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Message: 7700 Date: Wed, 22 Oct 2003 06:59:26 Subject: Re: naive question From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> > I'm not familiar with Ellis' work, believe it or not. > > I've always thought of Bosanquet as the first to really > > work with the idea of linear temperaments (c. 1876). > > Followed by Fokker, Wilson, Graham Breed, Dave Keenan, > > Paul Erlich, and Gene Smith. But Gene's work > > may actually come much earlier in this list...

> > I suspect Paul Erlich should come before me on that list. > In any case Breed-Erlich-Keenan was pretty much a simultaneous > cooperative effort that definitely built on earlier stuff by > Wilson.

i think the history of theoretical speculation on linear temperaments begins with writings concerning meantone, doesn't it? Woolhouse seems to have been the first to give a rigourously mathematical underpinning to it, c. 1835: W. S. B. Woolhouse's 'Essay on musical interva... * [with cont.] (Wayb.) but Zarlino gave a mathematically precise explanation of /7-comma meantone in 1558: Zarlino 1558 _Le institutione harmoniche_ part... * [with cont.] (Wayb.) Bosanquet mentioned Woolhouse in the original draft of his book, but all references to Woolhouse were removed when it was published. Rasch discusses this in his foreward to the Diapason Press edition of Bosanquet's book. REFERENCE Bosanquet, Robert Halford Macdowall. 1876. _An Elementary Treatise on Musical Intervals and Temperament_. MacMillan & Co., London. - 2nd edition, with introduction by Rudolf Rasch (ed.). 1987. - Tuning and temperament library vol. 4, - Diapason Press, Utrecht. i think that the correct recent chronology is: Woolhouse-Bosanquet-Wilson-Erlich-Breed-Keenan-Smith paul? -monz

Message: 7701 Date: Wed, 22 Oct 2003 21:42:45 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> Even if we perversely make a > >> scale with one ratio from each prime limit up to 23, it's still

a

> >> PB.

> > > >show me.

> > Can we not take 3/2 and make 9:8 a uv, 5/4 and make 25:16 a uv, > 7/4 and make 28:16 a uv, etc.? > > -Carl

28:16 = 7/4. you mean 49:32? anyway, it sounds like you're suggesting a fokker matrix with 2 along the diagonal and 0 everywhere else. the determinants are: 3-limit: 2 5-limit: 4 7-limit: 8 and so on . . . you're creating an euler-fokker genus in each case, not a scale with one ratio from each prime limit . . . but anyway what kind of unison vectors are these? notationally you want 'good' ones where the notes are in the correct order, i.e., small unison vectors.

Message: 7702 Date: Wed, 22 Oct 2003 00:05:19 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> Sure it does -- the interval J-A is not a secor. >> >> A B C D E F G H I J A >> s s s s s s s s s n

> >I know, but the alphabet doesn't go J A.

This alphabet does! J-A is a 2nd, just like I-J. -Carl

Message: 7703 Date: Wed, 22 Oct 2003 00:09:20 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Can you point me to a post from Kraig or whatever it was that made

you

> decide MOS were only single-chain?

a bunch of posts from kraig and daniel in late march of this year on the specmus list.

Message: 7704 Date: Wed, 22 Oct 2003 21:45:32 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> > >> If you think like Gene, there's no problem, because you don't > > >> think of tunings that aren't consistent in the first place. > > >> I must admit I'm rather fond of this approach, though jumping > > >> between approximations inside an et could also be fun. > > >> > > >> -Carl

> > > > > >as herman miller showed with 64-equal, to monz's delight.

> > > > In what piece? A warped canon? > > > > -Carl

> > > > in his "Pavane for a Warped Princess" ... his retunings > of a famous piece by Ravel. > > Pavane For a Warped Princess * [with cont.] (Wayb.) > > > my favorite is the 64edo version, despite the fact that it's > inconsistent.

which meaning of "inconsistent" are you referring to here -- the fact that the major triads are not all constructed the same way, or the technical meaning as on your dictionary?

Message: 7705 Date: Wed, 22 Oct 2003 00:06:59 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> > >the linear, pythagorean diatonic scale.

>> > >> > So you consider this scale a "5-limit PB" then?

//

>> yes, for example you can enclose it with a parallelogram of unison >> vectors 81:80 and 2187:2048.

//

>Perhaps that gives us a reason that will satisfy Carl, as to why, if >you are forced to choose one set of nominals as the best single set >for all purposes, it must be Pythagorean-7. Namely because it is the >only (conveniently sized) one that is a PB at every limit.

Care to flesh that a bit? -C.

Message: 7706 Date: Wed, 22 Oct 2003 00:10:00 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> Now, if you want to always use the same PB for 5-limit JI, > >> the diatonic PB may be the best choice...

> > > >which, the just major? i disagree, of course.

> > Of course? What would beat it out (with less than 11 tones)? > > -Carl

how about the linear, pythagorean diatonic scale?

Message: 7707 Date: Wed, 22 Oct 2003 21:46:18 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> >> If you think like Gene, there's no problem, because you don't > >> >> think of tunings that aren't consistent in the first place. > >> >> I must admit I'm rather fond of this approach, though jumping > >> >> between approximations inside an et could also be fun.

> >> > > >> >as herman miller showed with 64-equal, to monz's delight.

> >> > >> In what piece? A warped canon? > >> > >> -Carl

> > > >in his "Pavane for a Warped Princess" ... his retunings > >of a famous piece by Ravel. > > > >Pavane For a Warped Princess * [with cont.] (Wayb.) > > > >my favorite is the 64edo version, despite the fact that it's > >inconsistent. as paul just said earlier, "the ear is the > >final arbiter".

> > Yes, but is Herman's technique really going to vacillate between > approximations?

yes, he used a fixed 12-tone subset of 64-equal, so different triads are constructed differently.

Message: 7708 Date: Wed, 22 Oct 2003 00:26:25 Subject: Re: Why Johnston notation sucks (was: Polyphonic notation) From: Carl Lumma

>Consider this scenario: > >You're sight-reading a 5-limit strict-JI piece on the violin, or any >variable pitch (or fixed-pitch 5-comma distinguishing) instrument that >can play at least two notes at once. > >You see a major third on the staff, or at least it would be a normal >old 12-ET major third if you were allowed to ignore all those funny >new comma accidentals you've been forced to learn and the fact that >it's supposed to be in JI. You see that neither of the notes in this >dyad have any 5-comma accidentals. Do you play it Pythagorean or Just >or a comma narrower than just?

This doesn't strike me as a plausible thought process for a performer. You've got to know the scale, A1 priority. If you don't, you're screwed. Applying offsets from that is a natural way to think. Thinking in terms of the set of all just intervals doesn't strike me as the way it would like to go.

>With Johnston notation you would play it Just ... _except_ if the low >note happens to be a D (or Db or D# etc) in which case you play it a >comma lower than just.

The thing you're glossing over is that by using Johnston notation, the composer is requesting the 5-limit diatonic scale. So in Sagittal you've got lots of accidentals on basic scale members, which are superfluous from this point of view.

>You may be able to find some examples that make the Johnston notation >look good, but these will be pretty much confined to the 7-note JI >major scale itself.

Probably the bulk of music stays in the "scale itself", in the case of pythagorean and meantone. -Carl

Message: 7709 Date: Wed, 22 Oct 2003 00:34:10 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan Sorry I allowed that ambiguity. I should indeed have said "rational" rather than simply "ratio". When I am not considering strict JI then I am happy for the nominals to be contiguous on a uniform chain of some irrational generator, including linear-temperament-specific cases where the nominals are in a DE scale of that temperament with cardinality other than 7. I can't support notations that do otherwise, such as Johnstons, because as Paul said, it just adds more things you have to keep track of - commas that vanish between some pairs of nominals and not others. Even the JI folks that use it express reservations. Yes it is like the BF case, but one of those "wolves" per notation is already too many. Interestingly this problem doesn't occur at all in decimal miracle, or any other notation based on a DE scale that doesn't "wrap around".

Message: 7710 Date: Wed, 22 Oct 2003 15:57:04 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> Yes, but is Herman's technique really going to vacillate between >> approximations?

> >yes, he used a fixed 12-tone subset of 64-equal, so different triads >are constructed differently.

Fascinatin'. -Carl

Message: 7711 Date: Wed, 22 Oct 2003 07:27:48 Subject: [tuning] Re: Polyphonic notation From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >

> > It isn't ratios in general > > that you need to notate, but commas.

> > I disagree. Most musicians and composers couldn't care less > about what comma an accidental stands for, they are happy just > to know that if 1/1 is C then 5/4 is E\ and 7/4 is Bb< and so > on. How many could tell you, or would care, that in Pythagorean > a sharp or flat symbolises the comma 2187/2048? But they sure > know that F# is a fifth above B.

i think the basis of a notation is how a tuning conforms to the L,s concept: Definitions of tuning terms: L (Large), s (sma... * [with cont.] (Wayb.) the L (large) step determines the nominals, and the s (small) step determines the main pair of accidentals. other accidentals may be added to indicate other commatic inflections. so i'm not so sure that i can agree with you here, Dave. if someone is experienced with working a Pythagorean tuning, he/she certainly knows what 2187/2048 *sounds* like, even if not familiar with the mathematical description of it. i think the ancient were on to something here that has been rather lost among our modern theory and experience, in that the chromatic semitone is a transition from one type of genus to another. if someone is performing a piece in Pythagorean tuning, and it has been going along diatonically, then all of a sudden there is an accidental indicatinag a chromatic change, that comma (2187/2048) will have an *extremely* distinctive sound. labeling that distinctive sounds makes perfect sense ... as much sense as using different nominals to name the diatonic steps.

> > But of course it makes sense to notate ratios in such a way > that their symbols can be factored into nominal and accidental > parts such that the accidental has a constant meaning as a > certain comma no matter which nominal it is used with (and > vice versa).

hmmm ... can anyone comment on how consistency of notation is similar to or different from consistency in the erlichian sense? what i'm getting at is this: is it possible to come up with a notation for a scale which is inconsistent, such that it eliminates the consistency problem for that tuning? -monz

Message: 7712 Date: Wed, 22 Oct 2003 15:55:34 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> Can we not take 3/2 and make 9:8 a uv, 5/4 and make 25:16 a uv, >> 7/4 and make 28:16 a uv, etc.? >> >> -Carl

> >28:16 = 7/4. you mean 49:32?

Yes.

>anyway, it sounds like you're suggesting >a fokker matrix with 2 along the diagonal and 0 everywhere else. the >determinants are: > >3-limit: 2 >5-limit: 4 >7-limit: 8 > >and so on . . . you're creating an euler-fokker genus in each case,

Crud, you're right. But also, I suppose any non-convex structure would be non-PB. Are there any such structures we couldn't live without?

>not a scale with one ratio from each prime limit . . . but anyway >what kind of unison vectors are these?

"perverse" ones. -Carl

Message: 7713 Date: Wed, 22 Oct 2003 00:32:35 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>hmmm ... can anyone comment on how consistency of notation >is similar to or different from consistency in the erlichian >sense?

I'll guess you'll have to define what you mean by "consistency in notation". -Carl

Message: 7714 Date: Wed, 22 Oct 2003 00:35:58 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Sorry I allowed that ambiguity. I should indeed have said "rational" > rather than simply "ratio". > > When I am not considering strict JI then I am happy for the nominals > to be contiguous on a uniform chain of some irrational generator, > including linear-temperament-specific cases where the nominals are

in

> a DE scale of that temperament with cardinality other than 7. > > I can't support notations that do otherwise, such as Johnstons, > because as Paul said, it just adds more things you have to keep

track

> of - commas that vanish between some pairs of nominals and not

others.

> Even the JI folks that use it express reservations. Yes it is like

the

> BF case, but one of those "wolves" per notation is already too many. > Interestingly this problem doesn't occur at all in decimal miracle,

or

> any other notation based on a DE scale that doesn't "wrap around".

what do you mean? certainly there's a single interval of 6 nominals in decimal miracle that fails to be an approximate 2:3.

Message: 7715 Date: Wed, 22 Oct 2003 00:36:34 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> >> Now, if you want to always use the same PB for 5-limit JI, > >> >> the diatonic PB may be the best choice...

> >> > > >> >which, the just major? i disagree, of course.

> >> > >> Of course? What would beat it out (with less than 11 tones)? > >> > >> -Carl

> > > >how about the linear, pythagorean diatonic scale?

> > Ok, I meant 5-limit PB. > > -Carl

which one? (we're going around in circles)

Message: 7716 Date: Wed, 22 Oct 2003 07:52:17 Subject: [tuning] Re: Polyphonic notation From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> > So in HEWM D:F and D:F# aren't pure thirds?

> > no, they're pythagorean. >

> > If so, there's no reduction in the number of new and > > strange accidentals.

> > right, but their usage is more straightforward. >

> > I could > > believe that for the diatonic scale, pythagorean notation > > would be a more natural basis.

> > then you're agreeing with dave and me.

and me, of course. Carl, i suggest that you re-read not only my Dictionary page on HEWM: Definitions of tuning terms: HEWM, (c) 2001 by... * [with cont.] (Wayb.) but also the one on Johnston notation: Definitions of tuning terms: Johnston notation... * [with cont.] (Wayb.) the big problem with Johnston notation is that it requires the user to keep a 2-dimensional reference scale in mind, which complicates things far more than necessary. if one only need keep a 1-dimensional chain of notes in mind as either his set of nominals (i.e., the pythagorean diatonic scale) or his nominals-plus-accidentals (setting an arbitrary limit somewhat based on historical usage, the 35-tone pythagorean chain stretching from Fbb to Bx), matters are greatly simplified. (they would be simply even further if all nominals could be used for the entire chain, thus giving an absolute consistency to the meaning of the notational symbols.) so anyway, HEWM and sagittal have a pythagorean basis for this reason, and also because it reflects historical usage. indeed, if they were to ignore historical usage, they *would* employ a larger nominal set to represent a 12-tone pythagorean chain rather than a 7-tone chain, since the pythagorean comma (3^12) is a less-perceptible change in tuning than a 2187/2048 chromatic semitone (3^7). by forcing the user to remember a specific 2-dimensional set of notes which has significant exponents of both 3 and 5, Johnston notation is much harder to keep in mind, and also results in the kinds of sonic/notational inconsistencies pointed out by paul with his examples of "perfect-5ths". in HEWM and sagittal, there is one pair of symbols which represents respectively a sharpening or flattening of any pythagorean note by a syntonic comma. that system is much easier to remember conceptually. -monz -monz

Message: 7717 Date: Wed, 22 Oct 2003 07:56:03 Subject: [tuning] Re: Polyphonic notation From: monz --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> >> If so, there's no > >> >> reduction in the number of new and strange accidentals.

> >> > > >> >right, but their usage is more straightforward.

> >> > >> Only because the Pythagorean scale is actually a decent > >> temperament of the 5-limit diatonic scale.

> > > >no, it's because of the linearity (1-dimensionality) of the set of > >nominals.

> > Then one could just as well use a 7-tone chain-of-minor-thirds. > But I don't think that would work, do you?

of course it would! ... if one wanted to write a piece in which a 7-tone chain-of-minor-3rds was a prominent feature! in fact, a notation based on that would be a good notation for the diatonic subset of 19edo, if i'm not mistaken (but i might be ... this is just a passing thought).

>

> >> I don't think scales without a good series of fifths, such > >> as untempered kleismic[7], would work so well.

> > > >johnston notation does at least as poorly.

> > You mean generalized johnston notation, of the kind I've been > discussing with Dave, or actual Johnston notation?

i missed your discussion of "generalized johnston". please point me to more info. -monz

Message: 7718 Date: Wed, 22 Oct 2003 08:09:09 Subject: [tuning] Re: Polyphonic notation From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> > > Yes it is like the BF case, but one of those "wolves" > > > per notation is already too many.

> > > > It's never been a problem for me. I've never been reading > > a piece of music and stopped and said, "oh wait, there's > > that BF again; it's not a 2:3". Just me, I guess.

> > Well sure. Everyone's used to it. You learn it very early. > You _hear_ that it's different, then you count the black notes > as well as the white and you _see_ why it's different, and > eventually the nominals cease to even be letters of the > alphabet. They even start and end at C.

thanks, Dave, for taking us back to kindergarten music-theory. i mean that sincerely. i have had to struggle to figure out how to teach this stuff to my young students in a way that leaves them curious to learn more about tuning. i pretty much use exactly the approach you describe, counting first the letter-names to determine the name of the interval ("2nd", "3rd", etc.), then counting "half-steps" to determine from a chart i give them whether intervals of a certain name are "perfect", "major", "minor", "augmented", or "diminished". ... then later, when i think they're ready for it, i let them in on the fact that "back in the old days" composers wanted sharps to be different from flats, etc. etc. and by using their previous knowledge of how a certain number of "half-steps" in an interval determined its quality, then can extrapolate that now a certain number of commas differentiating intervals of the same name and quality indicates their prime-limit. i'm getting them on our side early. ;-)

> I was just deliberately adopting a naive stance to point > something out.

i think it worked. :) -monz

Message: 7719 Date: Wed, 22 Oct 2003 00:43:05 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> >> >> Now, if you want to always use the same PB for 5-limit JI, > >> >> >> the diatonic PB may be the best choice...

> >> >> > > >> >> >which, the just major? i disagree, of course.

> >> >> > >> >> Of course? What would beat it out (with less than 11 tones)? > >> >> > >> >> -Carl

> >> > > >> >how about the linear, pythagorean diatonic scale?

> >> > >> Ok, I meant 5-limit PB. > >> > >> -Carl

> > > >which one? (we're going around in circles)

> > Exactly! Which one would you use to notate 5-limit JI?

the linear, pythagorean diatonic scale.

Message: 7720 Date: Wed, 22 Oct 2003 01:09:02 Subject: standards From: Carl Lumma

>the big problem with Johnston notation is that it requires >the user to keep a 2-dimensional reference scale in mind, >which complicates things far more than necessary.

That seems to be what Dave and Paul are saying. But it's a fallacy that you can simplify music with notation. If the music really features a 2-D scale, the best notation is optimized for that scale. If the music wasn't featuring it, you wouldn't use such a notation, unless it happened to be some sort of standard. Making up standards is a little like putting the cart before the horse if you ask me, considering all the worthwhile extended-JI music ever made fits on a few cds. On the other hand, it's nice to have an engineered standard available, if someone wants one. But the idea of pushing for the adoption of a standard now, simply to prevent a non-engineered (evolved) one from later taking hold, is a very bad idea. Evolved standards may not always be optimal, but it's very hard to define optimal in advance. Evolved standards have the advantage that they beat competing approaches while the criteria for optimality were still up for grabs. It's important for people in a field to reach out and try different things, especially at first. -Carl

Message: 7721 Date: Wed, 22 Oct 2003 00:58:35 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:

> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>

> wrote:

> > > when the period is a fraction of an octave but the interval of > > > equivalence is still an octave (the latter, we tend to assume a > > > priori), we no longer have an MOS.

> > > > I'm with Carl on this one. > > > > Didn't you just say that Erv Wilson seems to have missed the > > fractional octave cases. If so, then he couldn't exactly have stated > > that these are not to be considered MOS, could he?

> > they are not, according to kraig and daniel.

You mean Daniel Wolf?

> they only would be if > the period were considered to be the interval of equivalence, which > is not how we view them in the context of tempering periodicity > block. instead, we keep the interval of equivalence at 1:2. >

> > Isn't it up to us whether or not to generalise it to these cases? We > > did so for quite some time. Or at least I did. I even thought Kraig > > supported this at one stage.

> > kraig turned around on this one -- you must have missed all that and > the ensuing discussion. >

> > Can you point me to a post from Kraig or whatever it was that made

> you

> > decide MOS were only single-chain?

> > i don't know where these posts are offhand, but there were quite a > few of them, and daniel got involved too (not on the tuning list of > course). must have happened while you were away from the list.

OK. You win. :-) I'll just avoid the term MOS entirely, and use DE scale. I don't see myself using MOS to mean single-chain DE scale. It would be too ambiguous given all the chopping and changing.

Message: 7722 Date: Wed, 22 Oct 2003 08:14:07 Subject: Re: naive question From: monz oops ... --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> i think the history of theoretical speculation on > linear temperaments begins with writings concerning meantone, > doesn't it? > > > Woolhouse seems to have been the first to give a rigourously > mathematical underpinning to it, c. 1835: > > W. S. B. Woolhouse's 'Essay on musical interva... * [with cont.] (Wayb.) > > > but Zarlino gave a mathematically precise explanation of > /7-comma meantone in 1558:

that should have been "2/7-comma meantone". sorry. -monz

Message: 7723 Date: Wed, 22 Oct 2003 01:12:18 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>

> > Interestingly this problem doesn't occur at all in decimal miracle,

> or

> > any other notation based on a DE scale that doesn't "wrap around".

> > what do you mean? certainly there's a single interval of 6 nominals > in decimal miracle that fails to be an approximate 2:3.

Yes, I should have explained more. If you consider the nominals circularly then you're right of course. In fact there are 6 of them 4-0, 5-1, 6-2, 7-3, 8-4, 9-5, so in that regard it's worse for fifths than 7 nominals in Pythagorean or Meantone. But if you only consider them alphabetically you don't _expect_ these 6 to be good fifths, whereas in meantone or Pythagorean, when given ABCDEFG and the fact that A-E and C-G are fifths, there is every reason to expect that B-F would be also.

Message: 7724 Date: Wed, 22 Oct 2003 08:17:50 Subject: [tuning] Re: Polyphonic notation From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:

> > > > But of course it makes sense to notate ratios in such a way > > that their symbols can be factored into nominal and accidental > > parts such that the accidental has a constant meaning as a > > certain comma no matter which nominal it is used with (and > > vice versa).

> > > hmmm ... can anyone comment on how consistency of notation > is similar to or different from consistency in the erlichian > sense? > > what i'm getting at is this: is it possible to come up with > a notation for a scale which is inconsistent, such that it > eliminates the consistency problem for that tuning?

i decided to re-word that last paragraph for better clarity: is it possible to come up with a consistent notation for an inconsistent tuning, such that it eliminates the consistency problem for scales in that tuning? -monz

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