probably something that I've put more weight on than anyone else, but when there is more than one comma being tempered out, *straightness* enters the picture and some combinations of the "best" commas will be worse than some combinations that include a "non-best" comma. Gene has always seemed fully cognizant of this fact. Still, I think Gene should reply for himself! --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith > <genewardsmith@j...>" <genewardsmith@j...> wrote: > ...>> The extra commas I suggested were all that was needed in the 7- limit> all had epimericity less than .46. I suggested .5 as a cutoff for>> the 7-limit and .3 for the 11-limit; I boosted this to .35, with a> 50 cent cutoff for size. This gave me the following list of 51 commas,>> in order of badness of the corresponding planar temperament: >> >> [9801/9800, 3025/3024, 3294225/3294172, 151263/151250, 441/440,> 385/384, 225/224, 2401/2400, 56/55, 176/175, 4375/4374, 540/539, > 64/63, 100/99, 250047/250000, 5632/5625, 36/35, 1375/1372, 126/125, > 45/44, 99/98, 43923/43904, 896/891, 81/80, 49/48, 50/49, 121/120, > 117649/117612, 55/54, 41503/41472, 1771561/1771470, 77/75, 4000/3993, > 6250/6237, 8019/8000, 6144/6125, 1029/1024,5120/5103, 3388/3375, > 3136/3125, 32805/32768, 245/242, 243/242, 128/125, 12005/11979, > 245/243, 1728/1715, 19712/19683, 625/616, 1331/1323, 2200/2187] >>>> Wedging these three at a time led to 6135 wedgies. Taking the best> 100 of these by geometric badness gave me my list. > ... > > Hi Gene, > > I was looking for names for linear temperaments I had found using > Graham's online finder, and I noticed this 11-limit one wasn't in your > list: > > Complex aug fourths > generator mapping [[1, ?, ?, ?, ?], [0, -7, -26, -25, 3]] > minimax generators [1200., 585.14] > minimax error 4.1 c > > Does this mean there is another 11-limit comma that should be added to > your list above? > > I called it "complex" in deference to this one in your list: > >> Tritonic>> [5, -11, -12, -3, -29, -33, -22, 3, 31, 33] [[1, 4, -3, -3, 2], [0,> -5, 11, 12, 3]] >>>> generators [1200., 580.274408364] >> bad 6158.168745 rms 5.154394 comp 70.204409
Message: 8781 - Contents - Hide Contents Date: Thu, 11 Dec 2003 00:26:06 Subject: Re: Digest Number 862 From: Paul Erlich Aaron, hopefully this post will make it up soon, because my posts have not appeared in order. One crucial post I made before this one has not yet appeared, and it seems we are on bad footing as a result. Please be patient until that post appears. --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" > <perlich@a...> wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" > <cfaah@e...> wrote: >>>>> imaginary because you invoked an army of like-minded >>> physicists in order to rebut my point of view, and this seemed >>> rather fantastical and unnecessary to me. >>>> Fantastical and imaginary as in untrue? >>>>> By the way, Paul, it wasn't necessary to go to the trouble. >>>> If it seemed untrue/imaginary/fantastical before, but now > seems true,>> then certainly *something* changed your mind . . . > >> Ah, I see the problem now. > > Paul, you have not told me anything I do not already know. You > started bringing in these legions of physicists to support your > argument from a physicists point of view and I told you that you > were missing my point completely. When I said that some > physicists would agree with me and some with you, I was talking > about agreement on the *real* issue at hand which is the > _mathematical _verses _the _musical _definition _of _the > _harmonic _series, which - I'm sorry - does not involve > acoustics! > > Sheesh, this has become a royal waste of time... > > Aaron
Message: 8782 - Contents - Hide Contents Date: Fri, 12 Dec 2003 17:32:14 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul George wrote:>(For example, why are 64/63 and >63/32 being replaced by 45/44 and 88/45?)Because they have a lower van Prooijen harmonic distance value. Also a lower Erlich complexity, which is easier: log2( max( num and den without factors 2 ) ). Manuel
Message: 8783 - Contents - Hide Contents Date: Fri, 12 Dec 2003 19:40:28 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > George wrote:>> (For example, why are 64/63 and >> 63/32 being replaced by 45/44 and 88/45?) >> Because they have a lower van Prooijen harmonic distance > value. Also a lower Erlich complexity, which is easier: > log2( max( num and den without factors 2 ) ). > > ManuelHowever useful those criteria may be, I consider 64/63 and 63/32 simpler because: 1) The prime numbers in the factors are lower; and 2) The range of numbers in the ratios (32 to 64) is lower (than 44 to 88). Paul, if you're objecting to my use of ratios of 19 in my diagram because 72-ET is not 19-limit consistent, may I point out that the only 19-limit consonances that participate in the inconsistency are 19/13 and 26/19, and neither of those appear in the diagram. Besides, if we're just mapping JI tones to an octave division, I don't see any problem with a minor inconsistency such as this, as long as constancy is maintained for every ratio. --George
Message: 8785 - Contents - Hide Contents Date: Fri, 12 Dec 2003 21:05:01 Subject: Re: An 11-limit linear temperament top 100 list From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> I was looking for names for linear temperaments I had found using > Graham's online finder, and I noticed this 11-limit one wasn't in your > list: > > Complex aug fourths > generator mapping [[1, ?, ?, ?, ?], [0, -7, -26, -25, 3]] > minimax generators [1200., 585.14] > minimax error 4.1 c > > Does this mean there is another 11-limit comma that should be added to > your list above?I later added 12 more to my list of 100, but still don't find it on my top 112 list. The reason seems to be that it is above the badness cutoff. Here's some information on Complex Augmented Fourths: Wedgie: [7, 26, 25, -3, 25, 20, -29, -15, -97, -95] Mapping: [[1, 5, 15, 15, 2], [0, -7, -26, -25, 3]] MT basis: <540/539, 896/891, 1375/1372> ets: 41, 80, 121 rms error: 2.583842867 geometric complexity (natural log style): 124.3706717 badness: 8006.869167
Message: 8786 - Contents - Hide Contents Date: Fri, 12 Dec 2003 22:37:36 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul>However useful those criteria may be, I consider 64/63 and 63/32 >simpler because: >1) The prime numbers in the factors are lower; and >2) The range of numbers in the ratios (32 to 64) is lower (than 44 to >88).Still there are more consonant chords in the scale with the original pitches. Manuel
Message: 8787 - Contents - Hide Contents Date: Fri, 12 Dec 2003 22:45:10 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Gene Ward Smith After fixing my program, here is what I am getting for Prooijen and geometric 11-limit reductions: ! red72_11pro.scl Prooijen 11-limit reduced scale 72 ! 81/80 64/63 33/32 25/24 21/20 128/121 16/15 27/25 12/11 11/10 10/9 9/8 25/22 8/7 297/256 7/6 33/28 32/27 6/5 40/33 11/9 99/80 5/4 81/64 14/11 32/25 128/99 21/16 160/121 4/3 27/20 15/11 11/8 25/18 7/5 512/363 10/7 36/25 16/11 22/15 40/27 3/2 121/80 32/21 99/64 25/16 11/7 128/81 8/5 160/99 18/11 33/20 5/3 27/16 56/33 12/7 512/297 7/4 44/25 16/9 9/5 20/11 11/6 50/27 15/8 121/64 40/21 48/25 64/33 63/32 160/81 2 ! red72_11geo.scl Geometric 11-limit reduced scale 72 ! 100/99 56/55 33/32 25/24 21/20 35/33 15/14 27/25 12/11 11/10 10/9 9/8 112/99 8/7 231/200 7/6 33/28 25/21 6/5 40/33 11/9 99/80 5/4 44/35 14/11 9/7 35/27 21/16 33/25 4/3 27/20 15/11 11/8 25/18 7/5 140/99 10/7 36/25 16/11 22/15 40/27 3/2 50/33 32/21 54/35 14/9 11/7 35/22 8/5 160/99 18/11 33/20 5/3 42/25 56/33 12/7 400/231 7/4 99/56 16/9 9/5 20/11 11/6 50/27 28/15 66/35 40/21 48/25 64/33 55/28 99/50 2
Message: 8788 - Contents - Hide Contents Date: Fri, 12 Dec 2003 22:49:05 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:>>> However useful those criteria may be, I consider 64/63 and 63/32 >> simpler because: >> 1) The prime numbers in the factors are lower; and >> 2) The range of numbers in the ratios (32 to 64) is lower (than 44 to >> 88). >> Still there are more consonant chords in the scale with the original > pitches. > > ManuelThe next thing that I found was that I would have 28/27 and 27/14 instead of 25/24 and 48/25 (for which I would imagine that your reply would be the same). Another question is: why 15/14 and 15/8 (when 16/15 would have been the inversion of 15/8)? I may have so many questions regarding what the other pitches in the scale should be, that to choose ratios on the basis of consonant chords being produced with them could have us going around in circles. --George
Message: 8789 - Contents - Hide Contents Date: Fri, 12 Dec 2003 14:55:53 Subject: Re: Question for Manuel, Gene, Kees From: Carl Lumma>After fixing my program, here is what I am getting for Prooijen and >geometric 11-limit reductions:Thanks for the follow-up, Gene. I wonder what you and Manuel are doing differently? -Carl
Message: 8790 - Contents - Hide Contents Date: Fri, 12 Dec 2003 23:13:35 Subject: Re: An 11-limit linear temperament top 100 list From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> I later added 12 more to my list of 100, but still don't find it on my > top 112 list. The reason seems to be that it is above the badness cutoff. > > Here's some information on Complex Augmented Fourths: > > Wedgie: [7, 26, 25, -3, 25, 20, -29, -15, -97, -95] > > Mapping: [[1, 5, 15, 15, 2], [0, -7, -26, -25, 3]] > > MT basis: <540/539, 896/891, 1375/1372> > > ets: 41, 80, 121 > > rms error: 2.583842867 > geometric complexity (natural log style): 124.3706717 > badness: 8006.869167OK. Thanks. While it isn't anything to write home about, it doesn't seem as bad to me as the above complexity figure makes it. Can someone please explain what geometric complexity is, and how the badness figure is obtained?
Message: 8791 - Contents - Hide Contents Date: Sat, 13 Dec 2003 00:10:17 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul George wrote:>Another question is: why 15/14 and 15/8 (when 16/15 would have been >the inversion of 15/8)?Then it wouldn't be epimorphic anymore, nor a constant structure. The alternatives are limited to changes by the unison vectors of the PB. Manuel
Message: 8792 - Contents - Hide Contents Date: Sat, 13 Dec 2003 00:11:05 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul Gene, your geometric reduced scale isn't epimorphic. Is that a mistake? Manuel
Message: 8793 - Contents - Hide Contents Date: Sat, 13 Dec 2003 00:12:52 Subject: Re: Question for Manuel, Gene, Kees From: Manuel Op de Coul Carl wrote:>Thanks for the follow-up, Gene. I wonder what you and Manuel are >doing differently?We used different periodicity blocks to optimise. At least that's what I think. Manuel
Message: 8795 - Contents - Hide Contents Date: Mon, 15 Dec 2003 16:12:33 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > George wrote:>> Another question is: why 15/14 and 15/8 (when 16/15 would have been >> the inversion of 15/8)? >> Then it wouldn't be epimorphic anymore, nor a constant structure. > The alternatives are limited to changes by the unison vectors of > the PB. > > ManuelIf 15/14 were changed to 16/15, the tuning would still be a constant structure. But I still haven't worked my way through all of the intricacies involved in figuring out exactly what epimorphism is supposed to mean. Is there now a definition that does not require a degree in mathematics to comprehend? I don't mean to be giving you guys a hard time, but I can't even begin to consider changing the ratios on the decimal keyboard diagram unless I can get the number of required changes reduced to something that isn't going to eat up a lot of time for other (increasingly urgent) projects. (I'm presently trying to finish up the rest of the sagittal graphics for Scala.) --George
Message: 8796 - Contents - Hide Contents Date: Mon, 15 Dec 2003 17:59:11 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul George wrote:>If 15/14 were changed to 16/15, the tuning would still be a constant >structure.Sorry, yes. I must have made a typo when I tried it. Those changes are ok indeed, since 225/224 is one of the unison vectors, the others are 3025/3024, 1375/1372 and 4375/4374. I'll change it in the archive too.>Is there now a definition that does not require a >degree in mathematics to comprehend? See Definitions of tuning terms: epimorphic, (c) 2... * [with cont.] (Wayb.)Considering all the higher than 11-limit ratios, I can imagine it would take a lot of time to change the diagram.>(I'm presently trying to finish up the rest of the >sagittal graphics for Scala.)Splendid, by the way I now don't use xpm files anymore, but png, but that doesn't matter to you. Manuel
Message: 8797 - Contents - Hide Contents Date: Mon, 15 Dec 2003 17:44:59 Subject: Re: Question for Manuel, Gene, Kees From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > Carl wrote:>> Thanks for the follow-up, Gene. I wonder what you and Manuel are >> doing differently? >> We used different periodicity blocks to optimise. > At least that's what I think.That should make no difference.
Message: 8798 - Contents - Hide Contents Date: Mon, 15 Dec 2003 17:46:03 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:> > Gene, your geometric reduced scale isn't epimorphic. Is that > a mistake?I just checked this, and I get that it is epimorphic. Can you tell me where you think there is a problem?
Message: 8799 - Contents - Hide Contents Date: Mon, 15 Dec 2003 19:11:19 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul> Can you tell me >where you think there is a problem?I checked and there's a numerical problem in Scala which was silently ignored. Thanks, I'll see if I can fix it. Manuel
8000 8050 8100 8150 8200 8250 8300 8350 8400 8450 8500 8550 8600 8650 8700 8750 8800 8850 8900 8950
8750 - 8775 -