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Message: 7650 Date: Tue, 21 Oct 2003 11:43:57 Subject: Re: A 13-limit JI scale From: Manuel Op de Coul And a known one, it's a transposition of the 13-limit Partch diamond. Manuel

Message: 7651 Date: Tue, 21 Oct 2003 21:00:15 Subject: Re: hippopothesis From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> So if T[n] is a linear temperament when n is a MOS, what > >> is T[n] when n isn't?

> > > >T[n] is never a temperament--it is a scale.

> > Right, because temperaments are infinite or some such. But > then the hypothesis needs to be rephrased.

why?

Message: 7652 Date: Tue, 21 Oct 2003 10:06:58 Subject: Re: TM reduced chromatic commas From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > I assumed you meant that we should, for example, have a unique symbol > > pair for each of the following: > > the TM-reduced chromatic comma for meantone > > the TM-reduced chromatic comma for schismic > > the TM-reduced chromatic comma for diaschismic > > the TM-reduced chromatic comma for kleismic > > the TM-reduced chromatic comma for miracle > > etc. > > even though the first two could clearly use the same symbol.

> > The first order of business would be to define what the "TM reduced > chromatic comma" would be for a given linear temperament. It seems to > me this is really a property of a MOS for that temperament.

I totally agree. I was assuming (probably mistakenly) that we would be able to agree on which size MOS to use in each case. e.g. the proper one with cardinality closest to 8.7 or some such. But maybe it doesn't have to be proper.

Message: 7653 Date: Tue, 21 Oct 2003 21:04:41 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>

> >> The list of commas defines a finite region of the lattice. Every > >> pitch within the region gets a nominal. The lattice is tiled

with

> >> such regions. The commas in the list are assigned symbols...

really

> >> all this is covered by Paul, in numerous posts and his paper.

> > > >but i assume that the commatic unison vectors are irrelevant to

the

> >musician, as is true in the vast majority of western music. if the > >81:80s *are* being kept track of, in a piece in JI or something > >similar, then i would *not* go along with the johnston notation, > >which seems to be what you're inferring from my presentations.

> > Aha! > > And no, I did not say you would. :) > > What would you do?

i've argued strongly against johnston's system and for hewm or equivalents on the tuning list. maybe you weren't around then. some argued oppositely, saying that remembering the extra comma between D and A and between Bb and F (or B and F#) was no different in quality than remembering the extra accidental between B and F required to make a perfect fifth in standard western musical notation, being simply two extra things to remember. i'd rather not.

> >> >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!

pi:e is a ratio.

Message: 7654 Date: Tue, 21 Oct 2003 14:35:19 Subject: Re: hippopothesis From: Carl Lumma

>> >> So if T[n] is a linear temperament when n is a MOS, what >> >> is T[n] when n isn't?

>> > >> >T[n] is never a temperament--it is a scale.

>> >> Right, because temperaments are infinite or some such. But >> then the hypothesis needs to be rephrased.

> >why?

What is the official phrasing? -Carl

Message: 7655 Date: Tue, 21 Oct 2003 17:35:10 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> >> 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

Message: 7656 Date: Tue, 21 Oct 2003 14:47:34 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>i've argued strongly against johnston's system and for hewm or >equivalents on the tuning list. maybe you weren't around then. some >argued oppositely, saying that remembering the extra comma between D >and A and between Bb and F (or B and F#) was no different in quality >than remembering the extra accidental between B and F required to >make a perfect fifth in standard western musical notation, being >simply two extra things to remember. i'd rather not.

I'm not sure what you mean. I looked at the HEWM page, and it has an 81:80 accidental...

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

//

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

>> >> He said ratio!

> >pi:e is a ratio.

Yes, but there's also a Partchian definition in use aroundabouts. And if he really meant "all real numbers", he wouldn't have said "ratio". -Carl

Message: 7657 Date: Tue, 21 Oct 2003 21:47:30 Subject: Re: hippopothesis From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> >> So if T[n] is a linear temperament when n is a MOS, what > >> >> is T[n] when n isn't?

> >> > > >> >T[n] is never a temperament--it is a scale.

> >> > >> Right, because temperaments are infinite or some such. But > >> then the hypothesis needs to be rephrased.

> > > >why?

> > What is the official phrasing? > > -Carl

i don't know if there's an "official" one, but do any of them need to be rephrased? i don't think so.

Message: 7658 Date: Tue, 21 Oct 2003 17:40:16 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>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. Maybe you want a notation based on ets. You seem to say miracle[10] doesn't have a wolf, but all non-equal scales of course do. -Carl

Message: 7659 Date: Tue, 21 Oct 2003 21:58:08 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >i've argued strongly against johnston's system and for hewm or > >equivalents on the tuning list. maybe you weren't around then.

some

> >argued oppositely, saying that remembering the extra comma between

D

> >and A and between Bb and F (or B and F#) was no different in

quality

> >than remembering the extra accidental between B and F required to > >make a perfect fifth in standard western musical notation, being > >simply two extra things to remember. i'd rather not.

> > I'm not sure what you mean. > > I looked at the HEWM page, and it has an 81:80 > accidental...

right, but D:A is 2:3, Bb:F is 2:3, and B:F# is 2:3. they're not in the johnston notation.

>

> >> >> >I'm assuming that our nominals will be contiguous on a

uniform

> >> >> >chain of some ratio.

> //

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

> >> > >> He said ratio!

> > > >pi:e is a ratio.

> > Yes, but there's also a Partchian definition in use aroundabouts. > And if he really meant "all real numbers", he wouldn't have said > "ratio". >

i'll let dave speak for himself.

Message: 7660 Date: Tue, 21 Oct 2003 14:58:52 Subject: Re: hippopothesis From: Carl Lumma

>i don't know if there's an "official" one, but do any of them need to >be rephrased? i don't think so.

What's one of them? -Carl

Message: 7661 Date: Tue, 21 Oct 2003 17:41:26 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> >> >> 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? :) -Carl

Message: 7662 Date: Tue, 21 Oct 2003 22:03:20 Subject: Re: hippopothesis From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >i don't know if there's an "official" one, but do any of them need

to

> >be rephrased? i don't think so.

> > What's one of them? > > -Carl

when all but one of the defining unison vectors of a fokker periodicity block are regularly tempered out, the result is a distributionally even scale.

Message: 7663 Date: Tue, 21 Oct 2003 15:15:37 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> I'm not sure what you mean. >> >> I looked at the HEWM page, and it has an 81:80 >> accidental...

> >right, but D:A is 2:3, Bb:F is 2:3, and B:F# is 2:3. they're not in >the johnston notation.

So this is pythagorean notation then? In 5-limit music, doesn't this just change the balance of 25:24 and apotome accidentals vs. Johnston notation? -Carl

Message: 7664 Date: Tue, 21 Oct 2003 15:17:17 Subject: Re: hippopothesis From: Carl Lumma

>> What's one of them? >> >> -Carl

> >when all but one of the defining unison vectors of a fokker >periodicity block are regularly tempered out, the result is a >distributionally even scale.

Cool (no mention of "linear temperament"). -Carl

Message: 7665 Date: Tue, 21 Oct 2003 22:17:52 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> I'm not sure what you mean. > >> > >> I looked at the HEWM page, and it has an 81:80 > >> accidental...

> > > >right, but D:A is 2:3, Bb:F is 2:3, and B:F# is 2:3. they're not

in

> >the johnston notation.

> > So this is pythagorean notation then?

not sure what you mean exactly . . . isn't the hewm page sufficiently clear?

> In 5-limit music, doesn't this just change the balance of 25:24 > and apotome accidentals vs. Johnston notation?

don't know what you're referring to exactly, but in the johnston notation, D:A, Bb:F, and B:F# are not 2:3, while C:G, E:B, F:C, G:D, and A:E are.

Message: 7666 Date: Wed, 22 Oct 2003 01:24:34 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- 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. I was just deliberately adopting a naive stance to point something out.

> Maybe you want a notation based on ets.

No. I want one based on rationals, which as I see it, leaves me no choice but to make my nominals pythagorean in order to minimise the nominal "wolf" problem.

> 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. Explained further in Yahoo groups: /tuning-math/message/7113 * [with cont.] Not important.

Message: 7667 Date: Wed, 22 Oct 2003 08:39:17 Subject: Re: standards From: monz hi Carl, --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >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.

i'm sorry, but i disagree -- at least in the case of complex music which uses a larger-than-diatonic pitch set, which BTW is exactly the kind of music Johnston and his many students (all of whom use his notation) compose. for 5-limit JI music which only uses a diatonic pitch-set, sure, A B C D E F G is fine. but try to modulate into almost any other key, and you'll start having problems. for example, here's a portion of the Johnston lattice: (view in "Expand Messages" mode on the Yahoo website) F# .. C# .. G# .. D# / \ / \ / \ / \ / \ / \ / \ / \ D-... A ... E ... B ... F#+ \ / \ / \ / \ / \ \ / \ / \ / \ / \ F ... C ... G ... D ... A+ this subset is the diatonic C-major scale, upon which the notation is based: A ... E ... B / \ / \ / \ / \ / \ / \ F ... C ... G ... D notes which have a sharp (#) are a 24:25 higher than those 7 nominals. simply modulating from C-major to its *nearest* relative, G-major, invokes this monstrosity: E ... B ... F#+ / \ / \ / \ / \ / \ / \ C ... G ... D ... A+ yuck. in my HEWM, they are simply: C-major: A-... E-... B- / \ / \ / \ / \ / \ / \ F ... C ... G ... D G-major: E-... B-... F#- / \ / \ / \ / \ / \ / \ C ... G ... D ... A entire example lattice: F#--..C#--..G#--..D#-- / \ / \ / \ / \ / \ / \ / \ / \ D-... A-... E-... B-... F#- \ / \ / \ / \ / \ \ / \ / \ / \ / \ F ... C ... G ... D ... A simple. -monz

Message: 7668 Date: Wed, 22 Oct 2003 19:22:58 Subject: Re: standards From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> 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. i think that daniel wolf's modification of this standard (and hence hewm or sagittal) is a clear improvement.

Message: 7669 Date: Wed, 22 Oct 2003 01:27:25 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- 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.

for musicians, the nominals are too engrained in their musical meaning to ever be thought of non-circularly.

Message: 7670 Date: Wed, 22 Oct 2003 01:52:01 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>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?

What's a consistent notation? 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

Message: 7671 Date: Wed, 22 Oct 2003 19:27:21 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.

> You're not likely to ever hear me discuss something with an infinite > number of notes. The rest of the world uses "temperament" to mean > scale, and so can I.

except for a default assumption of *12* (not 7!) note scales on keyboard instruments, temperament means temperament. you must live on a different planet.

Message: 7672 Date: Wed, 22 Oct 2003 01:57:30 Subject: Re: standards From: Carl Lumma

>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

Message: 7673 Date: Wed, 22 Oct 2003 19:28:30 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >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?

> > What's a consistent notation? > > 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.

Message: 7674 Date: Wed, 22 Oct 2003 01:36:43 Subject: [tuning] Re: Polyphonic notation From: Gene Ward Smith --- 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.

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