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Message: 7600 Date: Fri, 17 Oct 2003 09:13:44 Subject: Re: [5-11]-prime-space p-block and 13edo From: monz hi Manuel, --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:

> > I will add this scale to the archive.

thanks! i can easily create tons more if you want them.

> Speaking of lattices, I have improved the lattice player > in Scala. Now it will also display nonrational intervals, > so one can use it for the "bingocard" type. > Go to Analyse:Lattice and player, click the 2D play button, > tick the "Nearest in emtpy positions" checkbox and set an > amount of rows and columns.

cool. -monz

Message: 7601 Date: Fri, 17 Oct 2003 11:22:32 Subject: Re: [5-11]-prime-space p-block and 13edo From: Manuel Op de Coul

>thanks! i can easily create tons more if you want them.

Me too, so thanks but not really. But if you have ones you think are nice or special they can of course be sent. Cheers, Manuel

Message: 7602 Date: Fri, 17 Oct 2003 10:48:43 Subject: Re: can someone check this data? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Awesome! Sorry I can't help check it. > > How'd you get this html? 'd be nice to have a database > with column-sort.

the database on the tuning list is just that.

> Oh, and where are the TM-reduced bases?

the comma, silly goose!

Message: 7603 Date: Fri, 17 Oct 2003 13:26:17 Subject: Re: can someone check this data? From: Manuel Op de Coul

>> Oh, and where are the TM-reduced bases?

>the comma, silly goose!

Don't you need more than one comma for a basis of these temperaments? Manuel

Message: 7604 Date: Fri, 17 Oct 2003 11:27:50 Subject: Re: can someone check this data? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:

>

> >> Oh, and where are the TM-reduced bases?

>

> >the comma, silly goose!

> > Don't you need more than one comma for a basis of > these temperaments? > > Manuel

no, they're 5-limit linear temperaments, and 2 minus 1 equals 1.

Message: 7605 Date: Sat, 18 Oct 2003 12:08:49 Subject: Re: can someone check this data? From: Manuel Op de Coul Thanks Graham, good to know. Manuel

Message: 7606 Date: Sun, 19 Oct 2003 14:31:47 Subject: Re: T[n] revisited From: Carl Lumma

>> Did you use one of your maple routines to do this?

> >Of course--gf7 or gf11 of transpose(mapping to primes).

Sweet; this works on my end. -Carl

Message: 7607 Date: Mon, 20 Oct 2003 19:53:36 Subject: Re: Chains of fifths and notation From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> This is a long overdue reply to Gene in "Re: Polyphonic notation" on > the tuning list > Yahoo groups: /tuning/message/47497 * [with cont.] > > --- In tuning@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>

wrote:

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

wrote:

> >

> > > We have also used a temperament to help decide on the actual

symbols

> > > to be used for the comma ratios in the superset. This is an > > > 8-dimensional temperament with a maximum error of 0.39 cents.

The 8

> > > dimensions relate to the 9 flags (including the accent mark)

that make

> > > up the symbols, less one degree of freedom because a certain > > > combination is set equal to the apotome.

> > > > What commas are being tempered out?

> > I'm afraid I don't know. I just specified certain combinations of > generators to approximate certain 23-limit ratios (which were > themselves commas, but were not being tempered out) and solved > numerically for the generators. If you're still interested, maybe > there's some other information I could give you if you wanted to

work

> them out. But it may have to wait some time.

I can give Gene some information, since I figured this out some time ago. Gene, what Dave was describing above applies to the extreme-precision version of the notation (i.e., extremely high precision), which I don't claim to understand completely. I do have some data that applies to the medium-precision and high- precision versions (areas in which I have been working), which for the 11 limit probably also applies to the extreme-precision version. In these, the symbol for the 7th harmonic [0, 0, 1] is also used for [7, -4]; the resulting schismina is 4374:4375 (2*3^7:5^4*7, ~0.396 cents). The symbol for the 11th harmonic [0, 0, 0, 1] is also used for [10, 5]; the resulting schismina is 184528125:184549376 (3^10*5^5:2^24*11, ~0.199 cents). The schismina for the 13th harmonic in medium and high-precision sagittal (which does *not* vanish in extreme-precision sagittal) is 4095:4096 (3^2*5*7*13:2^12, ~0.423) cents, such that [0, 0, 0, 0, 1] is equivalent to [-9, 4]. I hope this will be of some help. --George

Message: 7608 Date: Mon, 20 Oct 2003 03:59:24 Subject: Re: can someone check this data? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:

> > Now I checked more and found a few more differences in RMS values. > I'm beginning to worry about my square root routine. Could someone > else verify, for example for counterschismic, is the RMS error > 0.026391 as Paul gives or 0.026394? Gene, Graham? > Thanks, > > Manuel

i got my rms values from gene or graham originally -- it was the et/period/generator numbers especially i hoped would be checked.

Message: 7609 Date: Mon, 20 Oct 2003 20:33:44 Subject: A 13-limit JI scale From: Gene Ward Smith I'm presenting this simply because I'm writing something in it. It is irregular and has a lot of notes (41), but is useful because it has fourteen complete 13-limit septads, 7 otonal and 7 utonal. It could be tempered with (tempering out 10648/10647 is hardly likely to do any damage, for instance) but I don't because the septads are all I need. ! sk13.scl 13-limit JI scale with 14 complete septads 41 ! 33/32 27/26 21/20 15/14 13/12 12/11 9/8 15/13 7/6 33/28 6/5 39/32 27/22 5/4 33/26 9/7 21/16 4/3 27/20 15/11 11/8 18/13 39/28 3/2 21/13 13/8 18/11 33/20 5/3 27/16 12/7 7/4 39/22 9/5 11/6 24/13 15/8 21/11 27/14 39/20 2

Message: 7611 Date: Mon, 20 Oct 2003 16:20:19 Subject: Re: naive question From: Carl Lumma

>I am curious to know who's work you all would cite as the first >serious efforts to explore linear temperaments in depth. Put >another way, who was it that laid the groundwork for all of the >exploration into linear temperaments you are involved with now? >This is not my central interest, and from what little I know, I >would guess that Ellis was really the first to size things up in >this area. How far off is that?

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

Message: 7612 Date: Mon, 20 Oct 2003 21:10:54 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>What more do you want me to agree to? Are you saying you want us to >desist from ever notating a non-fifth-generated linear temperament >using 7 nominals in a chain of fifths - to strike this from the set of >"allowable" uses of Sagittal? I can't imagine you would want that, so >I am at a loss to understand the problem here.

I was just stating my point of view on this. You'd already agreed that the Sagittal accidentals are portable, and that's all I wanted from you.

>> >> >the semantic foundations of Sagittal notation have absolutely >> >> >nothing to do with any temperament.

>> >> >> >> I should have said, "good PBs" there. [I think of PBs as >> >> temperaments, which always gets me into trouble.]

> >Indeed! This may have been a source of a good deal of my confusion >and impatience over what you were saying. As I understand it a PB is >strictly rational. What could be further from temperament?

To me, the very notion of 'finity' behind PBs implies ignoring commas, whether they're distributed or left be. In certain cases I'd argue that as one takes a PB and tempers it down through planar, linear, and finally et, there's something essential that doesn't change. Thus, I think of the PB as the defining thing...

>Do you maybe mean good MOS (tempered PB)?

No, and it may be worth noting that MOS with rational generators are also 1-D untempered PBs.

>>>>>So what's a _good_ PB for notational purposes?

>>>> >>>>The same kind that are good for composition purposes!

>>> >>>That sounds even less >>>likely to be agreed upon than a good linear temperament. How about >>>we forget about this given our agreement below?

>> >>I thought it was well-agreed-upon: the simplicity of the commas vs. >>their size. There are different ways to calculate this, and the >>details of how to do so with planar and higher temperaments and raw >>PBs are not settled, but using any of the proposed methods is fine -- >>pick your fav.

> >Aha! So you're not talking about a good temperament with which to >notate JI and other temperaments. You're merely talking about agreeing >on the right size of MOS for the temperament-specific-nominals for >notating a given temperament. > >No Problem.

Yes. In fact, my assertion would be: "There is no established reason to believe that ideal PBs/temperaments for notation are different any any way from those that are ideal for music-making." In other words, if we don't restrict ourselves to chains-of-fifths for music-making, we shouldn't do so for notation. Note this is entirely independent of the portability question, which adds that the basis of the notation and the tuning of the music should match in each particular case.

>> >> With linear temperaments, you only need 1 accidental pair at a >> >> time, as I've pointed out.

>> > >> >But Carl, that's like saying you only need 6 pairs of accidentals to >> >notate 19-limit JI. One for each prime above 3. It becomes >> >essentially unreadable once you go past 2 accidentals per note.

>> >> How is saying you only need 1 like saying you only need 6?

> >Because in both cases a readability problem occurs when you need to >stack more than two of them against a note, and you find you want to >have some more accidentals. In other words: Sure you only "need" 1, >but some people, maybe not you, will end up wanting some enharmonics.

I don't follow. Any linear temperament can be notated with the same technology as common-practice music, which is time-tested and proven, and *cannot* be notated any more simply. http://lumma.org/tuning/erlich/erlich-tFoT.pdf - Type Ok * [with cont.] (Wayb.) The use of enharmonics implies further tempering, and is fine by me.

>> >And even ignoring these "enharmonics", you need other accidentals >> >when you have multiple parallel chains, i.e. when the period is >> >not the whole octave.

>> >> Isn't this refuted by Paul's single-accidental decatonic notation?

> >Yes. I already conceded that. Its fine to use more nominals in the >other chains provided the total number isn't much more than the >"Miller limit" of 9.

Again, the Miller limit doesn't apply any more here than it does to the actual music. If the music is palatable with 10 notes in the scale, the notation will be readable with 10 nominals on the page.

>> >> you could try assigning (an) accidental(s) for each *temperament*, >> >> with the understanding that it/they would take on TM-reduced >> >> value(s) for the limit and scale cardinality being used.

>> > >> >Eek! So then we would have to learn not only new nominals for every >> >temperament, but new accidentals too?

> >Based on your response. I now see that I misinterpreted what you were >proposing here. And I think a few other people may have done so too. > >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.

They could?

>But you were in fact (tentatively) suggesting that a single symbol >pair could be used in _all_ such cases even though the chromatic >commas (chromas?) are very different in some cases.

I also suggested this. Any conventional notation software could be used immediately. But temperaments could not be mixed freely in a score without some additional notation (like a 'key signature'). 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, and it would be different from any symbol ever used for any flavor of meantone. Thus, different temperaments are distinguished automatically -- they could be mixed in a score freely. Finally, my remaining (of 3) suggestion is the main one we've been discussing, the one symbol per comma master list. Which seems best to me if the list doesn't get unwieldy.

>> >> >Then if that's accepted, the second part is that it is best if the >> >> >simplest or most popular ratios have the simplest notations.

>> >> >> >> Right. And it's this aspect that makes the search more-or-less >> >> equivalent to the search for good PBs.

>> > >> >Nope. You've lost me there.

>> >> The simplest commas would be the most popular for a reason!

> >Now I'm still unsure whether to read the "PBs" above as "temperaments" >or "MOSs". (I'm allowing MOS = Myhills, i.e. 1 or more chains here.)

Here I really do mean PBs (good PBs are those that lead to good temperaments), but in many cases in this thread I don't think there's an established term that's correct. Are miracle-7 and miracle-19 the same "temperament"?

>But I think I understand now that you are only talking about the >search for a good notation for a given linear temperament, not for >everything. Whereas I was talking about the search for a good notation >for untempered ratios (and previously thought you were talking about >the same).

Nope, I was talking about everything re. the master list proposal. I only shortly drifted into the other two proposals, and I noted they only work for temperaments.

>> >Huh? I thought you just agreed that we would first decide how to >> >_precisely_ notate ratios?

>> >> Yup. In fact, you can think of a PB/temperament *as* a notation in >> my scheme.

> >Now what the heck is a "PB/temperament"?

A list of commas.

>Then when you've explained that, please explain how you would use one >of them to notate untempered ratios exactly. Please give some examples.

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.

>> Don't follow you here. But try to track me again. By saying you >> want to always keep the lowest primes the simplest ones in the map >> (by assuming 2-equiv. on the staff and by always using 3:2s for your >> nominals), you are effectively weighting your complexity measure. >> If you completely disallow temperaments like miracle (which do not >> have a 3:2 generator) from showing up in your notation search (think >> temperament search), it's a *very* strongly weighted function -- >> you're insisting that both generators be primes.

> >You claim to have been saying all along that it is good to have a >notation system whose semantics are based on precise ratios and then >use that to also notate temperaments.

Yep.

>So before saying anything about temperaments, maps, generators or >complexity weightings of temperaments, please explain how you propose >to notate ratios.

Hopefully this is clear by now. As you temper out commas from the list ("temperament/PB") you simply find that some of the symbols you assigned never show up in your score.

>If by complexity you only mean "ratio complexity" then I can maybe >explain further. We didn't actually used any ratio complexity formula >based on prime exponents or any such. We used ratio popularity >statistics obtained from the Scala archive. But we had already decided >to make our nominals in a chain of fifths, before we did that.

Searching the space of possible notations, linear temperaments, and PBs is all the same search. If you do a search for notations and only those with 3:2 generators come up, you've heavily weighted your complexity function, just as if you've searched for temperaments and only chains-of-fifths tunings came up. Searching the scala archive for popular ratios is a hare-brained idea, if you don't mind me saying so. It isn't ratios in general that you need to notate, but commas. Commas are small ratios. Further, on tuning-math we assume that commas that are simple for their size will be popular with composers. 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., and none have them have excluded everything but chains-of-fifths tunings.

>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).

>But that's a notation for a temperament, and a fine one at that. But >that means there are lots of ratios that it is incapable of >distinguishing. How is this an embodiment of "what you've been saying >all along", namely that we should first figure out how to notate ratios?

Paul's scales are 3-D blocks. Add a symbol for every comma you don't temper out. Can the untempered decatonic scales be notated in Sagittal with fewer than 3 accidentals? -Carl

Message: 7613 Date: Tue, 21 Oct 2003 12:35:26 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: Dave Keenan:

> >Indeed! This may have been a source of a good deal of my confusion > >and impatience over what you were saying. As I understand it a PB is > >strictly rational. What could be further from temperament?

Carl:

> To me, the very notion of 'finity' behind PBs implies ignoring commas, > whether they're distributed or left be.

Well to me, and I think to Fokker who coined the term, a periodicity block implies avoiding comma-sized steps in your JI scales, but not distributing them at all.

> In certain cases I'd argue > that as one takes a PB and tempers it down through planar, linear, and > finally et, there's something essential that doesn't change. Thus, I > think of the PB as the defining thing...

Maybe so, but I strongly suggest you don't use the term PB to refer to something that's tempered. I'd call the tempered ones hyper_MOS, MOS, ET as you go down, or simply tempered-PBs.

> >Do you maybe mean good MOS (tempered PB)?

> > No,

Why isn't a linearly tempered PB a MOS? Maybe this hasn't been formally proven, but I don't think many people who understand it seriously doubt it. (Allowing that MOS may have fractional-octave periods).

> 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.

> Yes. In fact, my assertion would be: "There is no established reason > to believe that ideal PBs/temperaments for notation are different any > any way from those that are ideal for music-making." In other words, > if we don't restrict ourselves to chains-of-fifths for music-making, > we shouldn't do so for notation.

As I seem to keep saying, no one's asking you to restrict yourself to chains of fifths. However I think the reasons why many people might want to do so, for their set of nominals, are pretty obvious, and I believe George and I and others have given these reasons several times.

> Note this is entirely independent of > the portability question, which adds that the basis of the notation and > the tuning of the music should match in each particular case.

I have no idea what this means. Or rather I could think of maybe 5 different things that it could mean, and based on recent experience probably none of them are what you intend.

> >> >> With linear temperaments, you only need 1 accidental pair at a > >> >> time, as I've pointed out.

> >> > > >> >But Carl, that's like saying you only need 6 pairs of accidentals to > >> >notate 19-limit JI. One for each prime above 3. It becomes > >> >essentially unreadable once you go past 2 accidentals per note.

> >> > >> How is saying you only need 1 like saying you only need 6?

> > > >Because in both cases a readability problem occurs when you need to > >stack more than two of them against a note, and you find you want to > >have some more accidentals. In other words: Sure you only "need" 1, > >but some people, maybe not you, will end up wanting some enharmonics.

> > I don't follow. Any linear temperament can be notated with the same > technology as common-practice music, which is time-tested and proven, > and *cannot* be notated any more simply. > > http://lumma.org/tuning/erlich/erlich-tFoT.pdf - Type Ok * [with cont.] (Wayb.)

Paul's excellent paper gives us one way of looking at the notation of families of scales based on periodicity blocks. I don't believe it claims to describe the "notation technology" of common practice music, which as I pointed out earlier, originated historically with Pythagorean, not meantone or 5-limit JI.

> Again, the Miller limit doesn't apply any more here than it does to > the actual music. If the music is palatable with 10 notes in the > scale, the notation will be readable with 10 nominals on the page.

Did Partch have 43 notes in a scale? Was his music written in this scale palatable? Would it have been readable if notated with 41 nominals? I think we're just getting sidetracked here.

> >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.

> > They could?

Yes, if you chose to use 7 nominals in both cases, in which case an accidental representing the apotome would work for both. But perhaps you would want either 5 or 12 nominals for schismic so the nominals form a proper scale.

> >But you were in fact (tentatively) suggesting that a single symbol > >pair could be used in _all_ such cases even though the chromatic > >commas (chromas?) are very different in some cases.

> > I also suggested this. Any conventional notation software could be > used immediately. But temperaments could not be mixed freely in > a score without some additional notation (like a 'key signature'). > > 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, and it would be different from any symbol ever used > for any flavor of meantone. Thus, different temperaments are > distinguished automatically -- they could be mixed in a score > freely. > > Finally, my remaining (of 3) suggestion is the main one we've been > discussing, the one symbol per comma master list. Which seems best > to me if the list doesn't get unwieldy.

Ok. Well obviously I favour this 3rd option, but you go ahead and do what you like. I'm still not sure what it was that you took offence to, as an "ivory tower pronouncement". It was probably only a result of me misunderstanding your unusual use of "PB" or something.

> Here I really do mean PBs (good PBs are those that lead to good > temperaments),

But the same MOS of a good linear temperament can be derived by tempering a number of different periodicity blocks. i.e. several different PBs can lead to the same MOS of the same temperament. And the same PB may lead to MOS of several different linear temperaments depending which UV is left untempered.

> but in many cases in this thread I don't think there's > an established term that's correct. > > Are miracle-7 and miracle-19 the same "temperament"?

Yes of course. They are different scales or tunings within the same linear temperament. See Definitions of tuning terms: linear temperamen... * [with cont.] (Wayb.)

> >But I think I understand now that you are only talking about the > >search for a good notation for a given linear temperament, not for > >everything. Whereas I was talking about the search for a good notation > >for untempered ratios (and previously thought you were talking about > >the same).

> > Nope, I was talking about everything re. the master list proposal. > I only shortly drifted into the other two proposals, and I noted > they only work for temperaments.

OK. Good. Let's discuss only the master list proposal for now?

> >> >Huh? I thought you just agreed that we would first decide how to > >> >_precisely_ notate ratios?

> >> > >> Yup. In fact, you can think of a PB/temperament *as* a notation in > >> my scheme.

> > > >Now what the heck is a "PB/temperament"?

> > A list of commas. >

> >Then when you've explained that, please explain how you would use one > >of them to notate untempered ratios exactly. Please give some examples.

> > 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...

That sounds like a PB, and has nothing to do with any _particular_ temperament. Any or none of the unison vectors might be distributed.

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

really I don't recall him ever calling anything a "PB/temperament" or saying that you can think of it *as* a "notation". And if you're talking about Gene's use of the term "notation" to refer to some mathematical object, I never did buy that. So was I supposed to know you were using the term "notation" in an unusual way, as well as "PB"?

> >You claim to have been saying all along that it is good to have a > >notation system whose semantics are based on precise ratios and then > >use that to also notate temperaments.

> > Yep.

OK. It seems we must be reading this differently. I'll try to be clearer. 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?

> >So before saying anything about temperaments, maps, generators or > >complexity weightings of temperaments, please explain how you propose > >to notate ratios.

> > Hopefully this is clear by now. As you temper out commas from the > list ("temperament/PB") you simply find that some of the symbols > you assigned never show up in your score.

No, not clear. There must be some serious misunderstanding here. Can anyone else see what it is, because obviously Carl and I can't. I said "without saying anything about temperaments" and the third word you use is "temper". I want to know how you propose to notate untempered rational pitches (in arbitrary scales, not necessarily PBs). You seem to be saying you would do it without reference to any temperament. But you can't seem to explain it without mentioning temperament.

> >If by complexity you only mean "ratio complexity" then I can maybe > >explain further. We didn't actually used any ratio complexity formula > >based on prime exponents or any such. We used ratio popularity > >statistics obtained from the Scala archive. But we had already decided > >to make our nominals in a chain of fifths, before we did that.

> > Searching the space of possible notations, linear temperaments, and > PBs is all the same search.

I don't follow this at all. Only some of the possible notations relate to linear temperaments and only some to PBs, and there is not a one-to-one relationship between PBs and linear temperaments by any stretch of the terminology.

> If you do a search for notations and > only those with 3:2 generators come up, you've heavily weighted your > complexity function, just as if you've searched for temperaments and > only chains-of-fifths tunings came up.

I'm sorry. I have no idea what you're talking about here. What does it mean to "search for notations". For notating ratios we didn't "search for notations" in any sense I can give meaning to. As far as we were concerned, and I suspect most people on this list, it is a no-brainer to start with 7 nominals in a chain of fifths. The only other contender with any chance at all was Johnston's but that ends up being a nightmare to keep track of.

> Searching the scala archive for popular ratios is a hare-brained > idea, if you don't mind me saying so.

Well of course I mind you saying so! Since it was my idea and I don't enjoy having my brain compared to that of a hare. I might think some of your ideas are moronic, but if I did I wouldn't say so, would I? Because I wouldn't want you to feel bad. :-) 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. Now the Scala archive doesn't necssarily tell us that exactly, but it's probably the best handle we've got on it. What's the problem with this idea?

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

I disagree. Most musicians and composers couldn't care less about what comma an accidental stands for, they are happy just to know that if 1/1 is C then 5/4 is E\ and 7/4 is Bb< and so on. How many could tell you, or would care, that in Pythagorean a sharp or flat symbolises the comma 2187/2048? But they sure know that F# is a fifth above B. But of course it makes sense to notate ratios in such a way that their symbols can be factored into nominal and accidental parts such that the accidental has a constant meaning as a certain comma no matter which nominal it is used with (and vice versa).

> Commas are small ratios.

Hey! Something we can agree on. :-)

> Further, on tuning-math we assume that commas that are simple for > their size will be popular with composers.

Like I say, most composers, including JI composers, don't give a stuff about commas, except maybe in the sense of not wanting to have too many very small steps in their scales.

> 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.

> and none > have them have excluded everything but chains-of-fifths tunings.

We haven't excluded everything but chains-of-fifths tunings either. We aim to notate practically anything. I'm still waiting to hear how you propose to notate ratios with a chain of something other than fifths for your nominals.

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

> > Heavens, no!

Oh dear. So _are_ you a Johnston notation supporter?

> You'll miss some of the most compact notations that way, > which have irrational intervals when viewed as a chain (ie miracle).

I'm talking about notating ratios here. How are you going to notate untempered ratios using miracle. Assume C is 1/1. Notate the 19-limit diamond for me using Miracle. Make up any old set of ASCII accidentals and tell me what they and the nominals mean.

> >But that's a notation for a temperament, and a fine one at that. But > >that means there are lots of ratios that it is incapable of > >distinguishing. How is this an embodiment of "what you've been saying > >all along", namely that we should first figure out how to notate

ratios?

> > Paul's scales are 3-D blocks. Add a symbol for every comma you don't > temper out.

I thought they were in a linear temperament called pajara or paultone, but I'll take them to be 7-limit JI for the sake of argument. So which of the many possible JI scales will the nominals correspond to? You seem to be proposing a 7-limit analogue of 5-limit Johnston notation. Do you really not understand why that sucks?

> Can the untempered decatonic scales be notated in Sagittal with fewer > than 3 accidentals?

I don't expect so. But why does this matter? Please show me how you would do so? So are you saying that your notation for a given ratio would depend on what other ratios it was being used with? Such that you would somehow find the best periodicity block to use for its nominals. If so, good luck, but you can count me out. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 7614 Date: Tue, 21 Oct 2003 15:46:22 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> So this is pythagorean notation then?

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

Considering it doesn't have even a single bar of music, no, it's pretty far from clear. By pythagorean I mean the 3-limit PB approach with 7 nominals on a chain of pure fifths plus a single apotome accidental.

>> 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.

So in HEWM D:F and D:F# aren't pure thirds? If so, there's no reduction in the number of new and strange accidentals. I could believe that for the diatonic scale, pythagorean notation would be a more natural basis. So how would you notate the untempered symmetrical decatonic? -Carl

Message: 7615 Date: Tue, 21 Oct 2003 19:18:11 Subject: Re: TM reduced chromatic commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Mercy, I don't know, but it does look interesting. How can you TM > reduce a generator?

It's really Tenney reduction, not TM reduction. Starting from any p- limit version, you find the Tenney minimal element by multiplying by the comma set.

> I was talking about the TM reduced basis for a temperament T[n], > after the usual fashion. The chromatic part is the part that's > not tempered out. You do call it a chroma, I think.

What I was calling a chroma were the things discussed below:

> >If we take instead the 5-limit meantone[7], we get 25/24 as the > >reduced chromatic comma, which is presumably what we want. However, > >reducing the meantone[12] comma instead gives us 125/128 (which we > >can invert to 128/125, of course.) If we want 5-limit schismic > >instead, we have the schismic[7] reducing to 16/15, schismic[12] > >reducing to 81/80 and schismic[17] reducing to 25/24.

> > This looks right, alright. But I'll be daft if I know what the > generator raised to the 21st power and divided by 4 has to do with > it.

There are 21 notes to Blackjack; after running 21 15/14 generators in a row, you have something which is about two octaves wide. Octave reduce it and you have a small interval; Tenney reduce that and you have 36/35. I was going on and on about this from the point of view of the resultant chords a while back.

Message: 7617 Date: Tue, 21 Oct 2003 18:35:19 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>the linear, pythagorean diatonic scale.

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

Message: 7618 Date: Tue, 21 Oct 2003 19:19:37 Subject: Re: hippopothesis From: Gene Ward Smith --- 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.

Message: 7619 Date: Tue, 21 Oct 2003 22:52:42 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> So this is pythagorean notation then?

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

> > Considering it doesn't have even a single bar of music, no, it's > pretty far from clear. > > By pythagorean I mean the 3-limit PB approach with 7 nominals on > a chain of pure fifths plus a single apotome accidental.

yes, then it is.

> >> 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.

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

no, they're pythagorean.

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

right, but their usage is more straightforward.

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

then you're agreeing with dave and me.

> So how would you notate the untempered symmetrical decatonic?

it's pretty straightforward to notate any set of ratios in hewm or sagittal. give the ratios and i'm sure the notation will be forthcoming.

Message: 7620 Date: Tue, 21 Oct 2003 16:01:49 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.

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

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

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

Message: 7621 Date: Tue, 21 Oct 2003 19:21:11 Subject: Re: A 13-limit JI scale From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:

> And a known one, it's a transposition of the 13-limit > Partch diamond.

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? Thanks for the info!

Message: 7622 Date: Tue, 21 Oct 2003 00:10:28 Subject: [tuning] Re: Polyphonic notation 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: >

> > But imagine if the chromatic comma you needed an accidental for was > > (for some bizarre reason) 5569:5801, we would not find a symbol for > > that, so rather than invent a new symbol, we would calculate the > > untempered size of this to be about 70.66 cents and find that the > > closest symbolised comma has an untempered size of about 70.67

> cents,

> > namely 24:25, and so we would use the symbol for that.

> > Not necessarily bizarre--I was proposing 6561/6250 = > (4374/4375)*(21/20) for 5-limit ennealimmal notation.

If you attempt to prime-factorize 5569:5801 you will find that it would indeed be a bizarre chromatic comma for any temperament.

Message: 7623 Date: Tue, 21 Oct 2003 18:40:52 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> 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

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

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

Message: 7624 Date: Tue, 21 Oct 2003 19:27:00 Subject: [tuning] Re: Polyphonic notation From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Why isn't a linearly tempered PB a MOS?

Do you define MOS as any chain-of-generators scale? If so, then clearly it is a MOS.

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