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Message: 7675 Date: Wed, 22 Oct 2003 02:21:32 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> >> >> 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!

Right you are. But in the context of the diatonic scale here...

>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). > A ... E ... B > / \ / \ / \ > / \ / \ / \ >F ... C ... G ... D

In kleismic... 1/1 = C 252.632 = D 315.789 = E 568.421 = F 631.579 = G 884.211 = A 947.368 = B

> A --- E^ -- B^^ > / \ / \ / \ > / \ / \ / \ >Fv -- C --- G^ -- Dv

Whaddya think? The situation is *much* worse when trying to use kleismic to notate the ji diatonic. Considering how 1-D and linear kleismic[7] is, we need to seriously question Paul's claim.

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

See my discussion with Dave. I didn't use that term, but I've certainly spent a lot of ink on it. -Carl

Message: 7676 Date: Wed, 22 Oct 2003 19:29:21 Subject: Re: standards From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >but try to modulate into almost any other key, and you'll > >start having problems.

> //

> >for example,

> //

> >simply modulating from C-major to its *nearest* relative, > >G-major, invokes this monstrosity: > > > > E ... B ... F#+ > > / \ / \ / \ > > / \ / \ / \ > > C ... G ... D ... A+

> //

> >in my HEWM, they are simply: > > > >C-major:

> > [Involves three non-standard accidentals > vs. Johnston notation.] >

> >G-major: > > > > E-... B-... F#- > > / \ / \ / \ > > / \ / \ / \ > > C ... G ... D ... A

> > How is this supposed to be better than the above? > > -Carl

it's way better . . . the notation much more clearly corresponds to the ratios, and what looks like a fifth quacks like a fifth.

Message: 7677 Date: Wed, 22 Oct 2003 02:24:07 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma I meant to add...

>In kleismic... > >1/1 = C >252.632 = D >315.789 = E >568.421 = F >631.579 = G >884.211 = A >947.368 = B >

>> A --- E^ -- B^^ >> / \ / \ / \ >> / \ / \ / \ >>Fv -- C --- G^ -- Dv

...that ^/v symbols here mean changes of 63-and-some cents. -C.

Message: 7678 Date: Wed, 22 Oct 2003 19:33:29 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > > So far I have been satisfied with using MOS to mean a scale

based

> > on

> > > the period+generator representation of a linear temperament

which

> > has

> > > Myhill's property.

> > > > the the period is not one octave, the scale doesn't have myhill's > > property.

> > Of course it does; forget the damned octave equivalence, it isn't > relevant.

of course it's relevant! when you start with a big ji lattice, and then start tempering out commas, observing octave equivalence correctly is crucial.

Message: 7679 Date: Wed, 22 Oct 2003 01:44:14 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >the linear, pythagorean diatonic scale.

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

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

Message: 7680 Date: Wed, 22 Oct 2003 11:28:09 Subject: Re: A 13-limit JI scale From: Manuel Op de Coul

>I presumed it was likely to be a known stellated something or other, >but Scala didn't tell me that. Is there I way I could have gotten it >to?

I used the compare command. Another slightly crude way is "iter/key lat/diam" and looking if you see a filled square. Manuel

Message: 7681 Date: Wed, 22 Oct 2003 19:36:18 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > > So far I have been satisfied with using MOS to mean a scale

based

> > on

> > > the period+generator representation of a linear temperament

which

> > has

> > > Myhill's property.

> > > > the the period is not one octave, the scale doesn't have myhill's > > property.

> > Of course it does; forget the damned octave equivalence, it isn't > relevant.

i hate appeals to authority, but 'distributionally even' was the only accepted 'academic' qualifier for my symmetrical decatonic scale according to john clough when i spoke to him. myhill it isn't.

Message: 7682 Date: Wed, 22 Oct 2003 01:44:44 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > 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?

> > So far I have been satisfied with using MOS to mean a scale based

on

> the period+generator representation of a linear temperament which

has

> Myhill's property.

the the period is not one octave, the scale doesn't have myhill's property.

Message: 7683 Date: Wed, 22 Oct 2003 09:53:43 Subject: [tuning] Re: Polyphonic notation From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> 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).

as an example, one could notate either 12edo or a 12-tone pythagorean chain nicely with the nominals A...L, and put it on my 12edo-staff: (the numerals at the left are either 12edo degrees or degrees of the pythagorean scale, and can be considered as a kind of clef) -0- --A-- 11 L 10 ----------------------------------K---------------- 9 J 8 ----------------------------I---------------------- 7 H 6 -----------------------G--------------------------- 5 F 4 ------------------E-------------------------------- 3 D 2 ------------C-------------------------------------- 1 B -0- --A-- in this convention, if the bottom --A-- is "middle-C", then the set of diatonic triads in C-major is: (read vertically for each triad) -0- 11 L L L 10 -------------------------------------------------- 9 J J J 8 -------------------------------------------------- 7 H H H 6 -------------------------------------------------- 5 F F F 4 -----E------------E-----------------E------------- 3 2 ------------C------------------C-----------C------ 1 -0- --A-- --A-- --A-- I ii iii IV V vi vii i know that looks strange, but you're already very familiar with a 12-tone scale where all steps are more-or-less equal, and i'd bet that just a little use of this system would make you feel quite comfortable working within it for those tunings. and anyway, i only chose the diatonic triads for their familiarity -- actually (as you're aware) they don't illustrate anything about a 12-tone notation particularly well, being, rather, eminently suited to illustrate a notation based on 7 nominals. ... but even there, standard Western staff-notation comes up short because it doesn't represent 8ve-equivalence. that could be done thus: steps -0- --A-- 7 -----------------------G-------- 6 F 4 -----------------E-------------- 3 D 2 -----------C-------------------- 1 B -0- --A-- --A-- 7 -----------------------G-------- 6 F 4 -----------------E-------------- 3 D 2 -----------C-------------------- 1 B -0- --A-- --A-- 7 -----------------------G-------- 6 F 4 -----------------E-------------- 3 D 2 -----------C-------------------- 1 B -0- --A-- of course, because this scale has an odd-number of nominals the replication of the 8ve becomes awkward, having the line for 7-steps directly under the line for 0-steps without any room (or necessity) for an intervening space. still, i like visual representation of 8ve-equivalence. -monz

Message: 7684 Date: Wed, 22 Oct 2003 12:42:43 Subject: Re: standards From: Carl Lumma

>> 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.

> >johnston's notation is presented as a standard for *all* strict-JI >music up to the 31-limit or something like that, *regardless* of what >scales may or may not be implied in the music.

Yuck. -Carl

Message: 7685 Date: Wed, 22 Oct 2003 10:17:06 Subject: new Dictionary page: 7-nominal 8ve-equivalent staff From: monz oops ... --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> > and anyway, i only chose the diatonic triads for their > familiarity -- actually (as you're aware) they don't > illustrate anything about a 12-tone notation particularly > well, being, rather, eminently suited to illustrate a > notation based on 7 nominals. > > ... but even there, standard Western staff-notation comes > up short because it doesn't represent 8ve-equivalence. > that could be done thus: > > > steps > > -0- --A-- > 7 -----------------------G-------- > 6 F > 4 -----------------E-------------- > 3 D > 2 -----------C-------------------- > 1 B > -0- --A-- --A-- > > 7 -----------------------G-------- > 6 F > 4 -----------------E-------------- > 3 D > 2 -----------C-------------------- > 1 B > -0- --A-- --A-- > > 7 -----------------------G-------- > 6 F > 4 -----------------E-------------- > 3 D > 2 -----------C-------------------- > 1 B > -0- --A--

ack! the numbering on the left is wrong -- i accidentally left out "5" and ended with "6" and "7" rather than the correct "5" and "6". i've redone it and also added an example of the diatonic triads in C-major, here: Definitions of tuning terms: 7-nominal "8... * [with cont.] (Wayb.) -monz

Message: 7686 Date: Wed, 22 Oct 2003 12:46:33 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> 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

Message: 7687 Date: Wed, 22 Oct 2003 03:31:07 Subject: Why Johnston notation sucks (was: Polyphonic notation) From: Dave Keenan --- 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?

I believe Paul means that he would not assign nominals to any particular PB to notate 5-limit JI because sometimes you would want to use the major block and sometimes the minor block and that would be just too confusing. 7-of-Pythagorean is at least neutral in that regard. Lets compare JI-major-based (Johnston) D-- A- E- B- F# C# G# D# A#+ E#+ B#+ F- C- G- D- A E B F#+ C#+ G#+ D#+ Db--Ab- Eb- Bb- F C G D A+ E+ B+ Db- Ab Eb Bb F+ where + and - are 81/80 up and down, and # b are 25/24 up and down, with Pythagorean-based (everyone else). D_ A_ E_ B_ F#_ C#_ G#_ D#_ A#_ E#_ B#_ F\ C\ G\ D\ A\ E\ B\ F#\ C#\ G#\ D#\ Db Ab Eb Bb F C G D A E B Db/ Ab/ Eb/ Bb/ F/ where / \ are 81/80 up and down and # b are 2187/2048 up and down. Also = and _ are (81/80)^2 up and down. These are represented by single symbols in Sagittal that look more like //| and \\! . 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? With Sagittal or HEWM, that's easy. You play it Pythagorean. Only if it had a 5-comma-down accidental on the top note (or 5-comma-up on the bottom note etc.) would you play it just. 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. Now major thirds are relatively well behaved in Johnston notation. Lets consider major sevenths. In Sagittal or HEWM an 8:15 major seventh will always have one less 5-comma against the top note than the bottom note. In Johnston notation a major seventh will be an 8:15 if there's no difference in 5-commas between the two notes and the low note is an F, C, A or E, or if there's one more comma on the top note and the low note is a G, D or B. Gimme a break! Sure, you can learn this stuff eventually, but why bother? 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. Merely notating a piece in the JI minor scale makes things messy. D- A E B F C G I believe Johnston notation (and any generalisation of it) is an evolutionary dead-end. But I can also understand how Paul's 'The Forms of Tonality' might have seemed to be making a case for such notations. -- Dave Keenan

Message: 7688 Date: Wed, 22 Oct 2003 10:17:54 Subject: Re: naive question From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > yet you had already made the keen observation about the

properties

> of

> > tunings with 81/80 in the kernel differing radically from those > > without it as concerns compatibility with western musical

thinking

> > and notation.

> > Yes, but my attitude was that linear temperaments can always be > expressed in terms of equal temperaments.

More precisely, in terms of the intersection of the kernels of two equal temperaments.

Message: 7689 Date: Wed, 22 Oct 2003 12:47:27 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> 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

Message: 7690 Date: Wed, 22 Oct 2003 03:57:15 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> You seem to say miracle[10] > >> doesn't have a wolf, but all non-equal scales of course do.

> > > >Of course. It just doesn't have an "alphabetic" wolf, for what that's > >worth.

> > 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. Whereas in the chain of fifths case, the alphabet has exactly as many letters between B and F as it has between A and E and between C and G.

> >Explained further in > >Yahoo groups: /tuning-math/message/7113 * [with cont.]

> > I have no idea what you're talking about there. Fifths or something.

Yes, fifths. It says so at the start of the message. It was in response to something of Paul's. Perhaps the confusion is due to my use of decimal digits for nominals (after Graham Breed). Anyway, I said it wasn't important, and that I was adopting a deliberately naive stance towards the circularity. Forget it.

Message: 7691 Date: Wed, 22 Oct 2003 10:21:21 Subject: [tuning] Re: Polyphonic notation From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> > So far I have been satisfied with using MOS to mean a scale based

> on

> > the period+generator representation of a linear temperament which

> has

> > Myhill's property.

> > the the period is not one octave, the scale doesn't have myhill's > property.

Of course it does; forget the damned octave equivalence, it isn't relevant.

Message: 7692 Date: Wed, 22 Oct 2003 12:50:32 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>i hate appeals to authority, but 'distributionally even' was the only >accepted 'academic' qualifier for my symmetrical decatonic scale >according to john clough when i spoke to him. myhill it isn't.

You don't need to appeal to authority. Since the period of 600 cents appears as the same generic interval in all modes, you violate Myhill if you interpret the word "octave" in Myhill's property to mean 2:1, and you don't if you interpret that word to mean "period". -Carl

Message: 7693 Date: Wed, 22 Oct 2003 00:00:00 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> >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?

> > > >it would work fine as long as there were a set of symbols that > >signified the 7-tone chain-of-minor-thirds clearly to musicians. > >right now, there isn't.

> > So it's familiarity, not linearity.

linearity is why hewm works better than johnston.

> >> >> 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?

> > > >actual johnston notation. would generalized johnston notation

notate

> >JI any differently than actual johnston notation?

> > Yeah, the nominals can be from any PB, not just the 5-limit diatonic > scale.

an embarrassment of riches?

> 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.

Message: 7694 Date: Wed, 22 Oct 2003 04:06:02 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

> > >the linear, pythagorean diatonic scale.

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

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

Why yes, of course. Why didn't I see that. :-) 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.

Message: 7695 Date: Wed, 22 Oct 2003 10:26:52 Subject: Re: standards From: monz hi Carl, --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >but try to modulate into almost any other key, and you'll > >start having problems.

> //

> >for example,

> //

> >simply modulating from C-major to its *nearest* relative, > >G-major, invokes this monstrosity: > > > > E ... B ... F#+ > > / \ / \ / \ > > / \ / \ / \ > > C ... G ... D ... A+

> //

> >in my HEWM, they are simply: > > > >C-major:

> > [Involves three non-standard accidentals > vs. Johnston notation.] >

> >G-major: > > > > E-... B-... F#- > > / \ / \ / \ > > / \ / \ / \ > > C ... G ... D ... A

> > How is this supposed to be better than the above?

simple: () any pair of plain letters which are adjacent on the horizontal axis are always 2:3 or 3:4 ratios () any pair of letters with minus signs which are adjacent on the horizontal axis are always 2:3 or 3:4 ratios () any interval with one plain letter and one letter-with-minus which are connected on the lattice, is always a 3:5, 4:5, 5:6, or 5:8 ratio in Johnston notation, you always have to remember that most pairs of plain letters are 2:3 or 3:4, but that D:A is a 20:27 or 27:40 and D:A+ is the 2:3 or 3:4. same deal with the 5-limit ratios. that's needlessly confusing, and for what? simply to emphasize the JI periodicity-block as the basis? i'll take a simple 1-dimensional pythagorean chain for my nominals, and indicate all other primes with accidentals ... which is exactly what Johnston does for primes higher than 5 anyway. -monz

Message: 7696 Date: Wed, 22 Oct 2003 20:26:57 Subject: [tuning] Re: Polyphonic notation From: monz --- 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. as paul just said earlier, "the ear is the final arbiter". paul, your memory is really incredible. just yesterday, i was referring Chris (my business partner) to this page, and i remembered that 40edo was my second-favorite version but had trouble remembering if 64edo or another tuning was my favorite. thanks! :) -monz

Message: 7697 Date: Wed, 22 Oct 2003 06:21:57 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >As I seem to keep saying, no one's asking you to restrict yourself to > >chains of fifths.

> > But you once stated that the only generator for a notation that we > need consider is the perfect fifth. And I'm trying to establish > constraints on a condition that could justify such a statement.

I'm not sure what I said exactly, but my belief remains that a chain of fifths is by far the best general purpose option. It lets you do everything, even if the result is sometimes awkward. So you don't "need" any others, but of course there may be scale-specific notations that would benefit from a different set of nominals. But these nominals would always be best if they form a distributionally even set based on a linear chain of some interval, so there's at most one "wolf". The only time I could see any value in applying the nominals to a more than one dimensional periodicity block is if the scale doesn't use any notes outside that PB. But too many single-purpose notations is something many of us would like to avoid. I understand that it would be enough for you if we just had a master list of comma accidentals. That certainly is part of the plan.

> And as I responded then, this is another example of the same > "technology". I suppose I'll quote this here: >

> >> and it may be worth noting that MOS with rational generators > >> are also 1-D untempered PBs.

> > > >True. But I don't understand the importance of this special case to > >the discussion.

OK. I understand now, thanks to Paul.

> You brought up the Miller limit but you haven't said why you think a > notation search should be any different from a PB search, or how PB > searches have been done wrong so far.

OK. Well I think you know by now (from previous messages), why I am not interested in PBs for notation at all, except for one dimensional PBs in the 3-limit. :-) It has nothing to do with weighting any complexities, but simply minimising the number of wolves and making the generator as simple as possible.

> True but it doesn't make what I said wrong. Good PBs and good > temperaments are what composers like best.

Assuming that to be true, it still doesn't imply that it is a good idea to apply the nominals to a more-than-1D PB which is a proper subset of the PB being used.

> And composers are > supposed to like the least number of notes and the lowest error > (we're providing the whole lattice). The error gets bigger as > the defining commas get bigger, whether you temper or ignore. > The notes get fewer as the commas get simpler, whether you temper > or ignore.

OK. But composers have used a lot of things that bear little reseblance to anything as ordered and regular as a PB or temperament thereof. We have to be prepared to notate these too.

> That is a PB; you asked how I'd notate JI! But if you want to notate > a temperament, you use the same procedure. My statement applies to > both, and that's why the /.

OK.

> Usually a / just means or. Maybe I should have used the word "or". > You asked what a "PB/temperament" was, so I gave you a definition. > If you don't like the definition, go back to using "or".

"PB or temperament" still doesn't make sense to me. A temperament may cover an infinite number of pitches where a PB is finite. I still think you mean "PB or tempering thereof".

> If you think my assertion is wrong, you can just say why you think > a notation ought to be based on different principles than a PB.

OK. Well I've explained that elsewhere by now. You can think of my derivation of nominals as being based on 1D PBs but I don't understand how that adds anything. The important thing is the 1D.

> >I'm saying that we should first decide how we are going to notate > >strict-JI scales and other scales containing only rational pitches > >(notating at least a few hundred of the most commonly used rational > >pitches). i.e. We need to decide what nominals we will use and what > >they will mean, and what accidental symbols we will use and what they > >will mean. And you can't assume the set of pitches will bear any > >resemblance to any particular PB. And the same ratio will always be > >notated the same no matter what other ratios are in the scale > >(provided the 1/1 is the same). > > > >Is that what you're agreeing with?

> > Everything but: >

> >And you can't assume the set of pitches will bear any > >resemblance to any particular PB.

> > Why? Maybe it's because of this:

No. I see the above as an independent statement. It's simply a statement about reality. Look in the Scala archive.

> >the same ratio will always be > >notated the same no matter what other ratios are in the scale

> > Maybe you'd care to explain your reasoning behind this, or point > me to where it was explained.

It was never explained. I simply failed to imagine that anyone would want to do anything else. But I started to get a glimmer that maybe that's what you were on about, so I added that condition to find out.

> Accepting this for a moment, it still only means that at the end > of the PB search, you say, "Only the top PB on this list shall > ever be used as the basis for a notation.". It still doesn't > change the PB search in any way.

OK. Well the winner is the chain of fifths for the reasons I gave above.

> 1. Choose a list of commas. > 2. Assign nominals to each pitch in the PB they enclose. > 3. Use the commas to mark up the nominals to reach pitches in > neighboring blocks. > 4. Assign symbols to the commas if you like.

Right. Clear at last. And rejected, as explained elsewhere.

> >I'm sorry. I have no idea what you're talking about here. What does it > >mean to "search for notations".

> > You can search the m-limit for comma lists that enclose proper > scales. You can rank them by the badness of the commas. If we accept > that you want a single master notation for all of pitch space, then > you're only interested in the top-ranking result. With a weighted > complexity function, you may indeed get a fifth-based tuning at the > top. Are you willing to say that's what you've done? If so, we can > retire.

You can say that I have weighted the _dimensionality_ such that there's no need to consider anything but 1D PBs, then just about any complexity function you like will give you the chain of fifths as the winner.

> >For notating ratios we didn't "search for notations" in any sense I > >can give meaning to.

> > That's what I'm complaining about!

Fair enough. Are you happy now, with the explanation regarding the dimensionality?

> I'm not schooled in Johnston's notation, but I thought it did > conform to the PB approach I've been advocating.

Yes. It certainly does.

> If so, accidental > pile-up would seem to me a reason for using temperament. But if > you've really figured out a way to avoid pile-up in strict JI, > I'd like to hear it.

Yes. We have. We advocate only one accidental against any note (although this is modified in the "impure" version to allow one sagittal and one conventional). We have lots of symbols so that we can uniquely notate lots of rationals (all the most popular ones). This is called the olympian set. But then we describe at least two subsets of these, with cardinality roughly halving each time, called herculean and athenian, so that one can choose to use a smaller more learnable set of symbols, while accepting that you cannot uniquely notate as many rationals with it. The choice of that tradeoff is yours. When a ratio cannot be uniquely notated with the chosen set, you either notate it as the nearest rational that _can_ be notated, and hope that context does the rest, or else you go to the next bigger set of symbols. The symbol set is manageable because each symbol is composed of 2 or 3 parts chosen from a small set. These parts themselves represent commas so for example the right barb /| represents 80:81 and the left arc |) represents 63:64, so the symbol with both /|) represents 35:36, although the parts do not always add precisely like this, and there is no way to _subtract_ commas when forming a symbol (with a small exception).

> >Since we have a limited number of symbols and an infinite number of > >rational pitches to notate, it makes perfect sense to me that we > >should concentrate on notating the most popular or most commonly > >occurring ones.

> > Indeed. Just as with tunings, we have an infinite number of > pitches to provide and we want to concentrate on the most popular > ones. But we don't search the Scala archive, we use a theoretical > principle such as comma badness.

In theory, theory and practice are the same, but in practice they're not. :-) In any case, you've given an analogy not an equivalence. Shouldn't we avail ourselves of empirical data if we have it, when deciding whether to trust the theory. In our case we tried to come up with a neat formula that would predict the observed popularity but they all looked too messy and so we just went with the data which was quite unequivocal in many cases.

> Keeping track of extended JI by ratio is a nightmare, at least for > me. Something like monzo notation seems a minimum, unless you > restrict yourself to a cross-set, diamond, harmonic series or other > fixed structure.

Yes sagittal restricts the uniquely notated ratios to one of a few fixed structures, but they are very large ones and we have not determined their boundaries as such.

> I've used vertical JI with moving roots. I've also > used a linear temperament (meantone). I assume using PBs and > temperaments like meantone -- ones that compactly tile the lattice, > giving me plenty of vertical JI to keep me rooted, with a < 9-tone, > roughly even melodic structure plus a limited number of small > alterations (commas) I can place within the structure -- will also > work. I don't have to know what the commas stand for in terms of JI > ratios.

I think it makes sense to use temperaments too rather than keeping track of extended JI, but we've tried to do our best with sagittal for those who insist on trying to keep track of strict JI.

> So why is Saggital better for the untempered > decatonic scale than the PB approach?

Because the nominals are linear - only one wolf. Same ratios and intervals have same notation across all scales.

Message: 7698 Date: Wed, 22 Oct 2003 13:38:09 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> >> 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? -Carl

Message: 7699 Date: Wed, 22 Oct 2003 00:08:16 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> But in this case you read me right the first time... ie, the > >> TM-reduced chromatic comma for miracle-7, miracle-19, would be the > >> same symbol,

> > > >you lost me there. how are these suitable scales for notation? i > >thought for sure they didn't satisfy your criteria, any more than > >diatonic-6 would.

> > Sorry, those are limits. Cards I put in [], following Gene.

Can't we just write "7-limit-miracle" etc. for these? I thought they were cardinalities too. e.g. miracle-21 has been a synonym for blackjack since forever. So you asked whether I consider 7-limit and 19-limit versions of miracle to be the same temperament. Well not exactly, since they may have different optimum generators, but clearly the fact that the prime mapping of one is a subset of the other's is highly significant, So I think you could use the term either way, and just spell out somewhere how you're using it.

> >> >I'm assuming that our nominals will be contiguous on a uniform > >> >chain of some ratio.

> >> > >> Heavens, no! You'll miss some of the most compact notations that > >> way, which have irrational intervals when viewed as a chain (ie > >> miracle).

> > > >dave didn't mean a rational ratio necessarily.

> > He said ratio!

Thanks Paul, but I really did mean rational here because it was in the context of exactly notating strict JI.

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