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Message: 7725 Date: Wed, 22 Oct 2003 01:19:30 Subject: Re: naive question From: Gene Ward Smith --- 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. It hadn't occurred to me that linear temperaments are crucial for understanding equal temperaments, however.

Message: 7726 Date: Wed, 22 Oct 2003 01:32:28 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>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.

By definition the scale doesn't. You mean the music? Maybe you're thinking of non-PB scales. I guess I give up on that point. I think we can safely discard them, if indeed there is any region of the lattice that can't be defined by a set of uvs (on first thought it certainly doesn't seem like there are). Even if we perversely make a scale with one ratio from each prime limit up to 23, it's still a PB. I'll need lots of accidentals to modulate outside of the block, but I don't how you'd be better off.

>> 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.

Making the generator as simple as possible tends to happen when you weight the complexity. Minimizing the number of wolves sounds interesting. But what's a wolf?...

>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 \\! .

...where are they here? You're saying that because D-A isn't a 3:2, it's a wolf? I could just as easily say C-E is a wolf because it isn't a 5:4. I'm not arguing that the pythagorean approach isn't better here (Johnston just picked a bad scale) but I'm skeptical of any general principle at work, other than: () Familiarity. This goes right out my window, for reasons stated. () Scales should have a low Rothenberg mean variety. No argument here. But once again, a notation can't change the mean variety of a scale once a composer has decided to use it! But it can discourage him (unconsiously) from ever doing so!

>> 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.

I think what this really means is, it's not a good idea to write music with more than a 1D PB in mind.

>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.

Such as?

>> 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".

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.

>> 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

//

>We have lots of symbols so that we can uniquely notate lots of >rationals (all the most popular ones).

Does that include the compound commas like (80:81)^3 you'll need to back up that affirmative?

>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.

So it's not JI anymore, and we're back to a search where generators of a perfect fifth are not necessarily at the top.

>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.

Yes, but consider the source. Most of those scales have never been composed in. I'm not aware of any empirical data on this whatever. You're well aware of the state of ethnomusicology, and the picture in the West... singularly standardized. The whole alternate tunings movement is possible only by generalizing existing music theory.

>> 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.

It'll be cool to see these. Consider me pumped. -Carl

Message: 7727 Date: Thu, 23 Oct 2003 12:56:18 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan Sorry folks, That great long message with all the u's and v's swapped was just too awful. The point was made less painfully on the tuning list. So I've deleted it from the archive and here's the English version. Sorry to those who got the other one by email already. --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >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.

> > By definition the scale doesn't. You mean the music?

OK. The music. Or the tuning. The pitch set.

> Maybe you're > thinking of non-PB scales.

Yes, but also PB tunings where the tuning is too large to have a nominal for euery note and you choose a smaller PB for the nominals.

> I guess I give up on that point. I think > we can safely discard them,

You mean safely discard all pitch sets that are not PBs. Maybe we wouldn't lose much of value by doing so, but I still find this a rather extreme position.

> if indeed there is any region of the > lattice that can't be defined by a set of uvs (on first thought it > certainly doesn't seem like there are). Even if we perversely make a > scale with one ratio from each prime limit up to 23,

Why perverse? Aren't harmonic series scales similar to that?

> it's still a PB.

But that fact isn't very significant.

> I'll need lots of accidentals to modulate outside of the block, but I > don't how you'd be better off.

We can at least avoid accidentals for many of the ratios of powers of 2 and 3, which _are_ fairly popular, or so I've heard.

> Making the generator as simple as possible tends to happen when you > weight the complexity. Minimizing the number of wolves sounds > interesting. But what's a wolf?... >

> >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 \\! .

> > ...where are they here? You're saying that because D-A isn't a > 3:2, it's a wolf?

Of course. I put "wolf" in scare-quotes to indicate that I wasn't using it with its standard meaning but was generalising. However in the D-A case it has its standard meaning.

> I could just as easily say C-E is a wolf > because it isn't a 5:4.

I'm sorry I don't have time to make this "wolf" business rigorous. So feel free to ignore it, unless someone else wants to do the job.

> I'm not arguing that the pythagorean approach isn't better here > (Johnston just picked a bad scale)

Phew. I'm relieved to have you say at least that.

> but I'm skeptical of any general > principle at work, other than:

Skepticism is the right approach ... ... believe me. :-)

> () Familiarity. This goes right out my window, for reasons > stated.

And yet elsewhere you say you favour "evolved" solutions over "engineered" ones. Pardon me if I find this a little inconsistent.

> () Scales should have a low Rothenberg mean variety. No argument > here. But once again, a notation can't change the mean variety > of a scale once a composer has decided to use it! But it can > discourage him (unconsiously) from ever doing so!

No argument here.

> >> 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.

> > I think what this really means is, it's not a good idea to write > music with more than a 1D PB in mind.

Not at all. You go ahead and make your scale-specific notations, and feel free to help yourself to sagittal accidentals from the eventual master list, but I think you should realise by now that the purpose of sagittal is as a harmonically based lingua-franca of microtonality.

> >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.

> > Such as?

The currently fashionable tuning of the great highland bagpipes, as measured by Ewan MacPherson, involving a 7/4 G and an octave narrowed by 20 or 30 cents.

> >> 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".

> > You're not likely to ever hear me discuss something with an infinite > number of notes.

Have you never used the word "miracle" to refer to the abstract entity to which various tunings such as blackjack and canasta belong? Similarly, is meantone always 12 notes per octave for you. If you were looking at a split-key organ in a museum and someone asked you what temperament it was in, would you decline to say "meantone"?

> The rest of the world uses "temperament" to mean > scale, and so can I.

The rest of the world uses "temperament" ambiguously in this regard, but here on the tuning and tuning-math lists we've been using it pretty consistently to apply to the abstract entity for years now, so you might at least have clarified your usage.

> >> 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

> //

> >We have lots of symbols so that we can uniquely notate lots of > >rationals (all the most popular ones).

> > Does that include the compound commas like (80:81)^3 you'll need > to back up that affirmative?

Yes. It includes symbols for many compound commas, but this particular one is _not_ the primary value of any symbol, for what we believe is a good reason. You chose a useful example. The comma (80:81)^3 is around 64.5 cents. There happens to be another comma only 0.4 cents away, which is used to notate simpler and more popular ratios. Even Johnny Reinhard thinks it's OK to notate JI to the nearest cent. The simpler ratios are those of 35 with various powers of 2 and 3, and the comma is 8192:8505 (64.9 c). These ratios occur in the Scala archive approximately _twice_ as often as ratios of 125 with powers of 2 and 3, so you can see what I mean about it being fairly unequivocal. Also the symbol in question is, in ASCII longhand, (|\ which combines the flags for the 77-comma 45056:45927 and the 55-comma 54:55. The 35-large-diesis 8192:8505 happens to be the exact sum(product) of these.

> >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.

> > So it's not JI anymore,

Was it JI to start with? (but that's another battle). It's certainly still rational, not tempered, unless you want to call it "tempering to other ratios" _and_ you want to consider it as having nearly as many independent generators as there are symbol pairs! (not merely as many as there are flags making up the symbols), and all the generators are rational, and none is ever iterated more than once (except the one that generates the nominals). This certainly isn't what most people would think of as a temperament.

> and we're back to a search where generators > of a perfect fifth are not necessarily at the top.

Oh but I believe they are. If you're looking for the best temperaments where the number of generators is into two figures, then I think it's a very safe bet that one of the generators is going to be within half a cent of a 2:3 (or its "period-equivalent").

> >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.

> > Yes, but consider the source. Most of those scales have never been > composed in.

Maybe so, but they at least represent what _many_ different people have thought were a good idea. This seems to me much better than say relying on what just you or I might think are the most important ratios (or commas) to notate. Better even than relying on a consensus of a few present-day contributors on the tuning or tuning-math list, even if we base it on some nice neat mathematical theory.

> I'm not aware of any empirical data on this whatever.

So the Scala archive is only semi-empirical, but it's the best data we've got. I still trust it more than pure theory.

> You're well > aware of the state of ethnomusicology, and the picture in the West... > singularly standardized. The whole alternate tunings movement is > possible only by generalizing existing music theory.

If so, then that's what's most likely to happen, and that's what will most need notating simply.

> >> 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.

> > It'll be cool to see these. Consider me pumped.

But I guess you are already disapointed to learn that it doesn't stack more than two 5-commas precisely. However 0.4 cents is pretty damn close, and the context will usually eliminate this tiny ambiguity entirely. In fact, just the single note to which the accidental applies may be sufficient context. We envisage that even a dumb automatic player could get it right most of the time as to whether the (|\ was representing a 35-comma or a 125-comma or a 13-comma (26:27 65.3 c) merely by looking at what nominal and sharps or flats it was associated with. But we'd better leave _something_ for others to figure out with sagittal. :-) You can sort of see the crudest of the JI symbol sets already in Scala. SET NOTATION SAJI1, then choose your scale and SHOW it or look at it on the Chromatic Clavier. The only problem is that the choice of nominals and sharps or flats is screwed, and therefore the best comma isn't necessarily chosen. We'll eventually get back to working with Manuel to fix these. But the Xenharmonicon-18 deadline is looming and I really shouldn't spend any more time on this list. I'm glad we worked out what our differences really were. -- Dave Keenan

Message: 7728 Date: Thu, 23 Oct 2003 13:03:08 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> i can't speak for paul, but in my own case, hey, i just > decided that capitalization really sucks and i don't want > to use it if i don't feel it's really necessary.

That's OK. Monz and Paul, go ahead and keep doing it if it makes you feel better. It's nowhere near as annoying as the u/v thing.

> it all started because i noticed that my egocentric self > uses the word "I" an awful lot in tuning-list posts, and > it really started bothering me to see all those capital "I"s > everywhere.

So now you can be egotistical and not notice? ;-)

> and in my case, it really shouldn't be too much of a problem, > because i also generally make each paragraph only one sentence > long. > > i decided awhile back that lots of white space helped to > make my messages clearer.

No worries.

Message: 7729 Date: Thu, 23 Oct 2003 17:24:51 Subject: Vanishing commas From: Manuel Op de Coul Don't know if anyone has used it, but I found a bug in the EQUALTEMP/VANISH command in Scala, causing results to be missed. So please upgrade if you'd like to use it: http://www.huygens-fokker.org/software/Scala_Setup.exe - Type Ok * [with cont.] (Wayb.) And in the last lattice player upgrade I forgot to allow nonrational values for the basis, that's fixed too. Manuel

Message: 7730 Date: Thu, 23 Oct 2003 15:37:45 Subject: [tuning] Re: Polyphonic notation From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Sorry folks, > > That great long message with all the u's and v's swapped > was just too awful. The point was made less painfully on > the tuning list. So I've deleted it from the archive and > here's the English version. Sorry to those who got the > other one by email already.

thanks. i really appreciate that you went thru the trouble to replace the old one. i keep these tuning-math posts, print them out, and bind them into volumes for future study. -monz

Message: 7732 Date: Thu, 23 Oct 2003 21:46:10 Subject: Re: Please remind me From: Graham Breed Paul G Hjelmstad wrote:

> I've studied your Linear Temperaments page. Just one quick question, > regarding the 1 1 over 4 4 matrix at the end of the page. Does this > correspond to 2^1, (g)^1 and 2^4 (g)^4 or am I misunderstanding it?

I don't know, what's g? What it seems to be telling you is that the tempered fifth measured in octaves is the fractional part of (log2(3) + log2(5))/5 and the tempered major third is the fractional part of (log2(3) + log2(5))*4/5. In fact, you need to add or subtract arbitrary 1/5s to get the right answer. So the tempered fifth is

>>> ((math.log(3) + math.log(5))/math.log(2)/5-0.2)%1

0.58137811912170378 octaves and the tempered major third is

>>> ((math.log(3) + math.log(5))/math.log(2) * 4/5 + 0.2)%1

0.32551247648681514 octaves. I get the feeling I didn't appreciate this sublety when I wrote that original page. Graham

Message: 7734 Date: Thu, 23 Oct 2003 03:12:35 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >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.

> > By definition the scale doesn't. You mean the music?

OK. The mvsic. Or the tvning. The pitch set.

> Maybe you're > thinking of non-PB scales.

Yes, bvt also PB tvnings where the tvning is too large to haue a nominal for every note and yov choose a smaller PB for the nominals.

> I guess I give up on that point. I think > we can safely discard them,

Yov mean safely discard all pitch sets that are not PBs. Maybe we wovldn't lose mvch of ualve by doing so, but I still find this a rather extreme position.

> if indeed there is any region of the > lattice that can't be defined by a set of uvs (on first thought it > certainly doesn't seem like there are). Even if we perversely make a > scale with one ratio from each prime limit up to 23,

Why peruerse? Aren't harmonic series scales similar to that?

> it's still a PB.

Bvt that fact isn't uery significant.

> I'll need lots of accidentals to modulate outside of the block, but I > don't how you'd be better off.

We can at least auoid accidentals for many of the ratios of powers of 2 and 3, which _are_ fairly popvlar, or so I'ue heard.

> Making the generator as simple as possible tends to happen when you > weight the complexity. Minimizing the number of wolves sounds > interesting. But what's a wolf?... >

> >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 \\! .

> > ...where are they here? You're saying that because D-A isn't a > 3:2, it's a wolf?

Of covrse. I pvt "wolf" in scare-qvotes to indicate that I wasn't vsing it with its standard meaning bvt was generalising. Howeuer in the D-A case it has its standard meaning.

> I could just as easily say C-E is a wolf > because it isn't a 5:4.

I'm sorry I don't haue time to make this "wolf" bvsiness rigorovs. So feel free to ignore it, vnless someone else wants to do the job.

> I'm not arguing that the pythagorean approach isn't better here > (Johnston just picked a bad scale)

Phew. I'm relieued to haue yov say at least that.

> but I'm skeptical of any general > principle at work, other than:

Skepticism is the right approach ... ... belieue me. :-)

> () Familiarity. This goes right out my window, for reasons > stated.

And yet elsewhere yov say yov fauovr "euolued" solvtions ouer "engineered" ones. Pardon me if I find this a little inconsistent.

> () Scales should have a low Rothenberg mean variety. No argument > here. But once again, a notation can't change the mean variety > of a scale once a composer has decided to use it! But it can > discourage him (unconsiously) from ever doing so!

No argvment here.

> >> 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.

> > I think what this really means is, it's not a good idea to write > music with more than a 1D PB in mind.

Not at all. Yov go ahead and make yovr scale-specific notations, and feel free to help yovrself to sagittal accidentals from the euentval master list, bvt I think yov shovld realise by now that the pvrpose of sagittal is as a harmonically based lingva-franca of microtonality.

> >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.

> > Such as?

The cvrrently fashionable tvning of the great highland bagpipes, as measvred by Ewan MacPherson, inuoluing a 7/4 G and an octaue narrowed by 20 or 30 cents.

> >> 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".

> > You're not likely to ever hear me discuss something with an infinite > number of notes.

Haue yov neuer vsed the word "miracle" to refer to the abstract entity to which uariovs tvnings svch as blackjack and canasta belong? Similarly, is meantone always 12 notes per octaue for yov. If yov were looking at a split-key organ in a mvsevm and someone asked yov what temperament it was in, wovld yov decline to say "meantone"?

> The rest of the world uses "temperament" to mean > scale, and so can I.

The rest of the world vses "temperament" ambigvovsly in this regard, bvt here on the tvning and tvning-math lists we'ue been vsing it pretty consistently to apply to the abstract entity for years now, so yov might at least haue clarified yovr vsage.

> >> 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

> //

> >We have lots of symbols so that we can uniquely notate lots of > >rationals (all the most popular ones).

> > Does that include the compound commas like (80:81)^3 you'll need > to back up that affirmative?

Yes. It inclvdes symbols for many compovnd commas, bvt this particvlar one is _not_ the primary ualve of any symbol, for what we belieue is a good reason. Yov chose a vsefvl example. The comma (80:81)^3 is arovnd 64.5 cents. There happens to be another comma only 0.4 cents away, which is vsed to notate simpler and more popvlar ratios. Euen Johnny Reinhard thinks it's OK to notate JI to the nearest cent. The simpler ratios are those of 35 with uariovs powers of 2 and 3, and the comma is 8192:8505 (64.9 c). These ratios occvr in the Scala archiue approximately _twice_ as often as ratios of 125 with powers of 2 and 3, so yov can see what I mean abovt it being fairly vneqviuocal. Also the symbol in qvestion is, in ASCII longhand, (|\ which combines the flags for the 77-comma 45056:45927 and the 55-comma 54:55. The 35-large-diesis 8192:8505 happens to be the exact svm(prodvct) of these.

> >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.

> > So it's not JI anymore,

Was it JI to start with? (bvt that's another battle). It's certainly still rational, not tempered, vnless yov want to call it "tempering to other ratios" _and_ yov want to consider it as hauing nearly as many independent generators as there are symbol pairs! (not merely as many as there are flags making vp the symbols), and all the generators are rational, and none is euer iterated more than once (except the one that generates the nominals). This certainly isn't what most people wovld think of as a temperament.

> and we're back to a search where generators > of a perfect fifth are not necessarily at the top.

Oh bvt I belieue they are. If yov're looking for the best temperaments where the nvmber of generators is into two figvres, then I think it's a uery safe bet that one of the generators is going to be within half a cent of a 2:3 (or its "period-eqviualent").

> >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.

> > Yes, but consider the source. Most of those scales have never been > composed in.

Maybe so, bvt they at least represent what _many_ different people haue thovght were a good idea. This seems to me mvch better than say relying on what jvst yov or I might think are the most important ratios (or commas) to notate. Better euen than relying on a consensvs of a few present-day contribvtors on the tvning or tvning-math list, euen if we base it on some nice neat mathematical theory.

> I'm not aware of any empirical data on this whatever.

So the Scala archiue is only semi-empirical, bvt it's the best data we'ue got. I still trvst it more than pvre theory.

> You're well > aware of the state of ethnomusicology, and the picture in the West... > singularly standardized. The whole alternate tunings movement is > possible only by generalizing existing music theory.

If so, then that's what's most likely to happen, and that's what will most need notating simply.

> >> 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.

> > It'll be cool to see these. Consider me pumped.

Bvt I gvess yov are already disapointed to learn that it doesn't stack more than two 5-commas precisely. Howeuer 0.4 cents is pretty damn close, and the context will vsvally eliminate this tiny ambigvity entirely. In fact, jvst the single note to which the accidental applies may be svfficient context. We enuisage that euen a dvmb avtomatic player covld get it right most of the time as to whether the (|\ was representing a 35-comma or a 125-comma or a 13-comma (26:27 65.3 c) merely by looking at what nominal and sharps or flats it was associated with. Bvt we'd better leaue _something_ for others to figvre ovt with sagittal. :-) Yov can sort of see the crvdest of the JI symbol sets already in Scala. SET NOTATION SAJI1, then choose yovr scale and SHOW it or look at it on the Chromatic Clauier. The only problem is that the choice of nominals and sharps or flats is screwed, and therefore the best comma isn't necessarily chosen. We'll euentvally get back to working with Manvel to fix these. Bvt the Xenharmonicon-18 deadline is looming and I really shovldn't spend any more time on this list. I'm glad we worked ovt what ovr differences really were. I svppose yov're wondering what all this swapping of "v" and "u" is abovt? Annoying isn't it? It was George's idea to make a point regarding Joseph's claim that euen the new improued uersion of the sagittal symbols for the 5 and 7 commas are too alike; one hauing a straight flag on the left and the other a rovnd-cornered flag on the right. I'ue probably jvst annoyed yov all needlessly, since Joseph is probably no longer reading tvning-math. Sorry. Jvst consider it an annoying eccentricity like Pavl and Monz neuer starting a sentence with a capital letter. -- Daue Keenan

Message: 7735 Date: Thu, 23 Oct 2003 06:55:20 Subject: [tuning] Re: Polyphonic notation From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> I svppose yov're wondering what all this swapping of "v" and > "u" is abovt? Annoying isn't it? It was George's idea to make > a point regarding Joseph's claim that euen the new improued > uersion of the sagittal symbols for the 5 and 7 commas are > too alike; one hauing a straight flag on the left and the > other a rovnd-cornered flag on the right. > > I'ue probably jvst annoyed yov all needlessly, since Joseph > is probably no longer reading tvning-math. Sorry. Jvst consider > it an annoying eccentricity like Pavl and Monz neuer starting > a sentence with a capital letter.

i got a real lavgh from it at first, on the main tvning list. bvt yes, now it *is* annoying. i can't speak for paul, but in my own case, hey, i just decided that capitalization really sucks and i don't want to use it if i don't feel it's really necessary. it all started because i noticed that my egocentric self uses the word "I" an awful lot in tuning-list posts, and it really started bothering me to see all those capital "I"s everywhere. and in my case, it really shouldn't be too much of a problem, because i also generally make each paragraph only one sentence long. i decided awhile back that lots of white space helped to make my messages clearer. -monz

Message: 7738 Date: Fri, 24 Oct 2003 21:39:39 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> But also, I suppose any non-convex structure would be non-PB.

non-fokker, but it may be a PB . . . for example the harmonic minor scale.

Message: 7739 Date: Fri, 24 Oct 2003 21:46:21 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> But the Xenharmonicon-18 deadline is looming

it is? last i heard from manuel, it wasn't. someone please e-mail me with the latest info.

Message: 7740 Date: Fri, 24 Oct 2003 21:56:31 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

> > But the Xenharmonicon-18 deadline is looming

> > it is? last i heard from manuel, it wasn't. someone please e-mail

me

> with the latest info.

sorry, i said manuel, but i meant john, of course. i better write some articles before my brain is completely useless . . .

Message: 7741 Date: Fri, 24 Oct 2003 00:15:18 Subject: Re: Please remind me From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> Thanks. I am looking to find a program to multiply and invert matrices > (short of buying Mathematica).

... or MATLAB. Microsoft Excel will do it. You can use formulas like =MMULT(A1:B2, C1:D2) and =MINVERSE(A1:B2) and =TRANSPOSE(A1:B2) and combinations thereof. But you may have to go back to your installation CD (which may be for Microsoft Office) and install the optional Add-In called "Analysis ToolPak", if it does not appear when you choose "Tools/Add-Ins..." from the menus. Once it is installed, then choose "Tools/Add-Ins...", check "Analysis ToolPak" and click "OK". The way you enter formulas that return multiple cells (called "array formulas") is a little strange. You have to remember to select the entire correct-sized region of cells for the result before you start typing the formula, then when you're finished typing the formula you have to remember to hold down the Ctrl and Shift keys while typing Enter. There's also MDETERM() which takes a matrix and returns a single value, the determinant. Maybe you don't need the Analysis ToolPak for these, I'm not sure. But one thing you do need it for is the GCD() function (greatest common divisor). This is very useful for doing prime factorisations. For example, the power of 3 contained in an integer in cell A1 can be obtained by the formula =GCD(A1,3^31) or if you just want the exponent =ROUND(LN(GCD(A1,3^31))/LN(3),0) That magic number 3^31 is the largest power of 3 that can be represented exactly in Excel. Since Excel has 15 (decimal) digit accuracy, for other primes P you can calculate the maximum exponent as INT(15/LOG(P)). So to find the exponent of 2 you would use =ROUND(LN(GCD(A1,2^49))/LN(2),0) These GCD-based formulas don't work if the argument is greater than 2^50-1. -- Dave Keenan

Message: 7742 Date: Fri, 24 Oct 2003 02:37:23 Subject: Re: Please remind me From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > <paul.hjelmstad@u...> wrote:

> > Thanks. I am looking to find a program to multiply and invert

matrices

> > (short of buying Mathematica).

> > ... or MATLAB.

Or Maple, if you would like to use my Maple routines. There are freeware programs that can do this, for example Pari.

Message: 7743 Date: Sat, 25 Oct 2003 20:30:13 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "dkeenanuqnetau" <d.keenan@u...> Yahoo groups: /tuning-math/message/3918 * [with cont.] wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > Joseph, I'm sorry to have to point out that these symbols bear a

> much

> > greater resemblance to the "European" symbols, than the Sims

> symbols.

> >

> > The one thing I've always found unjustifiable and now find > irredeemable about the Sims notation is the use of arrows with full > heads to indicate something smaller than the arrows with half

heads. I

> could almost make a version of this notation that is compatible

with

> the Sims notation, if it wasn't for the twelfth-tone arrows. > > Joseph, remind me what you don't like about slashes again, assuming > the up slash has a short vertical stroke thru the middle of it and

the

> down slash doesn't? >

***Basically, nothing... and I'm glad they are currently now being used for HEWM-S! jP

Message: 7744 Date: Sat, 25 Oct 2003 20:32:34 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "dkeenanuqnetau" <d.keenan@u...> Yahoo groups: /tuning-math/message/3919 * [with cont.] wrote:

> By the way, we can actually notate 311-ET with combinations of

these

> flags, so that no note has more than one arrow next to it in

addition

> to a sharp or flat. Not that this is of any particular importance.

The

> values of the flags in steps of 311-ET are: > > sL 6 > sR 8 > xL 9 > xR 7 > vL 3 > vR 3 > cO 1 > cI 1

***I believe combining symbols is the way to go... but not with lots of left-right reversibility.. jP

Message: 7745 Date: Sat, 25 Oct 2003 20:37:02 Subject: forwarding emails on oneself From: Joseph Pehrson Well, it's not optimal, but forwarding emails to onself, as recommended by Dave Keenan, works just fine for those posts that require an ascii graphic element... jP

Message: 7746 Date: Sun, 26 Oct 2003 10:52:06 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>>> Therefore, the master >>> list should be based on a search for simple and small commas. >>> Conveniently, such searches have been done at least through the 7- >>> limit, with various flavors of complexity functions, etc.,

>> >>Well send me the list when you conveniently get up to 23 limit.

> >Ok, I'll work on that. I've been meaning to write a comma-searcher >for a while now. It might take me a while more.

Here are the 10 lowest-badness ratios among all ratios in lowest terms between 0 and 600 cents with denominator not greater than 500, where badness is defined as log_2(ratio)^2 * prime-limit(ratio)... ((3.3295737438398616 3 256/243) (3.796875 3 9/8) (4.2139917695473255 3 32/27) (4.805419921875 3 81/64) (5.12578125 5 81/80) (5.24288 5 128/125) (5.292214940134465 5 250/243) (5.333333333333333 3 4/3) (5.425347222222222 5 25/24) (5.56182861328125 5 135/128)) ...each triple is (badness, prime-limit, ratio). The search took less than 10 minutes on a P3 600 laptop (code available). Performance would be drastically better by using anything other than the slowest conceivable factoring algorithm, which I chose for expediency. Since every prime limit contains an infinite number of ratios, and neither size nor complexity behave smoothly as one searches farther out, it seems we'll never know the top 10 lowest-badness ratios at any prime limit.... -Carl

Message: 7747 Date: Sun, 26 Oct 2003 11:10:44 Subject: comma search (was Re: Polyphonic notation) From: Carl Lumma

> where badness is defined as log_2(ratio)^2 * prime-limit(ratio)...

Any comments on this badness measure? The ^2 is a variable in my procedure, and I usually have it higher than 2. prime-limit is also a variable, which could instead be log_2(n*d) or just log_2(d). Using ^3 and log_2(n*d), the results are... (badness log_2(n*d) ratio)

>((8.44126581032688 4.321928094887363 5/4) > (8.47910694921152 4.906890595608519 6/5) > (8.497688890598296 3.5849625007211565 4/3) > (8.56280035191257 5.392317422778761 7/6) > (8.668704723304646 5.807354922057605 8/7) > (8.78491274619423 6.169925001442312 9/8) > (8.90514828028762 6.491853096329675 10/9) > (9.025989778701321 6.78135971352466 11/10) > (9.145539472765897 7.044394119358453 12/11) > (9.2627480757178 7.285402218862249 13/12))

Paul, any thoughts on a badness heuristic log(d) * |n-d|/log(d) = |n-d| ? Thanks, -Carl

Message: 7748 Date: Sun, 26 Oct 2003 22:21:10 Subject: [tuning] Re: Polyphonic notation From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Since every prime limit contains an infinite number of ratios, and > neither size nor complexity behave smoothly as one searches farther > out, it seems we'll never know the top 10 lowest-badness ratios at > any prime limit....

For any limit, zero will be an accumulation point of log2(q)^2, since p-limit commas are arbitrarily small; but this hardly matters, since whatever it is you are calculating, it clearly isn't log2(q)^2 primelimit(q). Can we start over?

Message: 7749 Date: Sun, 26 Oct 2003 22:22:56 Subject: comma search (was Re: Polyphonic notation) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> > where badness is defined as log_2(ratio)^2 * prime-limit(ratio)...

> > Any comments on this badness measure?

You haven't defined it yet.

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