Re: [tuning] Re: Polyphonic notation

>> What gets wrecked if one's basis isn't distrib. even?
>
>you yourself said that one begins a set of nominals forming a
>periodicity block (on the tuning-math list; maybe you should reply to
>this over there).

What does distrib. evenness have to do with PBs?

>that won't work once you get beyond the set of primes that you're
>allowing the linear temperament to approximate.

Obviously.

>of course, george and
>dave's proposal is not based on any temperament at all.

I don't mean to suggest that notations should be based on
temperaments. But I do assert that the problem of finding
notations is equivalent to the problem of finding PBs.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What gets wrecked if one's basis isn't distrib. even?
> >
> >you yourself said that one begins a set of nominals forming a
> >periodicity block (on the tuning-math list; maybe you should reply
to
> >this over there).
>
> What does distrib. evenness have to do with PBs?

a fokker periodicity block, when all but one of the unison vectors
are tempered out, becomes a distributionally even scale. that's my
Hypothesis, anyway.

i mentioned *altered* versions of DE scales because one may wish to
start with a "non-fokker" periodicity block.

At 04:57 PM 9/30/2003, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>> >> What gets wrecked if one's basis isn't distrib. even?
>> >
>> >you yourself said that one begins a set of nominals forming a
>> >periodicity block (on the tuning-math list; maybe you should reply
>to
>> >this over there).
>>
>> What does distrib. evenness have to do with PBs?
>
>a fokker periodicity block, when all but one of the unison vectors
>are tempered out, becomes a distributionally even scale. that's my
>Hypothesis, anyway.

So distrib. even and MOS are equivalent?

>i mentioned *altered* versions of DE scales because one may wish to
>start with a "non-fokker" periodicity block.

What's a non-fokker PB -- a shape other than a parallelepiped?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> At 04:57 PM 9/30/2003, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> What gets wrecked if one's basis isn't distrib. even?
> >> >
> >> >you yourself said that one begins a set of nominals forming a
> >> >periodicity block (on the tuning-math list; maybe you should
reply
> >to
> >> >this over there).
> >>
> >> What does distrib. evenness have to do with PBs?
> >
> >a fokker periodicity block, when all but one of the unison vectors
> >are tempered out, becomes a distributionally even scale. that's my
> >Hypothesis, anyway.
>
> So distrib. even and MOS are equivalent?

no, scales with a period equal to a 1/N octave, where N is an integer
greater than 1, are distributionally even but not MOS.

> >i mentioned *altered* versions of DE scales because one may wish
to
> >start with a "non-fokker" periodicity block.
>
> What's a non-fokker PB -- a shape other than a parallelepiped?

right, like the hexagons in the gentle introduction, or more
radically, something like a harmonic minor scale.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> no, scales with a period equal to a 1/N octave, where N is an
integer
> greater than 1, are distributionally even but not MOS.

i should have said "may be" instead of "are" and "but cannot be"
instead of "but not".

>no, scales with a period equal to a 1/N octave, where N is an integer
>greater than 1, are distributionally even but not MOS.

Oh, you're enforcing the 'new' definition of MOS. Who came up
with distrib. even?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >no, scales with a period equal to a 1/N octave, where N is an
integer
> >greater than 1, are distributionally even but not MOS.
>
> Oh, you're enforcing the 'new' definition of MOS. Who came up
> with distrib. even?
>
> -Carl

john clough and nora englesbrshmegegel . . . i forget her last name.
you can find the term in my 22 paper on your website.

>>Who came up with distrib. even?
>>
>> -Carl
>
>john clough and nora englesbrshmegegel . . . i forget her last name.
>you can find the term in my 22 paper on your website.

Wow, was this added in a later rev? All I remember is "maximal
evenness".

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>Who came up with distrib. even?
> >>
> >> -Carl
> >
> >john clough and nora englesbrshmegegel . . . i forget her last
name.
> >you can find the term in my 22 paper on your website.
>
> Wow, was this added in a later rev? All I remember is "maximal
> evenness".
>
> -Carl

yes, that was changed. it was silly to keep using "maximal evenness"
when

() the definition clough and others use for this term reflects a
philosophy i don't subscribe to
() their definition didn't agree with the definition i gave
() the definition i gave agreed with their term "distributional
evenness" -- John Clough and Nora Engebretsen wrote a paper probably
called _distributionally even scales_, which seems to be missing from
the tuning and temperament bibliography.

This is a followup to
/tuning/topicId_46826.html#47437

Me (Dave):
> >Why would I want to? With the fifth as the generator of nominals, the
> >natural number of nominals is 7. 6 is improper. You say "By doing X
> >you're doing a bad thing". And I say "But I'm not doing X". And you
> >say "Do X". This isn't making much sense to me. Sorry.

Carl:
> But you are (or were) advocating doing X, but trying to force 7
> nominals on other scales. It is, as you say, improper to do so. Which
> is what I was trying to point out. It looks as though I've succeeded!
>

Er Sorry. No. That was _Rothenberg_ improper. But of course it's worse
than that, as Paul said, its not distributionally even. It's not MOS,
or well-formed or whatever. When you introduce the Pythagorean-limma
(or diatonic semitone) accidental needed to make it work, you find you
have a redundant nominal.

The analogy to your example, would be if I tried to force 7 nominals
_in_a_chain_of_secors_ onto Blackjack. But that's not what I'm doing.
I'm forcing 7 nominals in a chain of 2:3 approximations.

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. And we basically have no idea how to notate ratios precisely
if the generator of the nominals is not itself a _very_ simple ratio.

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.

> >No. A little reflection allows me to explain that, as it stands now,
> >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.]

So what's a _good_ PB for notational 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?

> >We quickly got beyond 72-ET, but for a very long time there was a
> >constant tension between basing the notation on some equal temperament
> >versus basing it on ratios and thereby keeping it open. The problem
> >with a notation based on ratios is that, to keep everyone happy, you
> >need a huge number of different accidentals,
>
> With, say, 19-limit JI, I don't see a way around this.

Nor do I, unless you are willing to allow some symbols to be only
approximations (which in many cases could be less than 0.5 c away).
That's why we _are_ providing a huge number of symbols, for those who
think they need them.

> 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. For example, few
people even want to refer to a note one degree above C in 31-ET,
exclusively as Dbb for very long. They soon invent a new accidental
for this and call it C^ or some such. The more you extend your chain
of generators without closing, the more new accidentals you will want
for these "enharmonics". Graham has already done this once for decimal.

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.

> The particular comma involved will depend on
> the limit and the number of notes in the base scale. This could be
> handled two ways. The first way I suggested is to get a list of simple
> 19-limit commas and assign accidental pairs to them.

OK. Well that's nearly done (to at least 23-limit), but not quite
ready for publication. I keep letting myself get distracted by tuning
list posts. :-)

> The same
> accidentals could be used for planar temperaments, JI, whatever, with
> more than one pair in use at a time.

Sure.

> If average use ("gimme 9 notes of such-and-such temperament in the
> 13-limit") turns out to require more commas than can fit on a list,

I don't understand how average-use could require "more commas than can
fit on a list". What could this mean except "an infinite number of
commas"?

> 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?

There's definitely no need for this. We've got so many accidentals
available in the sagittal system that if you can't find a simple
enough one that fits, by using their primary comma interpretations,
you can just choose the one whose primary comma has its untempered
value nearest in cents to the untempered value of the comma you really
wanted (so-called secondary interpretations of the symbols), and then
you let it be known that, for that temperament only, the symbol
exactly represents that secondary comma. We've already done that for
some obscure multiples of 12-ET, notated using nominals in a chain of
12-ET-sized fifths. In this case we figure people don't care what
ratio is being approximated and will just think of them as particular
fractions of a tone or particular fractions of a sharp or flat.

> >We soon realised we could have our cake and eat it too - that every
> >symbol could represent a single unique comma ratio but that users who
> >want to notate rational tunings are free to choose larger or smaller
> >sets of symbols to trade off economy-of-symbols against
> >accuracy-of-representation (for those ratios which are not represented
> >exactly by the chosen symbols).
>
> Great. That's the master list idea.

OK. I though I made it clear long ago that's what we were doing. Sorry

> >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
>
> Representing how many harmonic dimensions?

There are primary commas on the list with primes up to 23 (maybe 29,
it isn't finalised yet).

> >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?

> >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.

> >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? 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.

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

> >Well I think the fact that we have Graham proposing MIRACLE
> >temperament with 10 nominals and Gene proposing ennealimmal
> >temperament with 9 nominals should make it clear that there is
> >unlikely to ever be agreement on which is the ultimate temperament
> >for notating everything else including ratios.
>
> It was the ultimate-temperament aspect of the project I objected
> to since the beginning!

OK. Well I'm glad that's cleared up.

> >You seem to have been assuming that George and I were merely
> >championing some other (fifth-generated) temperament as the
> >ultimate for notating everything else. I hope I have explained
> >why this is not so.
>
> Ok, ok, I think we're more on the same page now.

Great!

> 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.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>

> So what's a _good_ PB for notational purposes? That sounds even less
> likely to be agreed upon than a good linear temperament.

a sizable community, perhaps most ji composers, have agreed -- ben
johnston's notation has the nominals on the so-called "major block".
as you know, i prefer your choice, as do joe monzo and daniel wolf.

> > With linear temperaments, you only need 1 accidental pair at a
time,
> > as I've pointed out.
>
> For example, few
> people even want to refer to a note one degree above C in 31-ET,
> exclusively as Dbb for very long. They soon invent a new accidental
> for this and call it C^ or some such.

anyway, it's a beautiful note to play over a G major chord before
resolving to C major.

> 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.

that's where it's nice to have new nominals. in particular, i like a
half-octave from G to be written as an upside-down G, and perhaps
notated with the notehead *between* the positions for C and D, with a
slash through the notehead just so no one mistakes it for a C or a
D . . .

> > 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?

i think carl meant this as a way of deciding which of the accidentals
to use in a particular scenario, not as a way of introducing
*additional* accidentals . . . in other words, it means that for both
meantone diatonic and 10-tone pajara you'd use the symbol for 25:24
to mean the single accidental of the temperament, while if blackjack
qualified you might use 36:35 . . .

> you can just choose the one whose primary comma has its untempered
> value nearest in cents to the untempered value of the comma you
really
> wanted

no, carl's solution is better . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
>
> > So what's a _good_ PB for notational purposes? That sounds even less
> > likely to be agreed upon than a good linear temperament.
>
> a sizable community, perhaps most ji composers, have agreed -- ben
> johnston's notation has the nominals on the so-called "major block".
> as you know, i prefer your choice, as do joe monzo and daniel wolf.

I actually meant, what are the _criteria_ that make a PB _good_ in
this application (as a universal set of nominals).

> > 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.
>
> that's where it's nice to have new nominals. in particular, i like a
> half-octave from G to be written as an upside-down G, and perhaps
> notated with the notehead *between* the positions for C and D, with a
> slash through the notehead just so no one mistakes it for a C or a
> D . . .

Good point. Additional chains can use additional nominals rather than
additional accidentals.

> > you can just choose the one whose primary comma has its untempered
> > value nearest in cents to the untempered value of the comma you
> really
> > wanted
>
> no, carl's solution is better . . .

Which was .... ?

Maybe if _you_ explain it ...

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> > > you can just choose the one whose primary comma has its
untempered
> > > value nearest in cents to the untempered value of the comma you
> > really
> > > wanted
> >
> > no, carl's solution is better . . .
>
> Which was .... ?
>
> Maybe if _you_ explain it ...

you want the chromatic alteration symbol to correspond to the
simplest ratio it actually represents in the temperament. so for the
diatonic/meantone case, your choice is between { . . . ,2187:2048,
135:128, 25:24, 250:243, . . .}, and the simplest is 25:24, so you
use the symbol for 25:24 . . .

>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?

>> >No. A little reflection allows me to explain that, as it stands now,
>> >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.]
>
>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.

>> 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?

>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?

>> If average use ("gimme 9 notes of such-and-such temperament in the
>> 13-limit") turns out to require more commas than can fit on a list,
>
>I don't understand how average-use could require "more commas than can
>fit on a list". What could this mean except "an infinite number of
>commas"?

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

>> 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?

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.

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.

>There's definitely no need for this.

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

>> >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.

>> >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!

>> >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.

>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.

>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.

>> 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.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > > > you can just choose the one whose primary comma has its
> untempered
> > > > value nearest in cents to the untempered value of the comma you
> > > really
> > > > wanted
> > >
> > > no, carl's solution is better . . .
> >
> > Which was .... ?
> >
> > Maybe if _you_ explain it ...
>
> you want the chromatic alteration symbol to correspond to the
> simplest ratio it actually represents in the temperament. so for the
> diatonic/meantone case, your choice is between { . . . ,2187:2048,
> 135:128, 25:24, 250:243, . . .}, and the simplest is 25:24, so you
> use the symbol for 25:24 . . .

There's no conflict here with what I wrote above. You're saying that
"the comma you really want" is 25:24. So first we look to see if we
have a symbol for this (i.e. with 24:25 as its primary (exact)
interpretation). It turns out that we do, namely )||( , so the part
you've quoted above does not apply.

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.

We would say that we are using a secondary interpretation of the )||(
symbol.

I'm sorry Carl,

I just don't have time to persue this any further. But it sure seems
as though the ways we are misinterpeting each other are legion.

>I'm sorry Carl,
>
>I just don't have time to persue this any further. But it sure seems
>as though the ways we are misinterpeting each other are legion.

You don't have time to study Paul's decatonic notation, provide simple
examples of a sagittal alternative, explain your reasoning or realize
that you've made an unnecessary assumption about the generator, etc.
Too bad. Expect me to continue to balk when I see ivory tower
pronouncements about sagittal in the future.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I'm sorry Carl,
> >
> >I just don't have time to persue this any further. But it sure seems
> >as though the ways we are misinterpeting each other are legion.
>
> You don't have time to study Paul's decatonic notation, provide simple
> examples of a sagittal alternative, explain your reasoning or realize
> that you've made an unnecessary assumption about the generator, etc.
> Too bad. Expect me to continue to balk when I see ivory tower
> pronouncements about sagittal in the future.
>
> -Carl

Carl,

This has been taking me away from working on the explanations of the
sagittal system, that you agree are sorely needed, and from urgent
paying work.

I'm really sorry I made that crack about "wasting my time". Please
accept my humble apologies.

Maybe we'll communicate better when we've both had a chance to cool
off, and I'm not under so much time pressure.

Regards,
-- Dave Keenan

>This has been taking me away from working on the explanations of the
>sagittal system, that you agree are sorely needed, and from urgent
>paying work.
>
>I'm really sorry I made that crack about "wasting my time". Please
>accept my humble apologies.
>
>Maybe we'll communicate better when we've both had a chance to cool
>off, and I'm not under so much time pressure.

Ok. Pls. reference:

/tuning-math/message/6927

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >no, scales with a period equal to a 1/N octave, where N is an
integer
> >greater than 1, are distributionally even but not MOS.
>
> Oh, you're enforcing the 'new' definition of MOS.

Not on me, I hope. I don't like it and there are too many terms
floating about as it is. We could just stick to Myhill's property.

--- In tuning-math@yahoogroups.com, "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.

>> Oh, you're enforcing the 'new' definition of MOS.
>
>Not on me, I hope. I don't like it and there are too many terms
>floating about as it is. We could just stick to Myhill's property.

Yes, I must admit I don't think we should change our usage of
MOS, because of something Kraig said. I don't think Erv would
mind. He reminds me somewhat of Derrida in that he considers
names a necessary evil. He has said repeatedly, to me, and in
a public lecture, that he's simply struggling to communicate
what he's doing, and that he hopes only that someone will come
along and find something useful, and improve the names if possible.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Oh, you're enforcing the 'new' definition of MOS.
> >
> >Not on me, I hope. I don't like it and there are too many terms
> >floating about as it is. We could just stick to Myhill's property.
>
> Yes, I must admit I don't think we should change our usage of
> MOS, because of something Kraig said.

but the only reason i dragged it in in the first place was because of
something else kraig said!!! it came in on false pretenses, and now
it goes back out.

>but the only reason i dragged it in in the first place was because of
>something else kraig said!!! it came in on false pretenses, and now
>it goes back out.

If the pretenses don't matter, we can let history and/or sensicality
prevail.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >but the only reason i dragged it in in the first place was because
of
> >something else kraig said!!! it came in on false pretenses, and
now
> >it goes back out.
>
> If the pretenses don't matter, we can let history and/or sensicality
> prevail.
>
> -Carl

meaning?

>> If the pretenses don't matter, we can let history and/or sensicality
>> prevail.
>>
>> -Carl
>
>meaning?

Whatever we think it means. I tend to see history as in, the
last few years on this list -- our contribution is substantial.
Given the term MOS, I don't see much of a role for sensicality,
but it seems the term applies just as well to the fractional-
octave case.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> If the pretenses don't matter, we can let history and/or
sensicality
> >> prevail.
> >>
> >> -Carl
> >
> >meaning?
>
> Whatever we think it means. I tend to see history as in, the
> last few years on this list -- our contribution is substantial.
> Given the term MOS, I don't see much of a role for sensicality

what does sensicality mean?

> but it seems the term applies just as well to the fractional-
> octave case.

after kraig told us it doesn't, daniel and other chimed in in
agreement. these are people who know the originator of the term, erv
wilson, far better than i do. so i'm inclined to revert to my
previous usage as it'll make communicating with the erv-ites easier,
and these are pretty much the main people who use the term anyway.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "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.

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@yahoogroups.com, 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

>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

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

--- In tuning-math@yahoogroups.com, "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.

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

>> 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

>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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > I assumed you meant that we should, for example, have a unique symbol
> > pair for each of the following:
> > the TM-reduced chromatic comma for meantone
> > the TM-reduced chromatic comma for schismic
> > the TM-reduced chromatic comma for diaschismic
> > the TM-reduced chromatic comma for kleismic
> > the TM-reduced chromatic comma for miracle
> > etc.
> > even though the first two could clearly use the same symbol.
>
> The first order of business would be to define what the "TM reduced
> chromatic comma" would be for a given linear temperament. It seems to
> me this is really a property of a MOS for that temperament.

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

--- In tuning-math@yahoogroups.com, 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

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 http://sonic-arts.org/dict/lineartemp.htm

> >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
http://dkeenan.com/

--- In tuning-math@yahoogroups.com, 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.

--- In tuning-math@yahoogroups.com, "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.

>> 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
>
>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

>> 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

--- In tuning-math@yahoogroups.com, 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.

--- In tuning-math@yahoogroups.com, 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.

>> 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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "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.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >> The list of commas defines a finite region of the lattice. Every
> >> pitch within the region gets a nominal. The lattice is tiled
with
> >> such regions. The commas in the list are assigned symbols...
really
> >> all this is covered by Paul, in numerous posts and his paper.
> >
> >but i assume that the commatic unison vectors are irrelevant to
the
> >musician, as is true in the vast majority of western music. if the
> >81:80s *are* being kept track of, in a piece in JI or something
> >similar, then i would *not* go along with the johnston notation,
> >which seems to be what you're inferring from my presentations.
>
> Aha!
>
> And no, I did not say you would. :)
>
> What would you do?

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

> >> >I'm assuming that our nominals will be contiguous on a uniform
> >> >chain of some ratio.
> >>
> >> Heavens, no! You'll miss some of the most compact notations
that
> >> way, which have irrational intervals when viewed as a chain (ie
> >> miracle).
> >
> >dave didn't mean a rational ratio necessarily.
>
> He said ratio!

pi:e is a ratio.

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

I'm not sure what you mean.

I looked at the HEWM page, and it has an 81:80
accidental...

>> >> >I'm assuming that our nominals will be contiguous on a uniform
>> >> >chain of some ratio.
//
>> >dave didn't mean a rational ratio necessarily.
>>
>> He said ratio!
>
>pi:e is a ratio.

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

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >i've argued strongly against johnston's system and for hewm or
> >equivalents on the tuning list. maybe you weren't around then.
some
> >argued oppositely, saying that remembering the extra comma between
D
> >and A and between Bb and F (or B and F#) was no different in
quality
> >than remembering the extra accidental between B and F required to
> >make a perfect fifth in standard western musical notation, being
> >simply two extra things to remember. i'd rather not.
>
> I'm not sure what you mean.
>
> I looked at the HEWM page, and it has an 81:80
> accidental...

right, but D:A is 2:3, Bb:F is 2:3, and B:F# is 2:3. they're not in
the johnston notation.
>
> >> >> >I'm assuming that our nominals will be contiguous on a
uniform
> >> >> >chain of some ratio.
> //
> >> >dave didn't mean a rational ratio necessarily.
> >>
> >> He said ratio!
> >
> >pi:e is a ratio.
>
> Yes, but there's also a Partchian definition in use aroundabouts.
> And if he really meant "all real numbers", he wouldn't have said
> "ratio".
>
i'll let dave speak for himself.

>> I'm not sure what you mean.
>>
>> I looked at the HEWM page, and it has an 81:80
>> accidental...
>
>right, but D:A is 2:3, Bb:F is 2:3, and B:F# is 2:3. they're not in
>the johnston notation.

So this is pythagorean notation then?

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

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> I'm not sure what you mean.
> >>
> >> I looked at the HEWM page, and it has an 81:80
> >> accidental...
> >
> >right, but D:A is 2:3, Bb:F is 2:3, and B:F# is 2:3. they're not
in
> >the johnston notation.
>
> So this is pythagorean notation then?

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

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

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

>> 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

--- In tuning-math@yahoogroups.com, 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.

>> 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

--- In tuning-math@yahoogroups.com, 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.

>> >> 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

--- In tuning-math@yahoogroups.com, 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?

--- In tuning-math@yahoogroups.com, "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".

>> >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

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "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.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >no, it's because of the linearity (1-dimensionality) of the set
of
> >> >nominals.
> >>
> >> Then one could just as well use a 7-tone chain-of-minor-thirds.
> >> But I don't think that would work, do you?
> >
> >it would work fine as long as there were a set of symbols that
> >signified the 7-tone chain-of-minor-thirds clearly to musicians.
> >right now, there isn't.
>
> So it's familiarity, not linearity.

linearity is why hewm works better than johnston.

> >> >> I don't think scales without a good series of fifths, such
> >> >> as untempered kleismic[7], would work so well.
> >> >
> >> >johnston notation does at least as poorly.
> >>
> >> You mean generalized johnston notation, of the kind I've been
> >> discussing with Dave, or actual Johnston notation?
> >
> >actual johnston notation. would generalized johnston notation
notate
> >JI any differently than actual johnston notation?
>
> Yeah, the nominals can be from any PB, not just the 5-limit diatonic
> scale.

an embarrassment of riches?

> Now, if you want to always use the same PB for 5-limit JI,
> the diatonic PB may be the best choice...

which, the just major? i disagree, of course.

>> 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

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> But in this case you read me right the first time... ie, the
> >> TM-reduced chromatic comma for miracle-7, miracle-19, would be the
> >> same symbol,
> >
> >you lost me there. how are these suitable scales for notation? i
> >thought for sure they didn't satisfy your criteria, any more than
> >diatonic-6 would.
>
> Sorry, those are limits. Cards I put in [], following Gene.

Can't we just write "7-limit-miracle" etc. for these? I thought they
were cardinalities too. e.g. miracle-21 has been a synonym for
blackjack since forever.

So you asked whether I consider 7-limit and 19-limit versions of
miracle to be the same temperament. Well not exactly, since they may
have different optimum generators, but clearly the fact that the prime
mapping of one is a subset of the other's is highly significant, So I
think you could use the term either way, and just spell out somewhere
how you're using it.

> >> >I'm assuming that our nominals will be contiguous on a uniform
> >> >chain of some ratio.
> >>
> >> Heavens, no! You'll miss some of the most compact notations that
> >> way, which have irrational intervals when viewed as a chain (ie
> >> miracle).
> >
> >dave didn't mean a rational ratio necessarily.
>
> He said ratio!

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

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> Can you point me to a post from Kraig or whatever it was that made
you
> decide MOS were only single-chain?

a bunch of posts from kraig and daniel in late march of this year on
the specmus list.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Now, if you want to always use the same PB for 5-limit JI,
> >> the diatonic PB may be the best choice...
> >
> >which, the just major? i disagree, of course.
>
> Of course? What would beat it out (with less than 11 tones)?
>
> -Carl

how about the linear, pythagorean diatonic scale?

Sorry I allowed that ambiguity. I should indeed have said "rational"
rather than simply "ratio".

When I am not considering strict JI then I am happy for the nominals
to be contiguous on a uniform chain of some irrational generator,
including linear-temperament-specific cases where the nominals are in
a DE scale of that temperament with cardinality other than 7.

I can't support notations that do otherwise, such as Johnstons,
because as Paul said, it just adds more things you have to keep track
of - commas that vanish between some pairs of nominals and not others.
Even the JI folks that use it express reservations. Yes it is like the
BF case, but one of those "wolves" per notation is already too many.
Interestingly this problem doesn't occur at all in decimal miracle, or
any other notation based on a DE scale that doesn't "wrap around".

>> >> Now, if you want to always use the same PB for 5-limit JI,
>> >> the diatonic PB may be the best choice...
>> >
>> >which, the just major? i disagree, of course.
>>
>> Of course? What would beat it out (with less than 11 tones)?
>>
>> -Carl
>
>how about the linear, pythagorean diatonic scale?

Ok, I meant 5-limit PB.

-Carl

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> Sorry I allowed that ambiguity. I should indeed have said "rational"
> rather than simply "ratio".
>
> When I am not considering strict JI then I am happy for the nominals
> to be contiguous on a uniform chain of some irrational generator,
> including linear-temperament-specific cases where the nominals are
in
> a DE scale of that temperament with cardinality other than 7.
>
> I can't support notations that do otherwise, such as Johnstons,
> because as Paul said, it just adds more things you have to keep
track
> of - commas that vanish between some pairs of nominals and not
others.
> Even the JI folks that use it express reservations. Yes it is like
the
> BF case, but one of those "wolves" per notation is already too many.
> Interestingly this problem doesn't occur at all in decimal miracle,
or
> any other notation based on a DE scale that doesn't "wrap around".

what do you mean? certainly there's a single interval of 6 nominals
in decimal miracle that fails to be an approximate 2:3.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> Now, if you want to always use the same PB for 5-limit JI,
> >> >> the diatonic PB may be the best choice...
> >> >
> >> >which, the just major? i disagree, of course.
> >>
> >> Of course? What would beat it out (with less than 11 tones)?
> >>
> >> -Carl
> >
> >how about the linear, pythagorean diatonic scale?
>
> Ok, I meant 5-limit PB.
>
> -Carl

which one? (we're going around in circles)

>Yes it is like the BF case, but one of those "wolves" per notation
>is already too many.

It's never been a problem for me. I've never been reading a piece
of music and stopped and said, "oh wait, there's that BF again; it's
not a 2:3". Just me, I guess.

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

-Carl

>> >> >> Now, if you want to always use the same PB for 5-limit JI,
>> >> >> the diatonic PB may be the best choice...
>> >> >
>> >> >which, the just major? i disagree, of course.
>> >>
>> >> Of course? What would beat it out (with less than 11 tones)?
>> >>
>> >> -Carl
>> >
>> >how about the linear, pythagorean diatonic scale?
>>
>> Ok, I meant 5-limit PB.
>>
>> -Carl
>
>which one? (we're going around in circles)

Exactly! Which one would you use to notate 5-limit JI?

:)

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >> Now, if you want to always use the same PB for 5-limit JI,
> >> >> >> the diatonic PB may be the best choice...
> >> >> >
> >> >> >which, the just major? i disagree, of course.
> >> >>
> >> >> Of course? What would beat it out (with less than 11 tones)?
> >> >>
> >> >> -Carl
> >> >
> >> >how about the linear, pythagorean diatonic scale?
> >>
> >> Ok, I meant 5-limit PB.
> >>
> >> -Carl
> >
> >which one? (we're going around in circles)
>
> Exactly! Which one would you use to notate 5-limit JI?

the linear, pythagorean diatonic scale.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "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.

You mean Daniel Wolf?

> 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.

OK. You win. :-) I'll just avoid the term MOS entirely, and use DE
scale. I don't see myself using MOS to mean single-chain DE scale. It
would be too ambiguous given all the chopping and changing.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > Interestingly this problem doesn't occur at all in decimal miracle,
> or
> > any other notation based on a DE scale that doesn't "wrap around".
>
> what do you mean? certainly there's a single interval of 6 nominals
> in decimal miracle that fails to be an approximate 2:3.

Yes, I should have explained more. If you consider the nominals
circularly then you're right of course. In fact there are 6 of them
4-0, 5-1, 6-2, 7-3, 8-4, 9-5, so in that regard it's worse for fifths
than 7 nominals in Pythagorean or Meantone. But if you only consider
them alphabetically you don't _expect_ these 6 to be good fifths,
whereas in meantone or Pythagorean, when given ABCDEFG and the fact
that A-E and C-G are fifths, there is every reason to expect that B-F
would be also.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Yes it is like the BF case, but one of those "wolves" per notation
> >is already too many.
>
> It's never been a problem for me. I've never been reading a piece
> of music and stopped and said, "oh wait, there's that BF again; it's
> not a 2:3". Just me, I guess.

Well sure. Everyone's used to it. You learn it very early. You _hear_
that it's different, then you count the black notes as well as the
white and you _see_ why it's different, and eventually the nominals
cease to even be letters of the alphabet. They even start and end at C.

I was just deliberately adopting a naive stance to point something out.

> Maybe you want a notation based on ets.

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

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

Of course. It just doesn't have an "alphabetic" wolf, for what that's
worth. Explained further in
/tuning-math/message/7113
Not important.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > > Interestingly this problem doesn't occur at all in decimal
miracle,
> > or
> > > any other notation based on a DE scale that doesn't "wrap
around".
> >
> > what do you mean? certainly there's a single interval of 6
nominals
> > in decimal miracle that fails to be an approximate 2:3.
>
> Yes, I should have explained more. If you consider the nominals
> circularly then you're right of course. In fact there are 6 of them
> 4-0, 5-1, 6-2, 7-3, 8-4, 9-5, so in that regard it's worse for
fifths
> than 7 nominals in Pythagorean or Meantone. But if you only consider
> them alphabetically you don't _expect_ these 6 to be good fifths,
> whereas in meantone or Pythagorean, when given ABCDEFG and the fact
> that A-E and C-G are fifths, there is every reason to expect that B-
F
> would be also.

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

>the linear, pythagorean diatonic scale.

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

-Carl

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

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

So far I have been satisfied with using MOS to mean a scale based on
the period+generator representation of a linear temperament which has
Myhill's property.

>> 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
>/tuning-math/message/7113

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

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >the linear, pythagorean diatonic scale.
>
> So you consider this scale a "5-limit PB" then?
>
> -Carl

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > Didn't you just say that Erv Wilson seems to have missed the
> > fractional octave cases. If so, then he couldn't exactly have
stated
> > that these are not to be considered MOS, could he?
>
> So far I have been satisfied with using MOS to mean a scale based
on
> the period+generator representation of a linear temperament which
has
> Myhill's property.

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

>> 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

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >> Now, if you want to always use the same PB for 5-limit JI,
> >> >> >> the diatonic PB may be the best choice...
> >> >> >
> >> >> >which, the just major? i disagree, of course.
> >> >>
> >> >> Of course? What would beat it out (with less than 11 tones)?
> >> >>
> >> >> -Carl
> >> >
> >> >how about the linear, pythagorean diatonic scale?
> >>
> >> Ok, I meant 5-limit PB.
> >>
> >> -Carl
> >
> >which one? (we're going around in circles)
>
> Exactly! Which one would you use to notate 5-limit JI?

I believe Paul means that he would not assign nominals to any
particular PB to notate 5-limit JI because sometimes you would want to
use the major block and sometimes the minor block and that would be
just too confusing. 7-of-Pythagorean is at least neutral in that regard.

Lets compare JI-major-based (Johnston)

D-- A- E- B- F# C# G# D# A#+ E#+ B#+
F- C- G- D- A E B F#+ C#+ G#+ D#+
Db--Ab- Eb- Bb- F C G D A+ E+ B+
Db- Ab Eb Bb F+

where + and - are 81/80 up and down, and # b are 25/24 up and down,

with Pythagorean-based (everyone else).

D_ A_ E_ B_ F#_ C#_ G#_ D#_ A#_ E#_ B#_
F\ C\ G\ D\ A\ E\ B\ F#\ C#\ G#\ D#\
Db Ab Eb Bb F C G D A E B
Db/ Ab/ Eb/ Bb/ F/

where / \ are 81/80 up and down and # b are 2187/2048 up and down.
Also = and _ are (81/80)^2 up and down. These are represented by
single symbols in Sagittal that look more like //| and \\! .

Consider this scenario:

You're sight-reading a 5-limit strict-JI piece on the violin, or any
variable pitch (or fixed-pitch 5-comma distinguishing) instrument that
can play at least two notes at once.

You see a major third on the staff, or at least it would be a normal
old 12-ET major third if you were allowed to ignore all those funny
new comma accidentals you've been forced to learn and the fact that
it's supposed to be in JI. You see that neither of the notes in this
dyad have any 5-comma accidentals. Do you play it Pythagorean or Just
or a comma narrower than just?

With Sagittal or HEWM, that's easy. You play it Pythagorean. Only if
it had a 5-comma-down accidental on the top note (or 5-comma-up on the
bottom note etc.) would you play it just.

With Johnston notation you would play it Just ... _except_ if the low
note happens to be a D (or Db or D# etc) in which case you play it a
comma lower than just.

Now major thirds are relatively well behaved in Johnston notation.
Lets consider major sevenths. In Sagittal or HEWM an 8:15 major
seventh will always have one less 5-comma against the top note than
the bottom note.

In Johnston notation a major seventh will be an 8:15 if there's no
difference in 5-commas between the two notes and the low note is an F,
C, A or E, or if there's one more comma on the top note and the low
note is a G, D or B.

Gimme a break!

Sure, you can learn this stuff eventually, but why bother?

You may be able to find some examples that make the Johnston notation
look good, but these will be pretty much confined to the 7-note JI
major scale itself. Merely notating a piece in the JI minor scale
makes things messy.

D- A E B
F C G

I believe Johnston notation (and any generalisation of it) is an
evolutionary dead-end. But I can also understand how Paul's 'The Forms
of Tonality' might have seemed to be making a case for such notations.

-- Dave Keenan

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> You seem to say miracle[10]
> >> doesn't have a wolf, but all non-equal scales of course do.
> >
> >Of course. It just doesn't have an "alphabetic" wolf, for what that's
> >worth.
>
> Sure it does -- the interval J-A is not a secor.
>
> A B C D E F G H I J A
> s s s s s s s s s n

I know, but the alphabet doesn't go J A. Whereas in the chain of
fifths case, the alphabet has exactly as many letters between B and F
as it has between A and E and between C and G.

> >Explained further in
> >/tuning-math/message/7113
>
> I have no idea what you're talking about there. Fifths or something.

Yes, fifths. It says so at the start of the message. It was in
response to something of Paul's. Perhaps the confusion is due to my
use of decimal digits for nominals (after Graham Breed).

Anyway, I said it wasn't important, and that I was adopting a
deliberately naive stance towards the circularity. Forget it.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >the linear, pythagorean diatonic scale.
> >
> > So you consider this scale a "5-limit PB" then?
> >
> > -Carl
>
> yes, for example you can enclose it with a parallelogram of unison
> vectors 81:80 and 2187:2048.

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

Perhaps that gives us a reason that will satisfy Carl, as to why, if
you are forced to choose one set of nominals as the best single set
for all purposes, it must be Pythagorean-7. Namely because it is the
only (conveniently sized) one that is a PB at every limit.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >As I seem to keep saying, no one's asking you to restrict yourself to
> >chains of fifths.
>
> But you once stated that the only generator for a notation that we
> need consider is the perfect fifth. And I'm trying to establish
> constraints on a condition that could justify such a statement.

I'm not sure what I said exactly, but my belief remains that a chain
of fifths is by far the best general purpose option. It lets you do
everything, even if the result is sometimes awkward. So you don't
"need" any others, but of course there may be scale-specific notations
that would benefit from a different set of nominals. But these
nominals would always be best if they form a distributionally even set
based on a linear chain of some interval, so there's at most one "wolf".

The only time I could see any value in applying the nominals to a more
than one dimensional periodicity block is if the scale doesn't use any
notes outside that PB.

But too many single-purpose notations is something many of us would
like to avoid. I understand that it would be enough for you if we just
had a master list of comma accidentals. That certainly is part of the
plan.

> And as I responded then, this is another example of the same
> "technology". I suppose I'll quote this here:
>
> >> and it may be worth noting that MOS with rational generators
> >> are also 1-D untempered PBs.
> >
> >True. But I don't understand the importance of this special case to
> >the discussion.

OK. I understand now, thanks to Paul.

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

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

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

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

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

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

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

OK.

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

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

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

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

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

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

> >the same ratio will always be
> >notated the same no matter what other ratios are in the scale
>
> Maybe you'd care to explain your reasoning behind this, or point
> me to where it was explained.

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

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

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

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

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

> >I'm sorry. I have no idea what you're talking about here. What does it
> >mean to "search for notations".
>
> You can search the m-limit for comma lists that enclose proper
> scales. You can rank them by the badness of the commas. If we accept
> that you want a single master notation for all of pitch space, then
> you're only interested in the top-ranking result. With a weighted
> complexity function, you may indeed get a fifth-based tuning at the
> top. Are you willing to say that's what you've done? If so, we can
> retire.

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

> >For notating ratios we didn't "search for notations" in any sense I
> >can give meaning to.
>
> That's what I'm complaining about!

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

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

Yes. It certainly does.

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

Yes. We have. We advocate only one accidental against any note
(although this is modified in the "impure" version to allow one
sagittal and one conventional). We have lots of symbols so that we can
uniquely notate lots of rationals (all the most popular ones). This is
called the olympian set. But then we describe at least two subsets of
these, with cardinality roughly halving each time, called herculean
and athenian, so that one can choose to use a smaller more learnable
set of symbols, while accepting that you cannot uniquely notate as
many rationals with it. The choice of that tradeoff is yours.

When a ratio cannot be uniquely notated with the chosen set, you
either notate it as the nearest rational that _can_ be notated, and
hope that context does the rest, or else you go to the next bigger set
of symbols.

The symbol set is manageable because each symbol is composed of 2 or 3
parts chosen from a small set. These parts themselves represent commas
so for example the right barb /| represents 80:81 and the left arc |)
represents 63:64, so the symbol with both /|) represents 35:36,
although the parts do not always add precisely like this, and there is
no way to _subtract_ commas when forming a symbol (with a small
exception).

> >Since we have a limited number of symbols and an infinite number of
> >rational pitches to notate, it makes perfect sense to me that we
> >should concentrate on notating the most popular or most commonly
> >occurring ones.
>
> Indeed. Just as with tunings, we have an infinite number of
> pitches to provide and we want to concentrate on the most popular
> ones. But we don't search the Scala archive, we use a theoretical
> principle such as comma badness.

In theory, theory and practice are the same, but in practice they're
not. :-)

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

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

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

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

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

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

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

>> Sure it does -- the interval J-A is not a secor.
>>
>> A B C D E F G H I J A
>> s s s s s s s s s n
>
>I know, but the alphabet doesn't go J A.

This alphabet does! J-A is a 2nd, just like I-J.

-Carl

>> > >the linear, pythagorean diatonic scale.
>> >
>> > So you consider this scale a "5-limit PB" then?
//
>> yes, for example you can enclose it with a parallelogram of unison
>> vectors 81:80 and 2187:2048.
//
>Perhaps that gives us a reason that will satisfy Carl, as to why, if
>you are forced to choose one set of nominals as the best single set
>for all purposes, it must be Pythagorean-7. Namely because it is the
>only (conveniently sized) one that is a PB at every limit.

Care to flesh that a bit?

-C.

>Consider this scenario:
>
>You're sight-reading a 5-limit strict-JI piece on the violin, or any
>variable pitch (or fixed-pitch 5-comma distinguishing) instrument that
>can play at least two notes at once.
>
>You see a major third on the staff, or at least it would be a normal
>old 12-ET major third if you were allowed to ignore all those funny
>new comma accidentals you've been forced to learn and the fact that
>it's supposed to be in JI. You see that neither of the notes in this
>dyad have any 5-comma accidentals. Do you play it Pythagorean or Just
>or a comma narrower than just?

This doesn't strike me as a plausible thought process for a performer.
You've got to know the scale, A1 priority. If you don't, you're
screwed. Applying offsets from that is a natural way to think.
Thinking in terms of the set of all just intervals doesn't strike me
as the way it would like to go.

>With Johnston notation you would play it Just ... _except_ if the low
>note happens to be a D (or Db or D# etc) in which case you play it a
>comma lower than just.

The thing you're glossing over is that by using Johnston notation,
the composer is requesting the 5-limit diatonic scale. So in Sagittal
you've got lots of accidentals on basic scale members, which are
superfluous from this point of view.

>You may be able to find some examples that make the Johnston notation
>look good, but these will be pretty much confined to the 7-note JI
>major scale itself.

Probably the bulk of music stays in the "scale itself", in the case
of pythagorean and meantone.

-Carl

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > 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.

i think the basis of a notation is how a tuning conforms to
the L,s concept:

http://sonic-arts.org/dict/ls.htm

the L (large) step determines the nominals, and the s (small)
step determines the main pair of accidentals.

other accidentals may be added to indicate other commatic
inflections.

so i'm not so sure that i can agree with you here, Dave.

if someone is experienced with working a Pythagorean tuning,
he/she certainly knows what 2187/2048 *sounds* like, even if
not familiar with the mathematical description of it.

i think the ancient were on to something here that has been
rather lost among our modern theory and experience, in that
the chromatic semitone is a transition from one type of genus
to another.

if someone is performing a piece in Pythagorean tuning, and
it has been going along diatonically, then all of a sudden
there is an accidental indicatinag a chromatic change, that
comma (2187/2048) will have an *extremely* distinctive sound.

labeling that distinctive sounds makes perfect sense ... as
much sense as using different nominals to name the diatonic
steps.

>
> 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).

hmmm ... can anyone comment on how consistency of notation
is similar to or different from consistency in the erlichian
sense?

what i'm getting at is this: is it possible to come up with
a notation for a scale which is inconsistent, such that it
eliminates the consistency problem for that tuning?

-monz

>hmmm ... can anyone comment on how consistency of notation
>is similar to or different from consistency in the erlichian
>sense?

I'll guess you'll have to define what you mean by
"consistency in notation".

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> > 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.

and me, of course.

Carl, i suggest that you re-read not only my Dictionary
page on HEWM:

http://sonic-arts.org/dict/hewm.htm

but also the one on Johnston notation:

http://sonic-arts.org/dict/johnston.htm

the big problem with Johnston notation is that it requires
the user to keep a 2-dimensional reference scale in mind,
which complicates things far more than necessary.

if one only need keep a 1-dimensional chain of notes in
mind as either his set of nominals (i.e., the pythagorean
diatonic scale) or his nominals-plus-accidentals (setting
an arbitrary limit somewhat based on historical usage,
the 35-tone pythagorean chain stretching from Fbb to Bx),
matters are greatly simplified.

(they would be simply even further if all nominals could
be used for the entire chain, thus giving an absolute
consistency to the meaning of the notational symbols.)

so anyway, HEWM and sagittal have a pythagorean basis
for this reason, and also because it reflects historical
usage. indeed, if they were to ignore historical usage,
they *would* employ a larger nominal set to represent a
12-tone pythagorean chain rather than a 7-tone chain,
since the pythagorean comma (3^12) is a less-perceptible
change in tuning than a 2187/2048 chromatic semitone (3^7).

by forcing the user to remember a specific 2-dimensional
set of notes which has significant exponents of both
3 and 5, Johnston notation is much harder to keep in mind,
and also results in the kinds of sonic/notational
inconsistencies pointed out by paul with his examples
of "perfect-5ths".

in HEWM and sagittal, there is one pair of symbols which
represents respectively a sharpening or flattening of any
pythagorean note by a syntonic comma. that system is
much easier to remember conceptually.

-monz

-monz

--- In tuning-math@yahoogroups.com, 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?

of course it would! ... if one wanted to write a piece
in which a 7-tone chain-of-minor-3rds was a prominent
feature!

in fact, a notation based on that would be a good notation
for the diatonic subset of 19edo, if i'm not mistaken
(but i might be ... this is just a passing thought).

>
> >> 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?

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

-monz

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > > Yes it is like the BF case, but one of those "wolves"
> > > per notation is already too many.
> >
> > It's never been a problem for me. I've never been reading
> > a piece of music and stopped and said, "oh wait, there's
> > that BF again; it's not a 2:3". Just me, I guess.
>
> Well sure. Everyone's used to it. You learn it very early.
> You _hear_ that it's different, then you count the black notes
> as well as the white and you _see_ why it's different, and
> eventually the nominals cease to even be letters of the
> alphabet. They even start and end at C.

thanks, Dave, for taking us back to kindergarten music-theory.
i mean that sincerely.

i have had to struggle to figure out how to teach this stuff
to my young students in a way that leaves them curious to
learn more about tuning.

i pretty much use exactly the approach you describe, counting
first the letter-names to determine the name of the interval
("2nd", "3rd", etc.), then counting "half-steps" to determine
from a chart i give them whether intervals of a certain
name are "perfect", "major", "minor", "augmented", or
"diminished".

... then later, when i think they're ready for it, i let
them in on the fact that "back in the old days" composers
wanted sharps to be different from flats, etc. etc.
and by using their previous knowledge of how a certain
number of "half-steps" in an interval determined its quality,
then can extrapolate that now a certain number of commas
differentiating intervals of the same name and quality
indicates their prime-limit.

i'm getting them on our side early. ;-)

> I was just deliberately adopting a naive stance to point
> something out.

i think it worked. :)

-monz

>the big problem with Johnston notation is that it requires
>the user to keep a 2-dimensional reference scale in mind,
>which complicates things far more than necessary.

That seems to be what Dave and Paul are saying. But it's
a fallacy that you can simplify music with notation. If
the music really features a 2-D scale, the best notation
is optimized for that scale. If the music wasn't featuring
it, you wouldn't use such a notation, unless it happened
to be some sort of standard. Making up standards is a
little like putting the cart before the horse if you ask me,
considering all the worthwhile extended-JI music ever made
fits on a few cds.

On the other hand, it's nice to have an engineered standard
available, if someone wants one.

But the idea of pushing for the adoption of a standard now,
simply to prevent a non-engineered (evolved) one from later
taking hold, is a very bad idea. Evolved standards may not
always be optimal, but it's very hard to define optimal in
advance. Evolved standards have the advantage that they
beat competing approaches while the criteria for optimality
were still up for grabs. It's important for people in a
field to reach out and try different things, especially at
first.

-Carl

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> >
> > 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).
>
>
> hmmm ... can anyone comment on how consistency of notation
> is similar to or different from consistency in the erlichian
> sense?
>
> what i'm getting at is this: is it possible to come up with
> a notation for a scale which is inconsistent, such that it
> eliminates the consistency problem for that tuning?

i decided to re-word that last paragraph for better clarity:

is it possible to come up with a consistent notation for an
inconsistent tuning, such that it eliminates the consistency
problem for scales in that tuning?

-monz

>The only time I could see any value in applying the nominals to a more
>than one dimensional periodicity block is if the scale doesn't use any
>notes outside that PB.

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

>> You brought up the Miller limit but you haven't said why you think a
>> notation search should be any different from a PB search, or how PB
>> searches have been done wrong so far.
>
>OK. Well I think you know by now (from previous messages), why I am
>not interested in PBs for notation at all, except for one dimensional
>PBs in the 3-limit. :-) It has nothing to do with weighting any
>complexities, but simply minimising the number of wolves and making
>the generator as simple as possible.

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

>Lets compare JI-major-based (Johnston)
>
>D-- A- E- B- F# C# G# D# A#+ E#+ B#+
> F- C- G- D- A E B F#+ C#+ G#+ D#+
>Db--Ab- Eb- Bb- F C G D A+ E+ B+
> Db- Ab Eb Bb F+
>
>where + and - are 81/80 up and down, and # b are 25/24 up and down,
>
>with Pythagorean-based (everyone else).
>
>D_ A_ E_ B_ F#_ C#_ G#_ D#_ A#_ E#_ B#_
> F\ C\ G\ D\ A\ E\ B\ F#\ C#\ G#\ D#\
>Db Ab Eb Bb F C G D A E B
> Db/ Ab/ Eb/ Bb/ F/
>
>where / \ are 81/80 up and down and # b are 2187/2048 up and down.
>Also = and _ are (81/80)^2 up and down. These are represented by
>single symbols in Sagittal that look more like //| and \\! .

...where are they here? You're saying that because D-A isn't a
3:2, it's a wolf? I could just as easily say C-E is a wolf
because it isn't a 5:4.

I'm not arguing that the pythagorean approach isn't better here
(Johnston just picked a bad scale) but I'm skeptical of any general
principle at work, other than:

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

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

>> True but it doesn't make what I said wrong. Good PBs and good
>> temperaments are what composers like best.
>
>Assuming that to be true, it still doesn't imply that it is a good
>idea to apply the nominals to a more-than-1D PB which is a proper
>subset of the PB being used.

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

>OK. But composers have used a lot of things that bear little
>reseblance to anything as ordered and regular as a PB or temperament
>thereof.

Such as?

>> Usually a / just means or. Maybe I should have used the word "or".
>> You asked what a "PB/temperament" was, so I gave you a definition.
>> If you don't like the definition, go back to using "or".
>
>"PB or temperament" still doesn't make sense to me. A temperament may
>cover an infinite number of pitches where a PB is finite. I still
>think you mean "PB or tempering thereof".

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

>> if you've really figured out a way to avoid pile-up in strict JI,
>> I'd like to hear it.
>
>Yes. We have. We advocate only one accidental against any note
//
>We have lots of symbols so that we can uniquely notate lots of
>rationals (all the most popular ones).

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

>When a ratio cannot be uniquely notated with the chosen set, you
>either notate it as the nearest rational that _can_ be notated, and
>hope that context does the rest, or else you go to the next bigger
>set of symbols.

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

>In any case, you've given an analogy not an equivalence. Shouldn't we
>avail ourselves of empirical data if we have it, when deciding whether
>to trust the theory.

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

I'm not aware of any empirical data on this whatever. You're well
aware of the state of ethnomusicology, and the picture in the West...
singularly standardized. The whole alternate tunings movement is
possible only by generalizing existing music theory.

>> Keeping track of extended JI by ratio is a nightmare, at least for
>> me. Something like monzo notation seems a minimum, unless you
>> restrict yourself to a cross-set, diamond, harmonic series or other
>> fixed structure.
>
>Yes sagittal restricts the uniquely notated ratios to one of a few
>fixed structures, but they are very large ones and we have not
>determined their boundaries as such.

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

-Carl

hi Carl,

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >the big problem with Johnston notation is that it requires
> >the user to keep a 2-dimensional reference scale in mind,
> >which complicates things far more than necessary.
>
> That seems to be what Dave and Paul are saying. But it's
> a fallacy that you can simplify music with notation. If
> the music really features a 2-D scale, the best notation
> is optimized for that scale.

i'm sorry, but i disagree -- at least in the case of
complex music which uses a larger-than-diatonic pitch set,
which BTW is exactly the kind of music Johnston and his
many students (all of whom use his notation) compose.

for 5-limit JI music which only uses a diatonic pitch-set,
sure, A B C D E F G is fine.

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

for example, here's a portion of the Johnston lattice:
(view in "Expand Messages" mode on the Yahoo website)

F# .. C# .. G# .. D#
/ \ / \ / \ / \
/ \ / \ / \ / \
D-... A ... E ... B ... F#+
\ / \ / \ / \ / \
\ / \ / \ / \ / \
F ... C ... G ... D ... A+

this subset is the diatonic C-major scale, upon which
the notation is based:

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

notes which have a sharp (#) are a 24:25 higher than
those 7 nominals.

simply modulating from C-major to its *nearest* relative,
G-major, invokes this monstrosity:

E ... B ... F#+
/ \ / \ / \
/ \ / \ / \
C ... G ... D ... A+

yuck.

in my HEWM, they are simply:

C-major:

A-... E-... B-
/ \ / \ / \
/ \ / \ / \
F ... C ... G ... D

G-major:

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

entire example lattice:

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

simple.

-monz

>i decided to re-word that last paragraph for better clarity:
>
>is it possible to come up with a consistent notation for an
>inconsistent tuning, such that it eliminates the consistency
>problem for scales in that tuning?

What's a consistent notation?

If you think like Gene, there's no problem, because you don't
think of tunings that aren't consistent in the first place.
I must admit I'm rather fond of this approach, though jumping
between approximations inside an et could also be fun.

-Carl

>but try to modulate into almost any other key, and you'll
>start having problems.
//
>for example,
//
>simply modulating from C-major to its *nearest* relative,
>G-major, invokes this monstrosity:
>
> E ... B ... F#+
> / \ / \ / \
> / \ / \ / \
> C ... G ... D ... A+
//
>in my HEWM, they are simply:
>
>C-major:

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

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

How is this supposed to be better than the above?

-Carl

>> >> >> If so, there's no
>> >> >> reduction in the number of new and strange accidentals.
>> >> >
>> >> >right, but their usage is more straightforward.
>> >>
>> >> Only because the Pythagorean scale is actually a decent
>> >> temperament of the 5-limit diatonic scale.
>> >
>> >no, it's because of the linearity (1-dimensionality) of the set
>> >of nominals.
>>
>> Then one could just as well use a 7-tone chain-of-minor-thirds.
>> But I don't think that would work, do you?
>
>of course it would! ... if one wanted to write a piece
>in which a 7-tone chain-of-minor-3rds was a prominent
>feature!

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

>in fact, a notation based on that would be a good notation
>for the diatonic subset of 19edo, if i'm not mistaken
>(but i might be ... this is just a passing thought).

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

In kleismic...

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

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

Whaddya think?

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

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

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

-Carl

I meant to add...

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

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

-C.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> if one only need keep a 1-dimensional chain of notes in
> mind as either his set of nominals (i.e., the pythagorean
> diatonic scale) or his nominals-plus-accidentals (setting
> an arbitrary limit somewhat based on historical usage,
> the 35-tone pythagorean chain stretching from Fbb to Bx),
> matters are greatly simplified.
>
> (they would be simply even further if all nominals could
> be used for the entire chain, thus giving an absolute
> consistency to the meaning of the notational symbols.)
>
>
> so anyway, HEWM and sagittal have a pythagorean basis
> for this reason, and also because it reflects historical
> usage. indeed, if they were to ignore historical usage,
> they *would* employ a larger nominal set to represent a
> 12-tone pythagorean chain rather than a 7-tone chain,
> since the pythagorean comma (3^12) is a less-perceptible
> change in tuning than a 2187/2048 chromatic semitone (3^7).

as an example, one could notate either 12edo or
a 12-tone pythagorean chain nicely with the nominals
A...L, and put it on my 12edo-staff:

(the numerals at the left are either 12edo degrees
or degrees of the pythagorean scale, and can be considered
as a kind of clef)

-0- --A--
11 L
10 ----------------------------------K----------------
9 J
8 ----------------------------I----------------------
7 H
6 -----------------------G---------------------------
5 F
4 ------------------E--------------------------------
3 D
2 ------------C--------------------------------------
1 B
-0- --A--

in this convention, if the bottom --A-- is "middle-C",
then the set of diatonic triads in C-major is:
(read vertically for each triad)

-0-
11 L L L
10 --------------------------------------------------
9 J J J
8 --------------------------------------------------
7 H H H
6 --------------------------------------------------
5 F F F
4 -----E------------E-----------------E-------------
3
2 ------------C------------------C-----------C------
1
-0- --A-- --A-- --A--

I ii iii IV V vi vii

i know that looks strange, but you're already very familiar
with a 12-tone scale where all steps are more-or-less equal,
and i'd bet that just a little use of this system would
make you feel quite comfortable working within it for those
tunings.

and anyway, i only chose the diatonic triads for their
familiarity -- actually (as you're aware) they don't
illustrate anything about a 12-tone notation particularly
well, being, rather, eminently suited to illustrate a
notation based on 7 nominals.

... but even there, standard Western staff-notation comes
up short because it doesn't represent 8ve-equivalence.
that could be done thus:

steps

-0- --A--
7 -----------------------G--------
6 F
4 -----------------E--------------
3 D
2 -----------C--------------------
1 B
-0- --A-- --A--

7 -----------------------G--------
6 F
4 -----------------E--------------
3 D
2 -----------C--------------------
1 B
-0- --A-- --A--

7 -----------------------G--------
6 F
4 -----------------E--------------
3 D
2 -----------C--------------------
1 B
-0- --A--

of course, because this scale has an odd-number of
nominals the replication of the 8ve becomes awkward,
having the line for 7-steps directly under the line
for 0-steps without any room (or necessity) for an
intervening space.

still, i like visual representation of 8ve-equivalence.

-monz

oops ...

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

ack! the numbering on the left is wrong -- i accidentally
left out "5" and ended with "6" and "7" rather than the
correct "5" and "6".

i've redone it and also added an example of the diatonic
triads in C-major, here:

http://sonic-arts.org/dict/7-nominal-staff.htm

-monz

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > So far I have been satisfied with using MOS to mean a scale based
> on
> > the period+generator representation of a linear temperament which
> has
> > Myhill's property.
>
> the the period is not one octave, the scale doesn't have myhill's
> property.

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

hi Carl,

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >but try to modulate into almost any other key, and you'll
> >start having problems.
> //
> >for example,
> //
> >simply modulating from C-major to its *nearest* relative,
> >G-major, invokes this monstrosity:
> >
> > E ... B ... F#+
> > / \ / \ / \
> > / \ / \ / \
> > C ... G ... D ... A+
> //
> >in my HEWM, they are simply:
> >
> >C-major:
>
> [Involves three non-standard accidentals
> vs. Johnston notation.]
>
> >G-major:
> >
> > E-... B-... F#-
> > / \ / \ / \
> > / \ / \ / \
> > C ... G ... D ... A
>
> How is this supposed to be better than the above?

simple:

() any pair of plain letters which are adjacent
on the horizontal axis are always 2:3 or 3:4 ratios

() any pair of letters with minus signs which are
adjacent on the horizontal axis are always 2:3 or 3:4 ratios

() any interval with one plain letter and one letter-with-minus
which are connected on the lattice, is always a
3:5, 4:5, 5:6, or 5:8 ratio

in Johnston notation, you always have to remember that
most pairs of plain letters are 2:3 or 3:4, but that
D:A is a 20:27 or 27:40 and D:A+ is the 2:3 or 3:4.
same deal with the 5-limit ratios.

that's needlessly confusing, and for what? simply to
emphasize the JI periodicity-block as the basis?

i'll take a simple 1-dimensional pythagorean chain for
my nominals, and indicate all other primes with accidentals
... which is exactly what Johnston does for primes higher
than 5 anyway.

-monz

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> That seems to be what Dave and Paul are saying. But it's
> a fallacy that you can simplify music with notation. If
> the music really features a 2-D scale, the best notation
> is optimized for that scale. If the music wasn't featuring
> it, you wouldn't use such a notation, unless it happened
> to be some sort of standard. Making up standards is a
> little like putting the cart before the horse if you ask me,
> considering all the worthwhile extended-JI music ever made
> fits on a few cds.

johnston's notation is presented as a standard for *all* strict-JI
music up to the 31-limit or something like that, *regardless* of what
scales may or may not be implied in the music. i think that daniel
wolf's modification of this standard (and hence hewm or sagittal) is
a clear improvement.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Even if we perversely make a
> scale with one ratio from each prime limit up to 23, it's still a
>PB.

show me.

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

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

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >i decided to re-word that last paragraph for better clarity:
> >
> >is it possible to come up with a consistent notation for an
> >inconsistent tuning, such that it eliminates the consistency
> >problem for scales in that tuning?
>
> What's a consistent notation?
>
> If you think like Gene, there's no problem, because you don't
> think of tunings that aren't consistent in the first place.
> I must admit I'm rather fond of this approach, though jumping
> between approximations inside an et could also be fun.
>
> -Carl

as herman miller showed with 64-equal, to monz's delight.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >but try to modulate into almost any other key, and you'll
> >start having problems.
> //
> >for example,
> //
> >simply modulating from C-major to its *nearest* relative,
> >G-major, invokes this monstrosity:
> >
> > E ... B ... F#+
> > / \ / \ / \
> > / \ / \ / \
> > C ... G ... D ... A+
> //
> >in my HEWM, they are simply:
> >
> >C-major:
>
> [Involves three non-standard accidentals
> vs. Johnston notation.]
>
> >G-major:
> >
> > E-... B-... F#-
> > / \ / \ / \
> > / \ / \ / \
> > C ... G ... D ... A
>
> How is this supposed to be better than the above?
>
> -Carl

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > So far I have been satisfied with using MOS to mean a scale
based
> > on
> > > the period+generator representation of a linear temperament
which
> > has
> > > Myhill's property.
> >
> > the the period is not one octave, the scale doesn't have myhill's
> > property.
>
> Of course it does; forget the damned octave equivalence, it isn't
> relevant.

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > So far I have been satisfied with using MOS to mean a scale
based
> > on
> > > the period+generator representation of a linear temperament
which
> > has
> > > Myhill's property.
> >
> > the the period is not one octave, the scale doesn't have myhill's
> > property.
>
> Of course it does; forget the damned octave equivalence, it isn't
> relevant.

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

>> That seems to be what Dave and Paul are saying. But it's
>> a fallacy that you can simplify music with notation. If
>> the music really features a 2-D scale, the best notation
>> is optimized for that scale. If the music wasn't featuring
>> it, you wouldn't use such a notation, unless it happened
>> to be some sort of standard. Making up standards is a
>> little like putting the cart before the horse if you ask me,
>> considering all the worthwhile extended-JI music ever made
>> fits on a few cds.
>
>johnston's notation is presented as a standard for *all* strict-JI
>music up to the 31-limit or something like that, *regardless* of what
>scales may or may not be implied in the music.

Yuck.

-Carl

>> Even if we perversely make a
>> scale with one ratio from each prime limit up to 23, it's still a
>> PB.
>
>show me.

Can we not take 3/2 and make 9:8 a uv, 5/4 and make 25:16 a uv,
7/4 and make 28:16 a uv, etc.?

-Carl

>> If you think like Gene, there's no problem, because you don't
>> think of tunings that aren't consistent in the first place.
>> I must admit I'm rather fond of this approach, though jumping
>> between approximations inside an et could also be fun.
>>
>> -Carl
>
>as herman miller showed with 64-equal, to monz's delight.

In what piece? A warped canon?

-Carl

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

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

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> If you think like Gene, there's no problem, because you don't
> >> think of tunings that aren't consistent in the first place.
> >> I must admit I'm rather fond of this approach, though jumping
> >> between approximations inside an et could also be fun.
> >>
> >> -Carl
> >
> >as herman miller showed with 64-equal, to monz's delight.
>
> In what piece? A warped canon?
>
> -Carl

in his "Pavane for a Warped Princess" ... his retunings
of a famous piece by Ravel.

http://www.io.com/~hmiller/music/pavane.html

my favorite is the 64edo version, despite the fact that it's
inconsistent. as paul just said earlier, "the ear is the
final arbiter".

paul, your memory is really incredible. just yesterday,
i was referring Chris (my business partner) to this page,
and i remembered that 40edo was my second-favorite version
but had trouble remembering if 64edo or another tuning was
my favorite. thanks! :)

-monz

>> >> If you think like Gene, there's no problem, because you don't
>> >> think of tunings that aren't consistent in the first place.
>> >> I must admit I'm rather fond of this approach, though jumping
>> >> between approximations inside an et could also be fun.
>> >
>> >as herman miller showed with 64-equal, to monz's delight.
>>
>> In what piece? A warped canon?
>>
>> -Carl
>
>in his "Pavane for a Warped Princess" ... his retunings
>of a famous piece by Ravel.
>
>http://www.io.com/~hmiller/music/pavane.html
>
>my favorite is the 64edo version, despite the fact that it's
>inconsistent. as paul just said earlier, "the ear is the
>final arbiter".

Yes, but is Herman's technique really going to vacillate between
approximations?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Even if we perversely make a
> >> scale with one ratio from each prime limit up to 23, it's still
a
> >> PB.
> >
> >show me.
>
> Can we not take 3/2 and make 9:8 a uv, 5/4 and make 25:16 a uv,
> 7/4 and make 28:16 a uv, etc.?
>
> -Carl

28:16 = 7/4. you mean 49:32? anyway, it sounds like you're suggesting
a fokker matrix with 2 along the diagonal and 0 everywhere else. the
determinants are:

3-limit: 2
5-limit: 4
7-limit: 8

and so on . . . you're creating an euler-fokker genus in each case,
not a scale with one ratio from each prime limit . . . but anyway
what kind of unison vectors are these? notationally you want 'good'
ones where the notes are in the correct order, i.e., small unison
vectors.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> If you think like Gene, there's no problem, because you don't
> > >> think of tunings that aren't consistent in the first place.
> > >> I must admit I'm rather fond of this approach, though jumping
> > >> between approximations inside an et could also be fun.
> > >>
> > >> -Carl
> > >
> > >as herman miller showed with 64-equal, to monz's delight.
> >
> > In what piece? A warped canon?
> >
> > -Carl
>
>
>
> in his "Pavane for a Warped Princess" ... his retunings
> of a famous piece by Ravel.
>
> http://www.io.com/~hmiller/music/pavane.html
>
>
> my favorite is the 64edo version, despite the fact that it's
> inconsistent.

which meaning of "inconsistent" are you referring to here -- the fact
that the major triads are not all constructed the same way, or the
technical meaning as on your dictionary?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> If you think like Gene, there's no problem, because you don't
> >> >> think of tunings that aren't consistent in the first place.
> >> >> I must admit I'm rather fond of this approach, though jumping
> >> >> between approximations inside an et could also be fun.
> >> >
> >> >as herman miller showed with 64-equal, to monz's delight.
> >>
> >> In what piece? A warped canon?
> >>
> >> -Carl
> >
> >in his "Pavane for a Warped Princess" ... his retunings
> >of a famous piece by Ravel.
> >
> >http://www.io.com/~hmiller/music/pavane.html
> >
> >my favorite is the 64edo version, despite the fact that it's
> >inconsistent. as paul just said earlier, "the ear is the
> >final arbiter".
>
> Yes, but is Herman's technique really going to vacillate between
> approximations?

yes, he used a fixed 12-tone subset of 64-equal, so different triads
are constructed differently.

>> Yes, but is Herman's technique really going to vacillate between
>> approximations?
>
>yes, he used a fixed 12-tone subset of 64-equal, so different triads
>are constructed differently.

Fascinatin'.

-Carl

>> Can we not take 3/2 and make 9:8 a uv, 5/4 and make 25:16 a uv,
>> 7/4 and make 28:16 a uv, etc.?
>>
>> -Carl
>
>28:16 = 7/4. you mean 49:32?

Yes.

>anyway, it sounds like you're suggesting
>a fokker matrix with 2 along the diagonal and 0 everywhere else. the
>determinants are:
>
>3-limit: 2
>5-limit: 4
>7-limit: 8
>
>and so on . . . you're creating an euler-fokker genus in each case,

Crud, you're right.

But also, I suppose any non-convex structure would be non-PB.
Are there any such structures we couldn't live without?

>not a scale with one ratio from each prime limit . . . but anyway
>what kind of unison vectors are these?

"perverse" ones.

-Carl

hi Dave,

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

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

i got a real lavgh from it at first, on the main tvning list.
bvt yes, now it *is* annoying.

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

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

and in my case, it really shouldn't be too much of a problem,
because i also generally make each paragraph only one sentence
long.

i decided awhile back that lots of white space helped to
make my messages clearer.

-monz

Sorry folks,

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

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >The only time I could see any value in applying the nominals to a more
> >than one dimensional periodicity block is if the scale doesn't use any
> >notes outside that PB.
>
> By definition the scale doesn't. You mean the music?

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

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

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

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

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

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

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

> it's still a PB.

But that fact isn't very significant.

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

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

> Making the generator as simple as possible tends to happen when you
> weight the complexity. Minimizing the number of wolves sounds
> interesting. But what's a wolf?...
>
> >Lets compare JI-major-based (Johnston)
> >
> >D-- A- E- B- F# C# G# D# A#+ E#+ B#+
> > F- C- G- D- A E B F#+ C#+ G#+ D#+
> >Db--Ab- Eb- Bb- F C G D A+ E+ B+
> > Db- Ab Eb Bb F+
> >
> >where + and - are 81/80 up and down, and # b are 25/24 up and down,
> >
> >with Pythagorean-based (everyone else).
> >
> >D_ A_ E_ B_ F#_ C#_ G#_ D#_ A#_ E#_ B#_
> > F\ C\ G\ D\ A\ E\ B\ F#\ C#\ G#\ D#\
> >Db Ab Eb Bb F C G D A E B
> > Db/ Ab/ Eb/ Bb/ F/
> >
> >where / \ are 81/80 up and down and # b are 2187/2048 up and down.
> >Also = and _ are (81/80)^2 up and down. These are represented by
> >single symbols in Sagittal that look more like //| and \\! .
>
> ...where are they here? You're saying that because D-A isn't a
> 3:2, it's a wolf?

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

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

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

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

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

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

Skepticism is the right approach ...

... believe me.

:-)

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

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

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

No argument here.

> >> True but it doesn't make what I said wrong. Good PBs and good
> >> temperaments are what composers like best.
> >
> >Assuming that to be true, it still doesn't imply that it is a good
> >idea to apply the nominals to a more-than-1D PB which is a proper
> >subset of the PB being used.
>
> I think what this really means is, it's not a good idea to write
> music with more than a 1D PB in mind.

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

> >OK. But composers have used a lot of things that bear little
> >reseblance to anything as ordered and regular as a PB or temperament
> >thereof.
>
> Such as?

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

> >> Usually a / just means or. Maybe I should have used the word "or".
> >> You asked what a "PB/temperament" was, so I gave you a definition.
> >> If you don't like the definition, go back to using "or".
> >
> >"PB or temperament" still doesn't make sense to me. A temperament may
> >cover an infinite number of pitches where a PB is finite. I still
> >think you mean "PB or tempering thereof".
>
> You're not likely to ever hear me discuss something with an infinite
> number of notes.

Have you never used the word "miracle" to refer to the abstract entity
to which various tunings such as blackjack and canasta belong?

Similarly, is meantone always 12 notes per octave for you. If you were
looking at a split-key organ in a museum and someone asked you what
temperament it was in, would you decline to say "meantone"?

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

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

> >> if you've really figured out a way to avoid pile-up in strict JI,
> >> I'd like to hear it.
> >
> >Yes. We have. We advocate only one accidental against any note
> //
> >We have lots of symbols so that we can uniquely notate lots of
> >rationals (all the most popular ones).
>
> Does that include the compound commas like (80:81)^3 you'll need
> to back up that affirmative?

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

Also the symbol in question is, in ASCII longhand, (|\ which combines
the flags for the 77-comma 45056:45927 and the 55-comma 54:55. The
35-large-diesis 8192:8505 happens to be the exact sum(product) of these.

> >When a ratio cannot be uniquely notated with the chosen set, you
> >either notate it as the nearest rational that _can_ be notated, and
> >hope that context does the rest, or else you go to the next bigger
> >set of symbols.
>
> So it's not JI anymore,

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

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

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

> >In any case, you've given an analogy not an equivalence. Shouldn't we
> >avail ourselves of empirical data if we have it, when deciding whether
> >to trust the theory.
>
> Yes, but consider the source. Most of those scales have never been
> composed in.

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

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

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

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

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

> >> Keeping track of extended JI by ratio is a nightmare, at least for
> >> me. Something like monzo notation seems a minimum, unless you
> >> restrict yourself to a cross-set, diamond, harmonic series or other
> >> fixed structure.
> >
> >Yes sagittal restricts the uniquely notated ratios to one of a few
> >fixed structures, but they are very large ones and we have not
> >determined their boundaries as such.
>
> It'll be cool to see these. Consider me pumped.

But I guess you are already disapointed to learn that it doesn't stack
more than two 5-commas precisely. However 0.4 cents is pretty damn
close, and the context will usually eliminate this tiny ambiguity
entirely.

In fact, just the single note to which the accidental applies may be
sufficient context. We envisage that even a dumb automatic player
could get it right most of the time as to whether the (|\ was
representing a 35-comma or a 125-comma or a 13-comma (26:27 65.3 c)
merely by looking at what nominal and sharps or flats it was
associated with. But we'd better leave _something_ for others to
figure out with sagittal. :-)

You can sort of see the crudest of the JI symbol sets already in
Scala. SET NOTATION SAJI1, then choose your scale and SHOW it or look
at it on the Chromatic Clavier. The only problem is that the choice of
nominals and sharps or flats is screwed, and therefore the best comma
isn't necessarily chosen. We'll eventually get back to working with
Manuel to fix these. But the Xenharmonicon-18 deadline is looming and
I really shouldn't spend any more time on this list. I'm glad we
worked out what our differences really were.

-- Dave Keenan

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> i can't speak for paul, but in my own case, hey, i just
> decided that capitalization really sucks and i don't want
> to use it if i don't feel it's really necessary.

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

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

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

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

No worries.

hi Dave,

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

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

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

-monz

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

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

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

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> But the Xenharmonicon-18 deadline is looming

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

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > But the Xenharmonicon-18 deadline is looming
>
> it is? last i heard from manuel, it wasn't. someone please e-mail
me
> with the latest info.

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

>>> Therefore, the master
>>> list should be based on a search for simple and small commas.
>>> Conveniently, such searches have been done at least through the 7-
>>> limit, with various flavors of complexity functions, etc.,
>>
>>Well send me the list when you conveniently get up to 23 limit.
>
>Ok, I'll work on that. I've been meaning to write a comma-searcher
>for a while now. It might take me a while more.

Here are the 10 lowest-badness ratios among all ratios in lowest
terms between 0 and 600 cents with denominator not greater than 500,
where badness is defined as log_2(ratio)^2 * prime-limit(ratio)...

((3.3295737438398616 3 256/243)
(3.796875 3 9/8)
(4.2139917695473255 3 32/27)
(4.805419921875 3 81/64)
(5.12578125 5 81/80)
(5.24288 5 128/125)
(5.292214940134465 5 250/243)
(5.333333333333333 3 4/3)
(5.425347222222222 5 25/24)
(5.56182861328125 5 135/128))

...each triple is (badness, prime-limit, ratio). The search took
less than 10 minutes on a P3 600 laptop (code available). Performance
would be drastically better by using anything other than the slowest
conceivable factoring algorithm, which I chose for expediency.

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

-Carl

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

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

Using ^3 and log_2(n*d), the results are...

(badness log_2(n*d) ratio)
>((8.44126581032688 4.321928094887363 5/4)
> (8.47910694921152 4.906890595608519 6/5)
> (8.497688890598296 3.5849625007211565 4/3)
> (8.56280035191257 5.392317422778761 7/6)
> (8.668704723304646 5.807354922057605 8/7)
> (8.78491274619423 6.169925001442312 9/8)
> (8.90514828028762 6.491853096329675 10/9)
> (9.025989778701321 6.78135971352466 11/10)
> (9.145539472765897 7.044394119358453 12/11)
> (9.2627480757178 7.285402218862249 13/12))

Paul, any thoughts on a badness heuristic

log(d) * |n-d|/log(d) = |n-d|

?

Thanks,

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

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

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

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > where badness is defined as log_2(ratio)^2 * prime-limit(ratio)...
>
> Any comments on this badness measure?

You haven't defined it yet.

>> > where badness is defined as log_2(ratio)^2 * prime-limit(ratio)...
>>
>> Any comments on this badness measure?
>
>You haven't defined it yet.

What's lacking in the above definition?

-Carl

>> Since every prime limit contains an infinite number of ratios, and
>> neither size nor complexity behave smoothly as one searches farther
>> out, it seems we'll never know the top 10 lowest-badness ratios at
>> any prime limit....
>
>For any limit, zero will be an accumulation point of log2(q)^2, since
>p-limit commas are arbitrarily small; but this hardly matters, since
>whatever it is you are calculating, it clearly isn't log2(q)^2
>primelimit(q). Can we start over?

Whoops, I wasn't actually taking the log2 of q. The formula used
was...

q^2 * primelimit(q)

Probably I should use (log2(q) + 1)^2 * primelimit(q). In this case
the 10 lowest-scoring ratios <= 600 cents with denominator <= 500
are...

(badness, primelimit, ratio)
((3.468084457207407 3 256/243)
(4.106173526999384 3 9/8)
(4.6509153968061785 3 32/27)
(5.180825053903934 5 81/80)
(5.348010729259556 5 128/125)
(5.385594090689787 3 81/64)
(5.418111244777691 5 250/243)
(5.6062792235873795 5 25/24)
(5.797659150259687 5 135/128)
(5.974440849845497 5 16/15))

-Carl

I wrote...

>...each triple is (badness, prime-limit, ratio). The search took
>less than 10 minutes on a P3 600 laptop (code available). Performance
>would be drastically better by using anything other than the slowest
>conceivable factoring algorithm, which I chose for expediency.

I measured the slowdown due to factoring by comparing the prime-limit complexity mode with the n*d complexity mode. It's there, but the
main cause of slowness was sorting all the results just to get the
top r of them, using insertsort which is generally O(n^2). So I cooked
up a procedure that just gets the top r results and leaves the rest
unsorted in O(n*r). The above search only takes a few seconds now.

Anyway, the point of posting this here is to find out how you guys
(Gene, Paul, Graham) cook up commas. Gene, is there a particular
maple function I should look at? I see lists of commas doped into
the code in various places...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Anyway, the point of posting this here is to find out how you guys
> (Gene, Paul, Graham) cook up commas. Gene, is there a particular
> maple function I should look at? I see lists of commas doped into
> the code in various places...

Extremely small commas are easy to find using integer relation
algorithms. The trick comes if you want a complete list of them
satisfying certain conditions. One way to do that is via what I call
notations, where you pass to notations using progressively smaller
commas (or equivalently, larger equal temperaments) taking care while
doing so to ensure you don't miss anything fulfilling your conditions.

Carl Lumma wrote:

> Anyway, the point of posting this here is to find out how you guys
> (Gene, Paul, Graham) cook up commas. Gene, is there a particular
> maple function I should look at? I see lists of commas doped into
> the code in various places...

I don't cook up commas, because it looks like a difficult problem that I'll leave for those who care about it. If you have commas, I can find temperaments from them, but it may take a very long time. This is because the number of commas per temperament increases the more prime numbers you consider.

I cook up linear temperaments by combining linear temperaments. This is roughly O(n**2) in the number of equal temperaments, and although it can be slow, is never intolerably so, given the investment required to actually make music in a linear temperament.

For 5-limit linear temperaments it doesn't make any difference, as there is only one comma. But then the 5-limit case is easy however you go about it.

Graham

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

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

it's a good one, but how is it derived? it almost looks like the term
between the '*' and the '=' is the error heuristic, but it's missing
a factor of d in the denominator.

>> Paul, any thoughts on a badness heuristic
>>
>> log(d) * |n-d|/log(d) = |n-d|
>>
>> ?
>>
>> Thanks,
>>
>> -Carl
>
>it's a good one, but how is it derived? it almost looks like the term
>between the '*' and the '=' is the error heuristic, but it's missing
>a factor of d in the denominator.

Drat! Ok, howabout this...

log(d) * |n-d|/d*log(d) = |n-d|/d

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Paul, any thoughts on a badness heuristic
> >>
> >> log(d) * |n-d|/log(d) = |n-d|
> >>
> >> ?
> >>
> >> Thanks,
> >>
> >> -Carl
> >
> >it's a good one, but how is it derived? it almost looks like the
term
> >between the '*' and the '=' is the error heuristic, but it's
missing
> >a factor of d in the denominator.
>
> Drat! Ok, howabout this...
>
> log(d) * |n-d|/d*log(d) = |n-d|/d
>
> -Carl

another decent badness measure -- complexity times error.

>Drat! Ok, howabout this...
>
>log(d) * |n-d|/d*log(d) = |n-d|/d

Difficult to see how we could get some of our favorites like
135/128 to come out of this, without some kind of restriction
on prime limit.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Drat! Ok, howabout this...
> >
> >log(d) * |n-d|/d*log(d) = |n-d|/d
>
> Difficult to see how we could get some of our favorites like
> 135/128 to come out of this, without some kind of restriction
> on prime limit.
>
> -Carl

you just need to penalize complexity more.