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Message: 7625 Date: Tue, 21 Oct 2003 23:05:22 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

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

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

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

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

> >> 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.

johnston notation does at least as poorly.

Message: 7626 Date: Tue, 21 Oct 2003 02:01:42 Subject: Re: naive question From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

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

Nah. Before finding this list, I worked on JI and equal temperament, but nothing in between.

Message: 7627 Date: Tue, 21 Oct 2003 16:24:27 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

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

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

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

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

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

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

> >johnston notation does at least as poorly.

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

Message: 7628 Date: Tue, 21 Oct 2003 20:25:16 Subject: Re: naive question From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:

> 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? > > Thanks, > Aaron

i don't think ellis did any work in this area at all. erv wilson mentioned or implied quite a few linear temperaments but seemed to miss out entirely on the cases where the period turns out to be a fraction of an octave. bosanquet, if i'm not mistaken, did not even look outside the period=octave, generator~=fifth cases. if you're not familiar with erv wilson's work, please spend some time scratching your head over these: WILSON ARCHIVES * [with cont.] (Wayb.)

Message: 7629 Date: Tue, 21 Oct 2003 02:44:38 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan I figure I'd better respond to this now. I'm still under much the same time pressure, but I figure I've left it long enough. --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >As Paul kindly said, at least with fifths it's a manageable sort of > >mess. :-) And I would add: with many familiar landmarks, particularly > >in the harmony.

> > This sort of attitude vastly: > > () Underestimates the effect of notation on music. Use ordinary > notation, think up ordinary music. > > () Overestimates the difficulty of 'learning new nominals'. You've > got new pitches, new rules, new fingerings, new sounds, new accidentals. > The nominals matter so much?

Carl, The paragraph you quote was preceded by: "George and I long ago conceded that there are advantages to notating a linear temperament with an appropriate number of nominals for that temperament. Our claim is only that, for those who are not willing to learn a totally new set of nominals every time they change tunings, 7 nominals in a chain of approximate fifths is by far the best general solution." 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. Clearly you are not "one of those who are not willing to learn ...", and I hope you will find the sagittal master list of comma accidentals helpful in conjuction with your chosen nominals. If we ever finish it. :-)

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

> >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.

> >> 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.

> >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.

> I thought you said something about the list getting unwieldy. If > you've come up with 600 symbols, I think that should be plenty!

That's around 300 up/down pairs with untempered sizes reasonably evenly spread across the range from -227 to +227 cents. But yes, I think it will be plenty too.

> >> 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. 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.

> Instruments don't read accidentals; people do. I'm not sure how > learning 600 accidentals is any easier than learning tuning-specific > interpretations of existing accidentals. In both cases, once the > tonal system is learned, one should be able to hear the correct notes. > And this proposal has the added benefit of not requiring any new fonts > or eye training -- just use conventional sharps and flats.

The approximately 600 accidentals are composed of only 10 kinds of component (or only 5 if you ignore left-right mirroring) assembled in various combinations, with any given symbol using at most 3 components (not counting the shafts of the arrows). And of course that was not a valid comparison. The alternative to learning tuning-specific interpretations of existing accidentals is not "learning 600 accidentals" but only learning at most one new pair per linear temperament. Sometimes different temperaments will use the same symbol pair because the chroma has the same comma interpretation. I'm not sure I agree that this should always be the TM-reduced version of the chroma, although of course you are free to do that. For example, I personally would not use the 24:25 symbol for meantone, but would continue to use the apotome symbol (which is to say the conventional sharp or flat, since I would not be using the multishaft symbols of the pure sagittal notation).

> It's just a proposal. Drawbacks include: > > () Only works for linear temperaments. > > () It's kinda neat to not have to specify the temperament in advance. > One could mix "temperaments" in the same bar just by using the > appropriate accidentals from a master-list. Can't do this with the > present proposal.

OK.

> >There's definitely no need for this.

> > How do you know? Who can say what composers won't need?

I think that what you meant to say and what I thought you were saying (the referent of "this") were two quite different things. Indeed, who can say? Which is why we've made sagittal as comprehensive and flexible as we could, and have not closed off the possibility of additional "semantic radicals" being added in future.

> >> >So the first part of my belief is that it is far better to have a > >> >notation system whose semantics are based on precise ratios and then > >> >use that to also notate temperaments, rather than trying to find the > >> >ultimate temperament and then using a notation based on that to

notate

> >> >both ratios and other temperaments.

> >> > >> Wow; this is exactly what I've been saying all along!!

> > > >Really? Then how have I managed to waste so much of my time answering > >this thread?

> > Glad to see you have such a high opinion of peer review.

I understand this to be sarcasm. I'm not sure why you thought my question implied a lack of respect for peer review. In fact I value it highly. If George and I did not, we wouldn't have continued to post our deliberations on sagittal to tuning-math for as long as we did after the amount of feedback dropped to practically zero. I think we can see now that a valid answer to my question may in part be "Because I (Carl Lumma) caused a lot of confusion by saying "PB" when I meant "temperament" (or MOS of temperament or something?), and because we both managed to misread each others explanations in various ways".

> >> >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.) 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).

> >> >I understand that you agree with this, and so it should be obvious > >> >that the simplest accidental is no accidental at all and so the > >> >simplest ratios should be represented by nominals alone. When we > >> >agree that powers of 2 will not be represented at all, or will be > >> >represented by an octave number, or by a distance of N staff

positions

> >> >or a clef, then surely you agree that the next simplest thing is to > >> >represent powers of three by the nominals.

> >> > >> Well, that's a weighted-complexity approach. But even with most > >> weighted-complexity lists I've seen, non-rational-generator > >> temperaments appear.

> > > >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"? I'm getting confused all over again. Just when I though I understood what you were on about. :-) Then when you've explained that, please explain how you would use one of them to notate untempered ratios exactly. Please give some examples.

> >Therefore we don't care about weighted > >complexity, or any complexity (except at the 3-prime-limit), because > >we know we are going to represent ratios of the other primes as being > >_OFF_ the chain, by using accidentals.

> > 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. So before saying anything about temperaments, maps, generators or complexity weightings of temperaments, please explain how you propose to notate ratios. 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. If you are trying to say that we should have used the 5-limit diatonic syntonon for our 7 nominals when notating ratios, as Ben Johnston did, then that will be the end of this discussion. There's just no way we could ever consider that, and I think enough people have been over that ground before, that I don't need to do it again.

> >Whether we use rational or irrational generators we can only represent > >powers of _ONE_ ratio _EXACTLY_, _ON_ the chain, (modulo our interval > >of equivalence).

> > "Ratio" obviously. Did you mean "prime"? Then your statement is false. >

I meant what I wrote. So we agree. I'm assuming that our nominals will be contiguous on a uniform chain of some ratio. Then we add some accidentals to allow that chain to be extended somewhat (but not indefinitely). In our case we have 7 nominals in a chain of 3/1's and accidentals that let us extend that to 35 notes. Then we add some other accidentals that let us notate lots of 23-limit ratios as comma inflections off the notes of that chain, since only ratios of 3 can be notated _on_ that chain. How would you do this if the nominals were not in a chain of 3/1's. (Assuming octave equivalence for now).

> >> But certainly the project didn't start out this way, and even in > >> the last few days I saw a blurb for George and/or you looking very > >> confused about non-heptatonic systems.

> > > >I think we're only confused about how a notation whose nominals are > >related by an irrational generator could be used notate ratios > >precisely.

> > Just observe Paul's decatonic notation. It's the perfect embodiment > of everything I've been saying.

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? Regards, -- Dave Keenan

Message: 7630 Date: Tue, 21 Oct 2003 20:26:30 Subject: Re: naive question From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

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

> > Nah. Before finding this list, I worked on JI and equal

temperament,

> but nothing in between.

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.

Message: 7631 Date: Tue, 21 Oct 2003 23:30:35 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

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

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

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

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

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

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

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

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

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

actual johnston notation. would generalized johnston notation notate JI any differently than actual johnston notation?

Message: 7632 Date: Tue, 21 Oct 2003 02:57:04 Subject: Re: naive question From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

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

I suspect Paul Erlich should come before me on that list. In any case Breed-Erlich-Keenan was pretty much a simultaneous cooperative effort that definitely built on earlier stuff by Wilson. The answer to the original question really depends on what you mean by "in depth".

Message: 7633 Date: Tue, 21 Oct 2003 13:31:54 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> 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.

So far, I haven't.

>Why isn't a linearly tempered PB a MOS? Maybe this hasn't been >formally proven,

That's the hypothesis, for which Gene has sketched a proof, I think.

>> 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.

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

>> 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.

:) I've been pushing for notating pieces of music with nominals based on the tuning of the music. That's the portability question. The assertion that a good notation search and a good PB search are equivalent is weaker.

>> 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.

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

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

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

>> 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.

It does sound like a sidetrack, but no, Partch's music does not use a 43-tone "scale" in the above sense. 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.

>> 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.

We all do what we like; I don't think anyone's suggesting fascism here. But you and I are claiming to be exploring the possible ways to do notation, pointing out pros and cons of various approaches and even advocating some approaches over others. In the above case, we both seem to be in favour of the same one (the master list).

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

Ok.

>> 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.

True but it doesn't make what I said wrong. Good PBs and good temperaments are what composers like best. And composers are supposed to like the least number of notes and the lowest error (we're providing the whole lattice). The error gets bigger as the defining commas get bigger, whether you temper or ignore. The notes get fewer as the commas get simpler, whether you temper or ignore.

>>>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.

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

>> 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"?

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

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

Everything but:

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

Why? Maybe it's because of this:

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

Maybe you'd care to explain your reasoning behind this, or point me to where it was explained. Accepting this for a moment, it still only means that at the end of the PB search, you say, "Only the top PB on this list shall ever be used as the basis for a notation.". It still doesn't change the PB search in any way.

>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.

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

>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".

:) If you notate the sym. decatonic in JI, you need three accidentals. There's your JI notation. I mention temperament because I'm trying to point out that nothing in the notation procedure changes. Some of your symbols just go unused.

>> 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.

I meant "same" in the sense that the same criteria are used.

>> 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".

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

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

That's what I'm complaining about!

>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.

I am trying to point out that it is not a no-brainer, any more than it is a no-brainer to use 7-tone pythagorean tuning for all the music we write.

>The only other contender with any chance at all was Johnston's but >that ends up being a nightmare to keep track of.

I'm not schooled in Johnston's notation, but I thought it did conform to the PB approach I've been advocating. If so, accidental pile-up would seem to me a reason for using temperament. But if you've really figured out a way to avoid pile-up in strict JI, I'd like to hear it.

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

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

>> 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.

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

>> 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.

I should have said 'tunings based on such commas would 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.,

> >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.

>> >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.

You can't. You won't, in fact, have a chain of any single interval. But you will have a chain of, on average, secors.

>So which of the many possible JI scales will the nominals correspond >to?

Presumably the one contained in the TM-reduced block.

>You seem to be proposing a 7-limit analogue of 5-limit Johnston >notation.

Yep.

>Do you really not understand why that sucks?

Nope, I sure don't.

>> 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?

I don't think it's possible, if you want to retain any semblance of normal notation. So why is Saggital better for the untempered decatonic scale than the PB approach?

>So are you saying that your notation for a given ratio would depend on >what other ratios it was being used with?

That's the portability issue. But as I pointed out, my assertion is weaker. So I can give you the 'there can be only one' notation, and still ask you base it on the top-ranking result of a search. -Carl

Message: 7634 Date: Tue, 21 Oct 2003 18:49:03 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> 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.

In a perverse sense this is ok according to Paul, since scales with fractional periods don't have *exactly* two intervals per class. -Carl

Message: 7635 Date: Tue, 21 Oct 2003 23:54:05 Subject: [tuning] Re: Polyphonic notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

I'm with Carl on this one. Didn't you just say that Erv Wilson seems to have missed the fractional octave cases. If so, then he couldn't exactly have stated that these are not to be considered MOS, could he? Isn't it up to us whether or not to generalise it to these cases? We did so for quite some time. Or at least I did. I even thought Kraig supported this at one stage. Can you point me to a post from Kraig or whatever it was that made you decide MOS were only single-chain?

> DE, distributionally even, means that each generic interval (number > of steps in the scale) comes in *at most* two step sizes. this is in > fact what you always get when you linearly temper a PB. all DEs have > a period which is 1/n octave, n whole number, and a generator > iterated enough times within the period so that you get only two step > sizes.

Mind you, "Moment of Symmetry" is a pretty non-descriptive term for what it really is. But using DE as a noun doesn't seem right. We say "a MOS of a temperament" and that's fine when expanded, but I don't thing should we say "a DE of a temperament" but rather "a DE scale of a temperament".

Message: 7636 Date: Tue, 21 Oct 2003 13:34:26 Subject: Re: TM reduced chromatic commas From: Carl Lumma

>> 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.

Ok, that's better. (I think)

>> >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.

Aha! -Carl

Message: 7637 Date: Tue, 21 Oct 2003 16:56:17 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

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

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

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

So it's familiarity, not linearity.

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

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

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

> >actual johnston notation. would generalized johnston notation notate >JI any differently than actual johnston notation?

Yeah, the nominals can be from any PB, not just the 5-limit diatonic scale. Now, if you want to always use the same PB for 5-limit JI, the diatonic PB may be the best choice... -Carl

Message: 7638 Date: Tue, 21 Oct 2003 05:23:32 Subject: TM reduced chromatic commas From: Gene Ward Smith --- 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. There is no problem defining the TM reduced period and generator for a given prime-limit linear temperament. For instance, for 7-limit miracle, the TM reduced period is 2 and the TM reduced generator is 15/14. If we TM reduce (15/14)^21 / 4 we get 36/35, the chroma for septimal Blackjack. Is this sort of thing what you mean by a "TM reduced chromatic comma"? 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.

Message: 7639 Date: Tue, 21 Oct 2003 13:35:15 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. -Carl

Message: 7640 Date: Tue, 21 Oct 2003 23:57:56 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:

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

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

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

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

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

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

you

> decide MOS were only single-chain?

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

> > DE, distributionally even, means that each generic interval

(number

> > of steps in the scale) comes in *at most* two step sizes. this is

in

> > fact what you always get when you linearly temper a PB. all DEs

have

> > a period which is 1/n octave, n whole number, and a generator > > iterated enough times within the period so that you get only two

step

> > sizes.

> > Mind you, "Moment of Symmetry" is a pretty non-descriptive term for > what it really is. But using DE as a noun doesn't seem right. We say > "a MOS of a temperament" and that's fine when expanded, but I don't > thing should we say "a DE of a temperament" but rather "a DE scale

of

> a temperament".

you're right. i was just being quick and sloppy.

Message: 7641 Date: Tue, 21 Oct 2003 08:21:40 Subject: Re: [tuning] Re: Polyphonic notation From: Graham Breed Carl Lumma wrote:

> 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.

I disagree. I wrote my Magic piece using a hybrid notation with 10 nominals. You could do the same thing with 7 or 12, but I happened to be thinking decimally at the time. Anyway, the simplest Magic scale that would have worked is 19. A notation with 19 nominals wouldn't have been palatable. The music itself was supposed to sound free within 9-limit harmony, so the scale structure wasn't a problem. So maybe the Miller limit applies to both the music and the notation, but not the temperament. Graham

Message: 7642 Date: Tue, 21 Oct 2003 20:38:42 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

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

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

> 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.

>

> >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.

Message: 7643 Date: Tue, 21 Oct 2003 00:49:52 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

>> 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.

> >I disagree. I wrote my Magic piece using a hybrid notation with 10 >nominals.

Which piece?

>You could do the same thing with 7 or 12, but I happened to >be thinking decimally at the time. Anyway, the simplest Magic scale >that would have worked is 19.

Worked?

>A notation with 19 nominals wouldn't have >been palatable. The music itself was supposed to sound free within >9-limit harmony, so the scale structure wasn't a problem. > >So maybe the Miller limit applies to both the music and the notation, >but not the temperament.

It's only supposed to apply to things that require working memory to understand. If your piece doesn't feature a prominent melodic line then it probably wouldn't apply. If it ever does (I'm not aware of any experiments having been done about it). -Carl

Message: 7644 Date: Tue, 21 Oct 2003 17:04:39 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

Message: 7645 Date: Tue, 21 Oct 2003 20:43:15 Subject: Re: TM reduced chromatic commas From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >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.

> > It clearly depends on the number of notes, which is why I said: >

> >> 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.

>

> >There is no problem defining the TM reduced period and generator

for

> >a given prime-limit linear temperament. > > > >For instance, for 7-limit > >miracle, the TM reduced period is 2 and the TM reduced generator is > >15/14. If we TM reduce (15/14)^21 / 4 we get 36/35, the chroma for > >septimal Blackjack. Is this sort of thing what you mean by a "TM > >reduced chromatic comma"?

> > Mercy, I don't know,

i think the answer is yes.

> but it does look interesting.

yes, 36:35 is the simplest ratio for the chromatic unison vector of blackjack.

> How can you TM reduce a generator?

you guys are saying "TM reduce" but it's really just Tenney-reduce, as these are single intervals. you can do it to the generator just as easily as you can do it to any other interval class, including the unisons, of a temperament.

> >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.

blackjack has 21 notes per octave; divide by 4 to octave-reduce the resulting chroma.

Message: 7646 Date: Tue, 21 Oct 2003 01:14:39 Subject: Re: TM reduced chromatic commas From: Carl Lumma

>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.

It clearly depends on the number of notes, which is why I said:

>> 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.

>There is no problem defining the TM reduced period and generator for >a given prime-limit linear temperament. > >For instance, for 7-limit >miracle, the TM reduced period is 2 and the TM reduced generator is >15/14. If we TM reduce (15/14)^21 / 4 we get 36/35, the chroma for >septimal Blackjack. Is this sort of thing what you mean by a "TM >reduced chromatic comma"?

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

>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. -Carl

Message: 7647 Date: Tue, 21 Oct 2003 13:45:09 Subject: Re: [tuning] Re: Polyphonic notation From: Carl Lumma

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

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

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

>> 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'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! -Carl

Message: 7648 Date: Tue, 21 Oct 2003 01:18:43 Subject: 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? The wolf introduces exactly 1 extra comma? Do we get a planar version of the same temperament? I think not. In general is there any sense to thinking of chains of generators? Clearly. -Carl

Message: 7649 Date: Tue, 21 Oct 2003 20:54:47 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

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

> > Do you define MOS as any chain-of-generators scale?

no, it has to have two step sizes to be MOS.

> If so, then > clearly it is a MOS.

when the period is a fraction of an octave but the interval of equivalence is still an octave (the latter, we tend to assume a priori), we no longer have an MOS. DE, distributionally even, means that each generic interval (number of steps in the scale) comes in *at most* two step sizes. this is in fact what you always get when you linearly temper a PB. all DEs have a period which is 1/n octave, n whole number, and a generator iterated enough times within the period so that you get only two step sizes.

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