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Message: 6125 - Contents - Hide Contents Date: Tue, 21 Jan 2003 17:02:54 Subject: Re: Superparticular temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> heuristic complexity is merely log(d) . . . epimericity can be >> expressed as a function of *both* heuristic complexity and >> heuristic eror -- thus it's a kind of badness measure, though >> not log-flat. > //>> log(d) is the complexity heuristic, not the error heuristic. >> the error heuristic is proportional to |n-d|/(d*log(d)). >> I swear you had that backwards when you defined "step" and > "cent" for me recently. where? > Anyway, all is clear now!! phew!>>>> however, once a temperament is defined by more than one >>>> comma, this approach will miss, since straightness is being >>>> ignored. >>>>>> No problem! I think linear temperaments are the most >>> interesting. >>>> a 7-liimit linear temperament is defined by a *pair* of unison >> vectors, so straighness comes into play there already. >> When you say "temerament is defined by", you mean all the uvs, > only the commatic ones, ...?only the commatic ones, of course. the chromatic ones only come into defining your finite "diatonic block".> And don't you mean that straigtness would _not_ come into play, > by what you say in the first paragraph (more than one comma = > ignore straightness)?in what "first paragraph" do i say this?? i've been saying the exact opposite!!!>>> Two sets of accidentals would be too confusing >>> fer the likes of me! I'd just use JI. >>>> 5-limit JI has two sets of accidentals; >> Adaptive JI. It should go without saying with me. I didn't > even know about strict JI until joining this list, and it > took 2 years of confusion before we figured it out!!you're kidding me, right? well anyway, adaptive JI in general can be even more complex to specify . . . do you specifically mean adaptive JI on a meantone basis?>>> If I'm going to temper, give me error, low complexity, and >>> simple notation! >>>> it should be a book of temperaments. for each, a notation, a >> generalized keyboard, a halbestadt-imitation keyboard, a >> horogram . . . >

Message: 6126 - Contents - Hide Contents Date: Tue, 21 Jan 2003 17:05:05 Subject: Re: Superparticular temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> I can never figure out these Erv Wilson things which consist of >nothing but pictures, so if you can explain horograms that >would be nice.they're simply concentric "clock diagrams" of all the MOSs of a given generator. search the tuning list archives . . .

Message: 6127 - Contents - Hide Contents Date: Tue, 21 Jan 2003 17:07:04 Subject: Re: A 13-limit comma list From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> Here is a list of 70 13-limit commas, with epimericity less than 0.25 > and size less than 50 cents.why do you keep insisting on "size less than 50 cents"? pelogic, for example relies on a much larger comma; and even negri, with its very low error, has a comma of 51 cents.

Message: 6128 - Contents - Hide Contents Date: Tue, 21 Jan 2003 11:23:26 Subject: Re: A common notation for JI and ETs From: David C Keenan>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> >wrote:>> I believe we should not use a comma for notating temperaments if it >has an>> absolute value of slope greater than 8. The following have slopes >between 7>> and 8 and so I think they should also be avoided if possibleHere are their slopes:> '| as 5' comma Slope 7.88 > ~|( as 7:13 comma Slope -7.90But this doesn't matter much since the primary interpretation of ~|( is as 17' comma with slope 4.09. I just noticed that your quick reference doesn't give this interpretation of 7:13 comma 28431:28672 for the symbol ~|( I think this is one that Margo found recently.> |~ as 23 comma Slope -7.02 > )|~ as 19' comma Slope 7.76 > |)) as 49' diesis Slope -7.36 >Ever since we started this notation project I've been basing my >choice of commas primarily on musical considerations involving as >little math as possible,A very good idea.>and the results have been in general >agreement with those arrived at when all of the mathematical analysis >is brought in. I have two questions/comments in light of the above: > >1) What are the most common symbols that we have proposed involving a >slope greater than 8?There are none. Which is why I didn't list them (or did I? :-). Actually it's the other way 'round. I suggested a cutoff at 8 because there weren't any beyond it, so far. Sorry I didn't say so. Some of the comma combinations that are not actually used to notate have slopes outside +-8, such as ~)| as 17+19 comma with slope 9.25 (but as 17:19 comma 152:153 it's fine at slope 1.30), and (|~ as (13'-(11-5))+(19'-19) with slope 8.90 and (|~ as (7:11')+(19'-19) with slope 10.93.>2) The choice of 23 vs. 23' comma and 19' vs. 19 comma involves >respelling of notes using different nominals. For example for 16:19 >with 16 as C, 19 will be Eb)| using the 19 (512:513) comma and D#)!~ >using the 19' comma (19456:19683). Likewise, 23 will be F#|~ using >the 23 comma (729:736) but Gb~|\ using the 23' comma (16384:16767). >If one of these tones occurs in a heptatonic scale, we would >certainly wish to notate the tones of the scale using all 7 nominals, >so the choice of which comma applies should be automatic. I would >imagine that a common use of the 23rd harmonic would be as a leading >tone to the 24th harmonic (or dominant), in which case (with C as >tonic) I would expect it to be spelled as F#|~ leading to G. And I >would assume that a much less common usage would use it descending to >the 22nd (11th) or 21st harmonics, in which case I would expect it to >be spelled as Gb~|\. > >So I never expected that the status of 729:736 as the principal 23 >comma would be brought into question, especially after all this >time.Please note that I'm talking about avoiding high slope comma symbols for notating _temperaments_, particularly ETs, because they will tend to represent widely differing apotome-fractions, and hence their size ordering will vary widely from rational, in temperaments with different fifth sizes. This is all irrelevant to rational tunings where you do not have a choice of symbols relative to a given nominal. So all I'm saying is that we should not use any of those 5 symbols for steps of an ET, if there is any other choice.>What are the slopes for the five commas you listed above, >taken to two decimal places? Given above.>> This also says that we should equally allow both >> '/|) as 7' diesis 57344:59049 >> and its apotome complement >> .(|\ as 7" diesis 27:28 >> as you first suggested, since they have slopes of 6.9 and -6.9. In >fact>> true apotome complements always have the same absolute value of >slope, but >> opposite signs. >>This is good, since it wouldn't be very appropriate for a comma to be >permitted but for its apotome complement to be rejected if both are >notated with single-shaft symbols. Agreed.>> ... I'm happy to simply limit the absolute slope to 8 >>-- depending on your answers to my questions in points 1) and 2) >above -- >>> and the comma size to 70.17 cents, the >> largest that could conceivably be notated as '((| >>I don't know how you got something that large. Two 7:11 commas plus >a 5' comma are ~68.25c, and two 13:17 commas plus a 5' comma are >~69.19c.But then we can have a schisma up to half a 5'-comma (0.98 c) past that and still be a valid use of the symbol. That's what takes us to 70.17 c.>But I don't know what reason we would have to use anything >with two convex left flags,Nor do I. But I wanted to keep the option open.>so I don't think I would have any problem >with any of these as an upper limit.The point of all this is to limit the search for commas for the rational pitches in the popularity list (and revisit those ratios we thought we'd done). Limiting slope to 8 and size to 70.17 cents seems to give 2 (and occasionally 3) commas for any ratio. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 6129 - Contents - Hide Contents Date: Tue, 21 Jan 2003 18:57:16 Subject: Re: heuristic and straightness From: Carl Lumma>> >?? If straightness is maximal when the uvs are maximally >> orthogonal, how does this mean the uvs have gotten shorter? >>any basis for the temperament that uses much longer unison >vectors will have to have much less straightness, because the >area/volume/etc. enclosed by the UVs must remain constant. >otherwise, you have torsion.So is this right: Straightness...LengthUVs...Length+/-UV...Badness Down...........Up..........Down..........Same Up.............Down........Up............Same ? If so, you can't say that changing the straightness of a given temperament makes it "worse". But am I correct that across temperaments, 'crooked' ones will tend to be worse (assuming we've already considered the LengthUVs)? -Carl

Message: 6130 - Contents - Hide Contents Date: Tue, 21 Jan 2003 15:34:00 Subject: Re: A common notation for JI and ETs From: David C Keenan George, Persuant to the development of the outline font, could you put up a bitmap or gif with all the _up_ symbols we've found a need for so far, showing (in the single-shaft case) the keyboard character that you want them mapped to. This would be an update of your Symbols3.bmp which is the latest I can find on the tuning-math files area. Don't include any combinations with 5' accents, just one with the acute accent beside a plain shaft. I'm thinking that we no longer need any 3-flag symbols (other than those where the third "flag" is a 5' accent) and the only 2-flags-on-the-same-side symbols that we need are //| and |)) and |\), although I'm still wondering whether you think we'd be better off keeping (/| instead of |)), since ||\) seems to be the only choice for its complement. We could replace ~)| 17:19 comma and |~) 13:19 comma which are way down the popularity list anyway (Nos 71 and 45, Ocurrences 0.08% and 0.15%), with ')|( 17:19 comma and '//| 13:19 comma Whaddya think? I understand we will never have a use for /|( since it is a synonym for |) and we don't yet have a use for )|\ which is very close to ~|) and could be replaced by '(|. '(| also has no known use so far, but we get it for free. So I count 26 single-shaft up symbols in all. I'm thinking we may need to revisit the apotome-complement issue again, with this symbol reorganisation. I don't think that an un-accented symbol should ever have a complement that is accented or vice versa. Is this possible? It might be a good idea to try notating 612-ET and 624-ET before settling this. By the way, in your otherwise excellent quick reference, I must object to the line 7:17 diesis 448:459 ~41.995c (for 217 mapping) A correct symbol for this diesis would be either .~|\ or (not quite) '//| I don't think the fact that //| can represent this 7:17 diesis in 217-ET is relevant here, where the universal comma roles of the symbols are being defined.

Message: 6131 - Contents - Hide Contents Date: Tue, 21 Jan 2003 19:12:18 Subject: Re: Superparticular temperaments From: Carl Lumma>A set of poptimal generators,I thought something had cast doubt on the poptimal stuff. And didn't Paul say the entire p range tends to minimax?>chord indexing function.What's a chord indexing function? -Carl

Message: 6132 - Contents - Hide Contents Date: Tue, 21 Jan 2003 15:55:58 Subject: Re: A common notation for JI and ETs From: David C Keenan George, To help with the development of the outline font, could you put up a bitmap or gif with all the _up_ symbols we've found a need for so far, showing (in the single-shaft case) the keyboard character that you want them mapped to. This would be an update of your Symbols3.bmp which is the latest I can find on the tuning-math files area. Don't include any combinations with 5' accents, just one with the acute accent beside a plain shaft. I'm thinking that we no longer need any 3-flag symbols (other than those where the third "flag" is a 5' accent) and the only 2-flags-on-the-same-side symbols that we need are //| and |)) and |\), although I'm still wondering whether you think we'd be better off keeping (/| instead of |)), since ||\) seems to be the only choice for its complement. We could replace ~)| 17:19 comma and |~) 13:19 comma which are way down the popularity list anyway (Nos 71 and 45, Ocurrences 0.08% and 0.15%), with ')|( 17:19 comma and '//| 13:19 comma Whaddya think? I understand we will never have a use for /|( since it is a synonym for |) and we don't yet have a use for )|\ which is very close to ~|) and could be replaced by '(|. '(| also has no known use so far, but we get it for free. So I count 26 single-shaft up symbols in all. I'm thinking we may need to revisit the apotome-complement issue again, with this symbol reorganisation. I don't think that an un-accented symbol should ever have a complement that is accented or vice versa. Is this possible? It might be a good idea to try notating 612-ET and 624-ET before settling this. By the way, in your otherwise excellent quick reference, I must object to the line 7:17 diesis 448:459 ~41.995c (for 217 mapping) A correct symbol for this diesis would be either .~|\ or (not quite) '//| I don't think the fact that //| can represent this 7:17 diesis in 217-ET is relevant here, where the universal comma roles of the symbols are being defined.

Message: 6133 - Contents - Hide Contents Date: Tue, 21 Jan 2003 19:18:48 Subject: Re: A 13-limit comma list From: Carl Lumma>Here is a list of 70 13-limit commas, with epimericity less >than 0.25 and size less than 50 cents. There is a very good >chance it is complete,Great! 50 cents is plenty big, and that list is plenty small!! Epimericity rocks. -C.

Message: 6134 - Contents - Hide Contents Date: Tue, 21 Jan 2003 08:04:39 Subject: Re: Superparticular temperaments From: Carl Lumma>> > dunno Gene; looks like a bust to me. I wonder what >> other easy stuff can be done to fractions to study commas. >> There are jacks, but they're a subset of the above... >> While we would expect superparticulars to be the smallest >> intervals of a given complexity, there must be a cleaner >> way of doing this... has anything been done on 'badness for >> commas'? >> the heurstic. note that |n-d| is right in there, so if you > define badness function that falls off instead of being > log-flat, you can easily arrange for the superparticulars > to be the "least bad".Can you explain how the complexity heuristic differs with Gene's epimericity? Still don't know how log(d) [the error heuristic] is reckoned to tell RMS error.> however, once a temperament is defined by more than one > comma, this approach will miss, since straightness is being > ignored.No problem! I think linear temperaments are the most interesting. Two sets of accidentals would be too confusing fer the likes of me! I'd just use JI. If I'm going to temper, give me error, low complexity, and simple notation! -Carl

Message: 6135 - Contents - Hide Contents Date: Tue, 21 Jan 2003 19:42:33 Subject: Re: Superparticular temperaments From: Carl Lumma>>> >og(d) is the complexity heuristic, not the error heuristic. >>> the error heuristic is proportional to |n-d|/(d*log(d)). >>>> I swear you had that backwards when you defined "step" and >> "cent" for me recently. > > where?D'oh! You had it right.>>> however, once a temperament is defined by more than >>> one comma, this approach will miss, since straightness >>> is being ignored. // >>> a 7-liimit linear temperament is defined by a *pair* of unison >>> vectors, so straighness comes into play there already. //>> And don't you mean that straigtness would _not_ come into >> play, by what you say in the first paragraph (more than one >> comma = ignore straightness)? >>in what "first paragraph" do i say this?? i've been saying >the exact opposite!!!Look ye above.>> Adaptive JI. It should go without saying with me. I didn't >> even know about strict JI until joining this list, and it >> took 2 years of confusion before we figured it out!! >> you're kidding me, right?No, why would you say that? Certainly you remember all those heated arguments about whether something was 'possible in JI'. It was quite ironic, actually.> well anyway, adaptive JI in general can be even more complex > to specify . . . do you specifically mean adaptive JI on a > meantone basis?No. I'm primarily interested in three types of adaptive JI. One is painstakingly described here... * [with cont.] (Wayb.) ...and completely automatic. Another, writing intervals and roots on scores as ratios, is complicated to perform (unless you're a computer), but easy to notate, since I already know ratios. I was thinking I should scan in the score of Retrofit to show this technique. The third is painstakingly described here... * [with cont.] (Wayb.)>>>> If I'm going to temper, give me error, low complexity, and >>>> simple notation! >>>>>> it should be a book of temperaments. for each, a notation, >>> a generalized keyboard, a halbestadt-imitation keyboard, a >>> horogram . . . >>>> Sounds good, but what does this have to do with my >> comment? > > notation.But the book contains notation??? Anyway, all I was saying was, I'm not interested in the more accurate temperaments. There are already linear temperaments that are too accurate through the 7-limit. Since we don't really know what the 11- and 13- limits look like yet, I won't rule out my interest in planar temperaments there... yet. But a lot of accuracy is required there. And it's already so weird that tempering might be asking for trouble. :) -Carl

Message: 6136 - Contents - Hide Contents Date: Tue, 21 Jan 2003 10:40:16 Subject: Re: Superparticular temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> I dunno Gene; looks like a bust to me. I wonder what >>> other easy stuff can be done to fractions to study commas. >>> There are jacks, but they're a subset of the above... >>> While we would expect superparticulars to be the smallest >>> intervals of a given complexity, there must be a cleaner >>> way of doing this... has anything been done on 'badness for >>> commas'? >>>> the heurstic. note that |n-d| is right in there, so if you >> define badness function that falls off instead of being >> log-flat, you can easily arrange for the superparticulars >> to be the "least bad". >> Can you explain how the complexity heuristic differs with > Gene's epimericity?heuristic complexity is merely log(d) . . . epimericity can be expressed as a function of *both* heuristic complexity and heuristic eror -- thus it's a kind of badness measure, though not log-flat.> Still don't know how log(d) [the error heuristic] is > reckoned to tell RMS error.log(d) is the complexity heuristic, not the error heuristic. the error heuristic is proportional to |n-d|/(d*log(d)).>> however, once a temperament is defined by more than one >> comma, this approach will miss, since straightness is being >> ignored. >> No problem! I think linear temperaments are the most > interesting.a 7-liimit linear temperament is defined by a *pair* of unison vectors, so straighness comes into play there already.> Two sets of accidentals would be too confusing > fer the likes of me! I'd just use JI.5-limit JI has two sets of accidentals; other planar temperaments are no more complex, and can reach effectively into the 11-limit or whatever . . .> If I'm going to > temper, give me error, low complexity, and simple notation!it should be a book of temperaments. for each, a notation, a generalized keyboard, a halbestadt-imitation keyboard, a horogram . . .

Message: 6137 - Contents - Hide Contents Date: Tue, 21 Jan 2003 19:56:45 Subject: Re: heuristic and straightness From: Carl Lumma>>> >he heuristics are only formulated for the one-unison-vector >>> case (e.g., 5-limit linear temperaments), and no one has >>> bothered to figure out the metric that makes it work exactly >>> (though it seems like a tractable math problem). but they do >>> seem to work within a factor of two for the current "step" >>> and "cent" functions. "step" is approximately proportional to >>> log(d), and "cent" is approximately proportional to >>> (n-d)/(d*log(d)). >>>> Why are they called "step" and "cent"? How were they derrived? >>that's what gene used to call them. "step" is simply complexity, >and "cent" is simply rms error.Now, look here. Maybe this was obvious to everyone but me, but a single paragraph on the derrivation each of these would have saved us much heartache... "There are complexity and error heuristics. They approximate many different complexity and error functions (resp.) of temperaments in which one comma is tempered out, through simple math on the ratio, n/d (in lowest terms, n > d) representing the comma that is tempered out. "The complexity heuristic is log(d). It works because ratios sharing denominator d are confined to a certain radius on the harmonic lattice. blah blah blah "The error heuristic is |n-d|/d*log(d). It works because it reflects the size of the comma, per the number of consonant intervals over which it must vanish. That is, ratios whose n is far bigger than d are larger, and you'll recognize the complexity heuristic underneath. blah blah blah "To apply the heuristics to temperaments where more than one comma vanishes, we might consider each of them in turn, but we must be careful to include the difference/sum vector, because it too must vanish. A concept called "straightness" measures the angle between the commas on the harmonic lattice, and therefore the relative length of the difference/sum vector. blah blah blah" Just an example, and probably contains errors. I must fly to lunch! -Carl

Message: 6138 - Contents - Hide Contents Date: Tue, 21 Jan 2003 11:01:34 Subject: Re: Superparticular temperaments From: Carl Lumma> heuristic complexity is merely log(d) . . . epimericity can be > expressed as a function of *both* heuristic complexity and > heuristic eror -- thus it's a kind of badness measure, though > not log-flat. // > log(d) is the complexity heuristic, not the error heuristic. > the error heuristic is proportional to |n-d|/(d*log(d)).I swear you had that backwards when you defined "step" and "cent" for me recently. Anyway, all is clear now!!>>> however, once a temperament is defined by more than one >>> comma, this approach will miss, since straightness is being >>> ignored. >>>> No problem! I think linear temperaments are the most >> interesting. >> a 7-liimit linear temperament is defined by a *pair* of unison > vectors, so straighness comes into play there already.When you say "temerament is defined by", you mean all the uvs, only the commatic ones, ...? And don't you mean that straigtness would _not_ come into play, by what you say in the first paragraph (more than one comma = ignore straightness)?>> Two sets of accidentals would be too confusing >> fer the likes of me! I'd just use JI. >> 5-limit JI has two sets of accidentals;Adaptive JI. It should go without saying with me. I didn't even know about strict JI until joining this list, and it took 2 years of confusion before we figured it out!!>> If I'm going to temper, give me error, low complexity, and >> simple notation! >> it should be a book of temperaments. for each, a notation, a > generalized keyboard, a halbestadt-imitation keyboard, a > horogram . . .Sounds good, but what does this have to do with my comment? -Carl

Message: 6139 - Contents - Hide Contents Date: Tue, 21 Jan 2003 21:11:14 Subject: Re: A 13-limit comma list From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> why do you keep insisting on "size less than 50 cents"? pelogic, > for example relies on a much larger comma; and even negri, > with its very low error, has a comma of 51 cents.I'm not insisting on it; it's just a nice, round number. What would you suggest?

Message: 6140 - Contents - Hide Contents Date: Tue, 21 Jan 2003 11:10:06 Subject: Re: Superparticular temperaments From: Graham Breed Carl Lumma wrote:> When you say "temerament is defined by", you mean all the uvs, > only the commatic ones, ...?You only need the commatic UVs to define the temperament. From those, all you need is the wedge product. It's useful to have a chromatic UV for doing the calculations, but they're interchangeble. Choosing each prime axis in turn will eventually give you a valid chromatic UV. The choice is only important when you look at the periodicity block, or corresponding MOS. If the badness can be calculated solely from the wedge product of the commatic UVs, then it's naturally independent of the specific UVs chosen. I'm not sure if that's what Gene's proposing. Graham

Message: 6142 - Contents - Hide Contents Date: Tue, 21 Jan 2003 22:59:09 Subject: Re: heuristic and straightness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>> ??? If straightness is maximal when the uvs are maximally >>> orthogonal, how does this mean the uvs have gotten shorter? >>>> any basis for the temperament that uses much longer unison >> vectors will have to have much less straightness, because the >> area/volume/etc. enclosed by the UVs must remain constant. >> otherwise, you have torsion. >> So is this right: > > Straightness...LengthUVs...Length+/-UV...Badness > Down...........Up..........Down..........Same > Up.............Down........Up............Same > > ?if you replace "badness" with "error", it's right.> If so, you can't say that changing the straightness of a > given temperament makes it "worse". But am I correct > that across temperaments, 'crooked' ones will tend to be > worse (assuming we've already considered the LengthUVs)?if you're using TM reduction or something similar to define the basis, then yes, this will be the general trend.

Message: 6143 - Contents - Hide Contents Date: Tue, 21 Jan 2003 22:59:49 Subject: Re: Superparticular temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>> A set of poptimal generators, >> I thought something had cast doubt on the poptimal stuff. > And didn't Paul say the entire p range tends to minimax?didn't i say what?

Message: 6144 - Contents - Hide Contents Date: Tue, 21 Jan 2003 23:04:41 Subject: Re: Superparticular temperaments From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> however, once a temperament is defined by more than >>>> one comma, this approach will miss, since straightness >>>> is being ignored. > //>>>> a 7-liimit linear temperament is defined by a *pair* of unison >>>> vectors, so straighness comes into play there already. > //>>> And don't you mean that straigtness would _not_ come into >>> play, by what you say in the first paragraph (more than one >>> comma = ignore straightness)? >>>> in what "first paragraph" do i say this?? i've been saying >> the exact opposite!!! >> Look ye above.yes, it's the exact opposite of what you thought -- if there's only one comma, you're not missing or ignoring anything, because straightness doesn't even exist (let alone come into play).>>> Adaptive JI. It should go without saying with me. I didn't >>> even know about strict JI until joining this list, and it >>> took 2 years of confusion before we figured it out!! >>>> you're kidding me, right? >> No, why would you say that?you didn't refer to pitches as ratios?> Anyway, all I was saying was, > I'm not interested in the more accurate temperaments. There > are already linear temperaments that are too accurate through > the 7-limit. Since we don't really know what the 11- and 13- > limits look like yet, I won't rule out my interest in planar > temperaments there... yet. But a lot of accuracy is required > there. And it's already so weird that tempering might be asking > for trouble. :)i'm also not interested in the more accurate temperaments. they're mathematical curiosities. what interests me are temperaments where a "small-numbered" MOS already contains a good deal of the desired harmonies, and you can develop a grammar around the harmonic meaning of scalar alterations, etc.

Message: 6145 - Contents - Hide Contents Date: Tue, 21 Jan 2003 23:06:33 Subject: Re: heuristic and straightness From: wallyesterpaulrus sorry for the heartache, folk(s). save this post! --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> the heuristics are only formulated for the one-unison-vector >>>> case (e.g., 5-limit linear temperaments), and no one has >>>> bothered to figure out the metric that makes it work exactly >>>> (though it seems like a tractable math problem). but they do >>>> seem to work within a factor of two for the current "step" >>>> and "cent" functions. "step" is approximately proportional to >>>> log(d), and "cent" is approximately proportional to >>>> (n-d)/(d*log(d)). >>>>>> Why are they called "step" and "cent"? How were they derrived? >>>> that's what gene used to call them. "step" is simply complexity, >> and "cent" is simply rms error. >> Now, look here. Maybe this was obvious to everyone but me, but > a single paragraph on the derrivation each of these would have > saved us much heartache... > > "There are complexity and error heuristics. They approximate > many different complexity and error functions (resp.) of > temperaments in which one comma is tempered out, through simple > math on the ratio, n/d (in lowest terms, n > d) representing > the comma that is tempered out. > > "The complexity heuristic is log(d). It works because ratios > sharing denominator d are confined to a certain radius on the > harmonic lattice. blah blah blah > > "The error heuristic is |n-d|/d*log(d). It works because it > reflects the size of the comma, per the number of consonant > intervals over which it must vanish. That is, ratios whose > n is far bigger than d are larger, and you'll recognize the > complexity heuristic underneath. blah blah blah > > "To apply the heuristics to temperaments where more than one > comma vanishes, we might consider each of them in turn, but > we must be careful to include the difference/sum vector, > because it too must vanish. A concept called "straightness" > measures the angle between the commas on the harmonic lattice, > and therefore the relative length of the difference/sum vector. > blah blah blah" > > > Just an example, and probably contains errors. I must fly > to lunch! > > -Carl

Message: 6146 - Contents - Hide Contents Date: Tue, 21 Jan 2003 23:06:32 Subject: Re: heuristic and straightness From: wallyesterpaulrus sorry for the hearache, folk(s). save this post! --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> the heuristics are only formulated for the one-unison-vector >>>> case (e.g., 5-limit linear temperaments), and no one has >>>> bothered to figure out the metric that makes it work exactly >>>> (though it seems like a tractable math problem). but they do >>>> seem to work within a factor of two for the current "step" >>>> and "cent" functions. "step" is approximately proportional to >>>> log(d), and "cent" is approximately proportional to >>>> (n-d)/(d*log(d)). >>>>>> Why are they called "step" and "cent"? How were they derrived? >>>> that's what gene used to call them. "step" is simply complexity, >> and "cent" is simply rms error. >> Now, look here. Maybe this was obvious to everyone but me, but > a single paragraph on the derrivation each of these would have > saved us much heartache... > > "There are complexity and error heuristics. They approximate > many different complexity and error functions (resp.) of > temperaments in which one comma is tempered out, through simple > math on the ratio, n/d (in lowest terms, n > d) representing > the comma that is tempered out. > > "The complexity heuristic is log(d). It works because ratios > sharing denominator d are confined to a certain radius on the > harmonic lattice. blah blah blah > > "The error heuristic is |n-d|/d*log(d). It works because it > reflects the size of the comma, per the number of consonant > intervals over which it must vanish. That is, ratios whose > n is far bigger than d are larger, and you'll recognize the > complexity heuristic underneath. blah blah blah > > "To apply the heuristics to temperaments where more than one > comma vanishes, we might consider each of them in turn, but > we must be careful to include the difference/sum vector, > because it too must vanish. A concept called "straightness" > measures the angle between the commas on the harmonic lattice, > and therefore the relative length of the difference/sum vector. > blah blah blah" > > > Just an example, and probably contains errors. I must fly > to lunch! > > -Carl

Message: 6147 - Contents - Hide Contents Date: Tue, 21 Jan 2003 23:06:35 Subject: Re: heuristic and straightness From: wallyesterpaulrus sorry for the heartache, folk(s). save this post! --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:>>>> the heuristics are only formulated for the one-unison-vector >>>> case (e.g., 5-limit linear temperaments), and no one has >>>> bothered to figure out the metric that makes it work exactly >>>> (though it seems like a tractable math problem). but they do >>>> seem to work within a factor of two for the current "step" >>>> and "cent" functions. "step" is approximately proportional to >>>> log(d), and "cent" is approximately proportional to >>>> (n-d)/(d*log(d)). >>>>>> Why are they called "step" and "cent"? How were they derrived? >>>> that's what gene used to call them. "step" is simply complexity, >> and "cent" is simply rms error. >> Now, look here. Maybe this was obvious to everyone but me, but > a single paragraph on the derrivation each of these would have > saved us much heartache... > > "There are complexity and error heuristics. They approximate > many different complexity and error functions (resp.) of > temperaments in which one comma is tempered out, through simple > math on the ratio, n/d (in lowest terms, n > d) representing > the comma that is tempered out. > > "The complexity heuristic is log(d). It works because ratios > sharing denominator d are confined to a certain radius on the > harmonic lattice. blah blah blah > > "The error heuristic is |n-d|/d*log(d). It works because it > reflects the size of the comma, per the number of consonant > intervals over which it must vanish. That is, ratios whose > n is far bigger than d are larger, and you'll recognize the > complexity heuristic underneath. blah blah blah > > "To apply the heuristics to temperaments where more than one > comma vanishes, we might consider each of them in turn, but > we must be careful to include the difference/sum vector, > because it too must vanish. A concept called "straightness" > measures the angle between the commas on the harmonic lattice, > and therefore the relative length of the difference/sum vector. > blah blah blah" > > > Just an example, and probably contains errors. I must fly > to lunch! > > -Carl

Message: 6148 - Contents - Hide Contents Date: Tue, 21 Jan 2003 23:07:52 Subject: Re: A 13-limit comma list From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:>>> why do you keep insisting on "size less than 50 cents"? pelogic, >> for example relies on a much larger comma; and even negri, >> with its very low error, has a comma of 51 cents. >> I'm not insisting on it; it's just a nice, round number. What would >you suggest?i'm mystified as to why comma size would either be a convenient, or a useful, way to set a bound on the search . . .

Message: 6149 - Contents - Hide Contents Date: Tue, 21 Jan 2003 23:12:09 Subject: Re: Calculating geometric complexity and badness From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> > I get a slightly different result calculating geometric complexity for > meantone using the 5 limit comma formula. Should it be 4.132030727? Thanks. > Also I was wondering why complexity for a planar temperment doesn't include > u4 (power of 11).probably gene's typos . . .> Thanks, its neat to get the formulas for complexity and > badness. If its not asking too much, could I also get the formula for rms > (in the general sense, of course, I know the formula for the 2- optimal > generator )you mean p=2? that *is* rms -- "root mean square". are you looking for the formula for the p-norm? it's (|x1|^p + |x2|^p + |x3|^p + . . . + |xn|^p)^(1/p), where n is the number of consonant intervals and xi is the error in interval i.

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