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Message: 4250 - Contents - Hide Contents Date: Wed, 13 Mar 2002 21:51:25 Subject: Re: Systematic naming of new temperaments (was: amt) From: paulerlich --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>> And we used to say "chain-of-minor-thirds" for kleismic! >> Oh yeah. How embarrassing. Not. >>> How do you differentiate two different mappings with the same >> generator? >> I just say 5-limit whatever versus 7-limit whatever, and so on.that won't necessarily do the trick. look at the fourth and fifth entries here: 4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)

Message: 4251 - Contents - Hide Contents Date: Wed, 13 Mar 2002 21:52:51 Subject: Re: Systematic naming of new temperaments (was: amt) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> Huh? All the one's we've been talking about for this paper _are_ >> octave based. >> Not in my head; maybe in yours. :)gene means that the period is not always one octave, even in your list, dave.

Message: 4252 - Contents - Hide Contents Date: Wed, 13 Mar 2002 22:40:59 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> oh yeah, i meant divided by . . . i said it the right way last time, > about two months ago . . .So if were to do yet another search, what should be the limits?

Message: 4253 - Contents - Hide Contents Date: Wed, 13 Mar 2002 22:56:59 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>> oh yeah, i meant divided by . . . i said it the right way last time, >> about two months ago . . . >> So if were to do yet another search, what should be the limits?what do you mean (who and what limits)?

Message: 4254 - Contents - Hide Contents Date: Wed, 13 Mar 2002 23:03:33 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> what do you mean (who and what limits)?I don't want to do another search, this time weighted, unless you tell me what you want to look at.

Message: 4255 - Contents - Hide Contents Date: Wed, 13 Mar 2002 06:23:35 Subject: Re: Systematic naming of new temperaments (was: amt) From: dkeenanuqnetau I've added proposed systematic names to my spreadsheet now, for all the non-historicals. I've included the previous names as well, even if they were only invented yesterday. http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.] (Wayb.) There would be no harm in giving more than one name for each temperament in THE PAPER. This has been full-on addictive for me the past few days. You'll all be relieved to hear I'm gonna have to go cold turkey for a while, or I won't have a job or a family. Carl, I believe I have provided the "crucial BS control" that you were hoping for. I hope that by the time I come back you will all have long-ago agreed on the final list for 5-limit and moved on to 7-limit. I suggest doing that before coming back to the incomplete ones, for the same reason we did 5-limit first; we're familiar with it.

Message: 4256 - Contents - Hide Contents Date: Wed, 13 Mar 2002 22:29:08 Subject: Octaves and periods From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> gene means that the period is not always one octave, even in your > list, dave.I mean more than that. My point of departure is that for example 5-limit meantone is defined by 81/80. I now decide to look at this in connection withoctaves, and take 2/\81/80 = [0,1,4], giving me my map to generators. Then [0,1,4]/\[a,b,c] = 2^(c-4b) 3^4a 5^-a so if a=1, b=1, and c=0 I get [0,1,4]/\[1,1,0] = 81/80. This may therefore be used as a map, givingme generators of an octave and meantone fifth. What if I look at 720 instead of 2? I now have 720/\81/80 = [-6,0,24]. This has a gcd of 6, so I divide out the 6, getting [-1,0,4]. I now wedge [-1,0,4]/\[a,b,c] = 2^(-b) 3^(4a+c) 5^(-b), so a=b=1, c=0 will work to give me 81/80: [-1,0,4]/\[1,1,0] = 81/80. I now use this as my map, setting the condition that 720 must be pure if I want. I get generators of about 3/2 and 3, and if 720 is pure, then the 3 is really 720^(1/6), and is my period, and the meantone fifth is my generator. Now meantone consists of six parallel lines of generators an approximate twelvth apart! My point is, meantone is not really defined by the choice to priviledge octaves, which is a separate consideration.

Message: 4257 - Contents - Hide Contents Date: Wed, 13 Mar 2002 01:59:51 Subject: Systematic naming of new temperaments (was: amt) From: dkeenanuqnetau --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> Actually, I don't care what you call it. > > Except "acute minor thirds". That's a terrible name: > > 1. Three words. Temperaments should have cool, single-word > names.Well yeah, but if possible, they should give you some clue as to what the temperament is. This can be (a) The size of the generator (and possibly how many periods to the octave when there is more than one): neutral thirds acute minor thirds hemithirds quadrafourths/bestonic chrome augmented (indirectly) diminished (indirectly) pelogic (indirectly) orwell (obscurely) and possibly semisuper if "semi" is meant to indicate the half-octave period and "super" is short for super fourth. However I don't think most people distinguish between semi and hemi, so I think "twin" (or "double") is a better indicator of a half-octave period, as in "twin chains", which generalises to triple, quadruple, quintuple, sextuple, septuple, octuple, ?... Also note that in lowest terms the generator for "semisuper" is in fact a better example of a chromatic semitone than the one for "chrome". or (b)The comma that vanishes: schismic kleismic diaschismic wuerschmidt tiny diesic minimal diesic limmal meantone (indirectly) or is "meantone" indicating the generator size indirectly? Unfortunately this method doesn't generalise well to higher limits since more than one comma vanishes. We could stick to using the single comma that vanishes in its 5-limit subset, but that won't work for temperaments that don't _have_ a 5-limit subset. A problem with having these two different systems is that when the comma is bigger than a chromatic semitone (71c) or the generator is smaller than a large limma (133 c) it isn't clear whether the name refers to the generator or the comma. limmal and chrome being examples. (c) No clue whatsoever: porcupine starling pajara magic miracle There's nothing particularly magical or miraculuos about Magic and Miracle at the 5-limit.> 1. Three words. Temperaments should have cool, single-word > names.I can't see any reason to stop calling neutral thirds by a two word three syllable name. I think number of syllable is more relevant than number of words. I think 2 words is ok if no more than 4 syllables. di-a-schis-mic, par-a-kleis-mic, sem-i-su-per. And a-cute mi-nor thirds isn't as bad as min-i-mal di-es-ic.> 2. I find it perverse to name temperaments by their relation to > diatonic intervals. I guess this counts against Amt too.I know what you mean, but it's not too perverse for 5-limit since the JI diatonic is 5-limit. But hey, that's simply how we name intervals, of any limit. I think we're stuck with it. I favour naming temperaments based on the size of the generator and the number of periods to the octave. That immediately tells someone how to make a scale from it with only basic math. I favour describing the size of the generator as the appropriate fraction of the interval (within the specified limit) that needs the fewest generators and only whole octaves, provided it is accurate enough and provided it can be done in four syllables or less. This is more-or-less the approach Gene used with: hemithirds quadrafourths (except this should be trithirds, my mistake) Using this system, AMT becomes pentelevenths I'm unsure whether we should put an "s" on the end or not. But I think we should, because "neutral third temperament" sounds odd to me. Porcupine would become hemiminorthirds. Too many syllables. We could shorten it to haemorrhoids. No? OK, I guess it stays porcupine. :-) Chrome would be quadraminorthirds. Too many syllables again. Is there a shorter way to say "a quarter of"? Is there a shorter way to say "minor thirds"? There _is_ a shorter way to say "half of", "bi" as in bicarbonate. Tiny diesic becomes hemisixths. Minimal diesic becomes quadrafifths. 4294967296/4271484375 becomes septathirds. Any valid generator could be used, not just the smallest one. So semisuper becomes twin tritenths or double tritenths. But it's no use calling pajara/twintone/paultone "twin fourths" or "twin fifths", since that could apply to twin meantone as well. And Magic and Wuerschmidt would both be "major thirds" so they had better stay as they are.

Message: 4258 - Contents - Hide Contents Date: Wed, 13 Mar 2002 03:52:01 Subject: Degenerate temperaments (was: A proposed list of 5-limit not-quite-Just-things) From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> twin meantone = garbage >> half meantone-fourth = ditto >> half meantone-fifth = dittoPaul, you say they are not temperaments because they do not map to a _the_ JI lattice, and are therefore merely "tuning systems". But of course they map to two (or more) disconnected JI lattices which is a hell of a lot better than a mere "tuning system". How about we call them degenerate temperaments? Gene, since you feel so strongly about it I'll go with Paul's earlier suggested compromise. Leave them out of the lists. Mention somewhere in the paper that these modifications are possible and result in a degenerate temperament having the same errors as the parent temperament but twice the complexity, and a different pair of (generator, period) and therefore different MOS. Use these meantone ones as the example, then say no more about them. Agreed? Anyone else object?

Message: 4260 - Contents - Hide Contents Date: Wed, 13 Mar 2002 05:15:36 Subject: Re: Systematic naming of new temperaments (was: amt) From: dkeenanuqnetau Does anyone have a list of the proper latin-derived prefixes for the various fractions. And shouldn't we use "semi" for 1/2 since we already have semitone. And hemi is greek, not latin, as are tetra, penta, hexa, hepta, (octa is both L and Gr). And please ignore my suggestion of "bi", it cleraly means 2, not 1/2. Similarly I realise we can't use "tri" to mean 1/3 since it already clearly means 3 in tritone. Is "tertia" clearly 1/3 and not 3? Do any of them beyond 2 clearly mean 1/n and not also n? I think "quarta" is more clearly 1/4 than "quadra" which sounds to me like it could equally mean 4. Is "quinta" clearly 1/5 and not 5? Or is it the case that you change them from a multiple to a fraction by changing the final vowel e.g. semi terti quarti quinti sexti septi octi Even if there's no such rule, we can make one, it's pretty suggestive, following on from semi. Certainly a quartile and a quintile are 1/4 and 1/5. I could only find that a tertian is 1/3 of a tun (liquid measure). And a sextary is 1/6 of a congius or modius. And there's sextant meaning 1/6 part. So we have the following: semiminorthirds (porcupine) tertiminorsixths (orwell) semisixths (tiny diesic) quintelevenths (AMT) quartififths (minimal diesic) tertithirds (quadrafourths) twin tertitenths (semisuper) septithirds (4294967296/4271484375) quartiminorthirds (chrome) semithirds (hemithirds) [not on my list for the paper] The only prefix I'm not real keen on there is "terti", maybe 'cause it sounds like "dirty". Other suggestions?

Message: 4261 - Contents - Hide Contents Date: Wed, 13 Mar 2002 06:07:39 Subject: Re: Systematic naming of new temperaments (was: amt) From: dkeenanuqnetau --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:>> Well yeah, but if possible, they should give you some clue as to what >> the temperament is. This can be >> Well, crazy names won't tell you as much, but are more memorable, > and might make up for it.I think some of these systematic names sound pretty crazy and if any of them had had any history I thing you'd remember them just fine. I been remembering several temperaments just fine up till now, simply by the generator size in cents. I just say to myself "oh that's 126 cent temperament" or whatever. Orwell was "272 cent temperament" or "subminor thirds temperament" to me for a long time.> See also my complaint about using > diatonic > intervals for temperaments where the diatonic scale may not even > be supported.That's your problem. I long ago ceased to think of them as in any way related to the diatonic scale, they are just particular sized intervals near particular ratios.>> hemithirds >> quadrafourths/bestonic >> chrome >> I still don't know the generators for these.I thought you said you'd rather know the comma. The generators are, as my most recent name suggestions approximately tell you, semithirds, half a major third, 193 c, (rms is 193 c) tertithirds, a third of a third, 129 c, (rms is 126 c) quartiminorthirds, a quarter of a minor third, 79c, (rms is 78 c)>> augmented (indirectly) >> diminished (indirectly) >> pelogic (indirectly) >> Where historical names exist, I think they should > be used. Absolutely! >> orwell (obscurely) >> But you'll never forget it once you've heard it.Oh yeah. If folks were told once that it was a 19/84 oct generator, I'll bet some would come back to it months later wondering "was that a temperament consistent with both 19-tET and 84-tET".>> and possibly semisuper if "semi" is meant to indicate the half-octave >> period and "super" is short for super fourth. However I don't think >> most people distinguish between semi and hemi, >> I'm not sure I do. I'd guess semi is partial, while hemi is > exactly half.Nah they're both half. Hemi is greek semi is latin. We need latin, since semi is already used for exactly the purpose I want to use it, in semi-tone.> Super is short for super fourth?Beats me. Gene?>> (b)The comma that vanishes: >> schismic >> kleismic >> diaschismic >> wuerschmidt >> tiny diesic >> minimal diesic >> limmal >> meantone (indirectly) >> These are my favs -- I consider unison vectors far more informative > than generator size. Except for stuff like "tiny" diesic vs. "minimal" > diesic, which I still don't know as I write this.Indeed. I think gene made up the "tiny diesis" to start with. But if I say semisixths and quartififths anyone who knows the system (and even the system is not too hard to figure out for yourself) can figure out the rest if they want to. Very few musicians and composers even know the difference between a pythagorean and a syntonic comma, let alone what a diaschisma or a kleisma is. The historical comma-based names can stay but I say we don't base any more on obscurely named commas. In contrast, everyone knows what thirds fourths etc. are, even if they will assume the 12-tET version instead of the just.>> or is "meantone" indicating the generator size indirectly? >> Meantone is just historical. It doesn't have to make sense.Good grief! I never meant to suggest we rename meantone.>> Unfortunately this method doesn't generalise well to higher limits >> since more than one comma vanishes. >> I was just going to mention that.A serious problem, yes?>> We could stick to using the single comma that vanishes in its >> 5-limit subset, but that won't work for temperaments that don't >> _have_ a 5-limit subset. >> Indeed. That's when we resort to shamelessly locking in all of our > surnames! :) Tee hee. >> starling >> When was this ratified?Sorry I mentioned it. It's a 7-limit planar temperament by Herman Miller where the 125:126 vanishes.>> There's nothing particularly magical or miraculuos about Magic and >> Miracle at the 5-limit. >> That's okay. These names are historical now, too.Good grief! Only months old and only on this list. That's not historical. But anyway I don't propose to change these.>>> 1. Three words. Temperaments should have cool, single-word >>> names. >>>> I can't see any reason to stop calling neutral thirds by a two word >> three syllable name. >> I don't think any established (as in, more than a month or two) > names should be changed at all. >>> I think number of syllable is more relevant than number of words. >> I don't care if it's hard to say, I just want people to want to say > it. Compare "Acute minor thirds" to "kleismic" here.Yes. I don't want to say "Acute minor thirds" because it is hard to say.> Characters from novels, breads, etc., are also good.They are fun when you're in the in-crowd who knows, but they are totally mystifying to newcomers. I dunno about you, but I get tired of explaining terms to newbies or telling them where to find the list or dictionary. I'd rather newbies had at laest some chance of figuring it out for themselves.>> I know what you mean, but it's not too perverse for 5-limit since the >> JI diatonic is 5-limit. But hey, that's simply how we name intervals, >> of any limit. I think we're stuck with it. >> I'm willing to accept this, but I don't have to like it. I try > to avoid it, at any rate. Understood.>> Tiny diesic becomes hemisixths. >> Minimal diesic becomes quadrafifths. >> 4294967296/4271484375 becomes septathirds. ...> There's not enough variety in this naming scheme for my taste. In > effect, I'm going to have to think about the name each time I hear > the temperament, whereas "orwell" lives in my mind as its own entity. >But Carl, that's only because it has a history with you. There is at least one temperament name that essentially uses this system that has been around for a long time. I'm sure you don't have any problem recognising it. neutral thirds. Sure the terti quarti quinti ones are a little more difficult but they will usually be the more complex and therefore less popular ones anyway. When we go to higher limits, the best ones often have a generator which is a whole consonant interval. Like "subminor thirds temperament", the name that was used before "Orwell", for that 7-limit temperament.

Message: 4262 - Contents - Hide Contents Date: Wed, 13 Mar 2002 06:35:10 Subject: Re: Degenerate temperaments (was: A proposed list of 5-limit not-quite-Just-things) From: dkeenanuqnetau I just added degeneracy detection to my spreadsheet. http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.] (Wayb.)

Message: 4264 - Contents - Hide Contents Date: Wed, 13 Mar 2002 08:07:52 Subject: Re: Degenerate temperaments (was: A proposed list of 5-limit not-quite-Just-things) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Paul, you say they are not temperaments because they do not map to a > _the_ JI lattice, and are therefore merely "tuning systems". But of > course they map to two (or more) disconnected JI lattices which is a > hell of a lot better than a mere "tuning system".Eh? Why is that? So does the 7-et+5-et system, which is the sort of thing I want to avoid; these degenerate temperaments are headed in that direction. How about we call> them degenerate temperaments?Sounds OK; I don't want to give the impression that they are the same kind of thing as a "regular" regular temperament. Mentioning them makes sense to me, dwelling on them doesn't.

Message: 4265 - Contents - Hide Contents Date: Wed, 13 Mar 2002 08:29:12 Subject: Re: Systematic naming of new temperaments (was: amt) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: I> been remembering several temperaments just fine up till now, simply by > the generator size in cents. I just say to myself "oh that's 126 cent > temperament" or whatever. Orwell was "272 cent temperament" or > "subminor thirds temperament" to me for a long time.That doesn't really work in any exact way, and it incorrectly leaves the impression that temperaments are octave-based; if you want to really be scientific about it I suggest thinking of them in terms of the wedgies. To me wedgies do the precise naming thing, and verbal labels do the "remember this" thing.>> Super is short for super fourth? >> Beats me. Gene?It refers to 3125/2304, which I claimed was a semisuper fourth for the purpose of coming up with a name for the temperament. This, by the way, shows why I prefer "hemi" to mean "half"--"semi" is much more vague.>> These are my favs -- I consider unison vectors far more informative >> than generator size.There you go--stick with wedgies. :)>>> Unfortunately this method doesn't generalise well to higher limits >>> since more than one comma vanishes. >>>> I was just going to mention that. >> A serious problem, yes?Write the wedgies as wedge products of an MT reduced basis?>>> starling >>>> When was this ratified? >> Sorry I mentioned it. It's a 7-limit planar temperament by Herman > Miller where the 125:126 vanishes.When was that discussed? I was going over 7-limit planars on tuning, and I recall someone saying that 126/125 had been looked at, but I thought starling was a scale!>

Message: 4266 - Contents - Hide Contents Date: Wed, 13 Mar 2002 08:30:03 Subject: Re: Systematic naming of new temperaments (was: amt) From: dkeenanuqnetau --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:> And we used to say "chain-of-minor-thirds" for kleismic!Oh yeah. How embarrassing. Not.> How do you differentiate two different mappings with the same > generator?I just say 5-limit whatever versus 7-limit whatever, and so on.

Message: 4267 - Contents - Hide Contents Date: Wed, 13 Mar 2002 08:39:23 Subject: Re: Systematic naming of new temperaments (was: amt) From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > I>> been remembering several temperaments just fine up till now, simply by >> the generator size in cents. I just say to myself "oh that's 126 cent >> temperament" or whatever. Orwell was "272 cent temperament" or >> "subminor thirds temperament" to me for a long time. >> That doesn't really work in any exact way,It doesn't need to, most of the time, and when it might lead to confusion between temperaments, _then_ we can use other names.>and it incorrectly leaves the impression that temperaments are octave-based;>Huh? All the one's we've been talking about for this paper _are_ octave based.> if you want toreally be scientific about it I suggest thinking of them in terms of the wedgies. To me wedgies do the precise naming thing, and verbal labels do the "remember this" thing.>I'm sorry. I've never managed to understand what a wedgie is, and frankly I don't see any need to, in order to understand temperaments.> Write the wedgies as wedge products of an MT reduced basis?Doesn't really sound like something you can base a snappy name on.> When was that discussed? I was going over 7-limit planars on tuning,and I recall someone saying that 126/125 had been looked at, but I thought starling was a scale!>>It is. Maybe I was the first to apply the term to the temperament the scale is in. In referring to Genes recent 8-noter as Starling-8.>>>> Tiny diesic becomes hemisixths. >>>> Minimal diesic becomes quadrafifths. >>>> 4294967296/4271484375 becomes septathirds. > > Oog.Thanks for your helpful comments.

Message: 4268 - Contents - Hide Contents Date: Wed, 13 Mar 2002 10:04:39 Subject: Re: ekmelic From: manuel.op.de.coul@xxxxxxxxxxx.xxx <<sounds like 'ekmelic' -- richter-herf's name for 72-equal.>>>Ekmelic is a generic German term used to describe prime harmonics greater >than 5.You mean ekmelisch; I don't know if that's exactly true. If I remember correctly it comes from the Greek words ek=out and melos=series so it means "out of the normal range". So in that sense it can be seen as the equivalent of Ivor Darreg's term "xenharmonic". The opposite term is emmelisch. Manuel

Message: 4269 - Contents - Hide Contents Date: Wed, 13 Mar 2002 09:20:12 Subject: Re: Systematic naming of new temperaments (was: amt) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Huh? All the one's we've been talking about for this paper _are_ > octave based.Not in my head; maybe in yours. :)> I'm sorry. I've never managed to understand what a wedgie is, and > frankly I don't see any need to, in order to understand temperaments.One thing they might help with is with this idea that things are octave-based. If 81/80 defines meantone, where's the octave?

Message: 4271 - Contents - Hide Contents Date: Thu, 14 Mar 2002 17:11 +0 Subject: Re: some output from Graham's cgi From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <OF5C0B1B93.CD1B3E17-ONC1256B7C.005B70BE@xxxxxx.xxxxxxxxx.xx>> Anyway, the minimax solution is 495.662963 cents. > Highest error 10.595435 cents.Yes, that's what I get.

Message: 4272 - Contents - Hide Contents Date: Thu, 14 Mar 2002 13:35:48 Subject: Re: Ferguson WHO??? ( From: Pierre Lamothe Marc wrote: I searched tuning AND tuning math and I didn't find either Ferguson or Forcade. What exactly is this algorithm? I'm serious, I have to hear this. It's not precisely tuning math but see 385 Ferguson-Forcade and PSLQ algorithm 386 PSLQ technical 384 PSLQ in Yacas and MuPAD Yahoo groups: /harmonic_entropy/message/385 * [with cont.] Pierre [This message contained attachments]

Message: 4273 - Contents - Hide Contents Date: Thu, 14 Mar 2002 22:53:53 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> Yes I have already posted them, and they are also in my spreadsheet. >> But here they are again. 35 cents rms, 12 weighted-rms generators (or >> 10 for the shorter list) and max 625 for Gene's badness (using >> weighted complexity). >> Here's some examples of weighted badness, using weighted complexity.Can you check your limits in terms of this?> > 81/80: 103.257 > 15625/15552: 187.010 > 250/243: 463.641 > 78732/78125: 525.371In terms of these, the badness limit would need to be around 915. You could have checked them yourself against my spreadsheet. http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.] (Wayb.) Are you able to read Excel spreadsheets? Somehow your badnesses are a factor of 1.46131 times those I calculate. How could that be? Are you still using complexity^3*error. Here are the weighted-rms complexities and errors I have for those temperaments (in case you can't read the spreadsheet). In the order above. error complexity 4.218 2.559 1.030 4.991 7.976 3.414 1.157 6.772

Message: 4274 - Contents - Hide Contents Date: Thu, 14 Mar 2002 18:21:14 Subject: Standardising temperament mappings From: David C Keenan Gene and Graham are using different ways of presenting both the basis vector for a temperament and its mapping matrix. This makes it tedious to compare temperaments generated by their respective algorithms. We should standardise. The standard should be based on other long-standing standards or conventions or be otherwise defensible, particularly if we are going to publish. We should consider equal temperaments and planar temperaments as well as the current linear temperaments, and we need to consider both octave-based and non-octave-based tuning systems. I see the following issues which need to be addressed in the following order. 1. Standardising the basis vector a. What units to use for generators (cents vs decimal octaves vs rational octaves) b. Choosing a canonical ordering of the generators c. How to determine a canonical set of generators 2. Standardising the mapping matrix 3. Standardising the format for flattening the mapping matrix Here's my take on these issues: 1a. What units to use for generators While octaves make good sense within a program for calculating temperaments, I think it's clear that either cents or rational octaves will be better understood by most readers. Since a basis nearly always includes a generator whose optimum value can't be exprtessed as a simple rational fraction of an octave, it seems cents should be the standard unit. We might however want to include supplementary information when the cents do not make it obvious that there are say exactly 19 generators (or periods) to the octave, even though this information will also be buried in the map matrix. 1b. Choosing a canonical ordering of the generators This can't really be addressed independenlty of 1c. "How to determine a canonical set of generators". But something has to come first in this text. It seems obvious that the canonical basis for an equal temperament (or more generally an EDn where n represents an interval of equivalence) is its step size (its period), which is the smallest generator that generates n. n is usually the lowest prime approximated by the EDn. The generator is always a whole number division of n. So it seems obvious to me that the first generator to be listed for a linear temperament should be the one that generates only the lowest prime (usually 2). We sometimes distinguish this generator by calling it the "period", particularly when we consider it to be generating an "interval of equivalence" (usually at least an approximate octave, but the tritave 1:3 also has a following). In this case we consider that the number of pitches and their spacing _within_ a single interval-of-equivalence is what constitutes the scale, and we consider the number of intervals-of-equivalence provided on any given instrument to be a detail irrelevant to our calculations. When we come to planar temperaments, it seems even more obvious that the first generator should relate to the lowest used prime (usu. 2), then the next generator should be related to the next higher prime (usu. 3) and the last generator will give the rest. 1c. How to determine a canonical set of generators. This has been partly assumed above. I suggest a generator is first found that generates only the lowest prime (but note that in some cases the interval of equivalence may not be related to the lowest prime, e.g. a tritave-based scale that includes some ratios of 2) and it should be the smallest such generator, which means that it will always be a whole-number division of (the approximation of) that lowest prime. Then the next generator is only what is needed to generate the next prime (in conjunction with the first generator if necessary). i.e. we are aiming for a triangular map matrix (an accepted canonical form). But in the end the last generator has to pick up all the remaining primes. At each step the generator given should be the smallest that will do the job. So at least in the case of a linear temperament, the second generator will always be less than half the first generator (or period). Graham currently gives the proposed canonical generators in the proposed order but does not give the first generator (period) in cents. Gene gives both in cents, but not in the proposed order and the generators are not always the smallest that will do the job. Note that a different choice of basis vector (generators) will lead to a different mapping matrix and make comparison tedious. 2. Standardising the mapping matrix Consider a planar 7-limit temperament described by this set of approximations: lg(2) ~= gen0*p2 + gen1*g2 + gen2*h2 lg(3) ~= gen0*p3 + gen1*g3 + gen2*h3 lg(5) ~= gen0*p5 + gen1*g5 + gen2*h5 lg(7) ~= gen0*p7 + gen1*g7 + gen2*h7 where lg(x) = ln(x)/ln(2)*1200, i.e. the primes in cents, and gen0, gen1, gen2 are the generators in cents, and p2, p3, g2, g3 etc are the integer coefficients. I've used "p"s for the gen0 coefficients to remind us that this is the generator we sometimes distinguish by calling it the period. There are various ways to express this in matrix form depending on whether our vectors are rows or columns. Since it is much easier to deal with row vectors in text, I propose we use: [If you're viewing this from Yahoo's dumb web interface, you'll need to click either Reply, or Message index then Expand Messages, to see it formatted properly.] [lg(2) lg(3) lg(5) lg(7)] ~= [gen0 gen1 gen2] [p2 p3 p5 p7] [g2 g3 g5 g7] [h2 h3 h5 h7] The canonical choice and ordering of generators should always result in the lower triangle being all zeros, as follows. [gen0 gen1 gen2] [p2 p3 p5 p7] [0 g3 g5 g7] [0 0 h5 h7] An equal temperament looks like [lg(2) lg(3) lg(5) lg(7)] ~= [gen0] [p2 p3 p5 p7] 3. Standardising the format for flattening the mapping matrix For 7-limit linear temperaments [lg(2) lg(3) lg(5) lg(7)] ~= [gen0 gen1] [p2 p3 p5 p7] [0 g3 g5 g7] we currently have flattenings: Gene [[0, g3, g5, g7], [p2, p3, p5, p7]] Graham [(p2, 0), (p3, g3), (p5, g5), (p7, g7)] The most obvious flattening of the above matrix is neither of these. I suggest we use: [[p2 p3 p5 p7] [0 g3 g5 g7]] (Why not omit the ","s?) It is particularly easy to find the most important information in this format. The number of periods per octave (p2) is at the start and the numbers of generators in the other primes are at the end. These are the only factors that affect the complexity of an octave-equivalent tuning. In fact, in the common case of an octave period, we often abbreviate the map to just [g3 g5 g7]. Can anyone see any problem with these proposals? If not, I'd appreciate it if Graham and Gene would both implement them, where they haven't already. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

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