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Message: 5800 - Contents - Hide Contents

Date: Sun, 29 Dec 2002 01:46:41

Subject: Re: Temperament notation

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:

> Whatever you do, I suggest you keep in mind that notating linear
temperaments is just as important as notating ets, if not more so. Good point.
> Note that what we have now is a linear temperament notation adapted
for use as a 12-et notation, and not the other way around. Good point. Thanks to the efforts of you and Graham Breed and others on this list we now have agreed lists of the most important 5-limit and 7-limit linear temperaments which we can attempt to notate. We also have Graham's lists of LTs for higher limits up to 21, Automatically generated temperaments * [with cont.] (Wayb.) maybe you could check that his algorithm hasn't missed any important ones, starting with 9-limit. We should try to develop rules for notating them which do not depend on predetermined ET notations. Ideally the notation would depend only on the LT's mapping from gens and periods to primes, and not on any particular value for the generator. The problem is that we can't specify a notation that will work for unlimited length chains of an LT while only allowing one or two symbols against each note. And what's more, for even moderate length chains we will typically need to use symbols for primes that do not appear in the LT's map. For example with a chain of more than 21 notes in 5-limit meantone, we wish to have a single-symbol alternative to double-sharps and double-flats, e.g. Fx may be written G^ so that in pitch order the letters can remain monotonic. This ^ symbol must relate to a prime greater than 5. In this case either 7 or 11 will do. For longer chains we may need to use symbols for 13-commas. So even though it might only be used for 5-limit, the notation ends up being for a particular 7, 11 or 13 limit mapping of meantone, when in fact the range of generator sizes that are of interest for 5-limit meantone, may encompass more than one 7 or 11 or 13-limit mapping. How do we choose which one to use?
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Message: 5801 - Contents - Hide Contents

Date: Sun, 29 Dec 2002 20:16:48

Subject: Re: Temperament notation

From: Carl Lumma

>> > don't know how you expect to do that. The most obvious approach >> seems to me to pick nominals for a MOS with 26 steps or fewer, >> pick the generator which has the lowest height in the correct >> p-limit, and use something like sharps and flats. That is >> completely at odds with what you are doing. If I understand how >> Graham's decimal notation works, it would be a generalization of >> that. >
>Yes, it is completely at odds. Yes, it is a generalisation of >multi-sharp/flat meantone notation and Graham's decimal notation >for miracle temperament, and the 4, 7 and 8 natural notations for >kleismic described in my "Chain of minor thirds" article.
I believe that the simplest way to notate 'diatonic' music is to put the transposition in the fingers and have the scale degrees make sense on paper. Handing the mind scale degrees is indispensible pre-processing for working with 'diatonic' music. The generalized keyboard would reduce the number of fingerings for each scale -- the musician could learn twelve tunings, reading from 'diatonic' notation in each case, with the same amount of effort needed to learn to read standard notation on the piano. But George's thoughts go a long way with me... could it be that for a strict performer, who had to cover lots of tunings, 'transpositionally invarient' notation is the way to go?
>By the way, although there are 26 letters in our alphabet, there >are good reasons, relating to human cognition, why one should aim >to have between 5 and 9 naturals if such a proper MOS exists for >the temperament, and otherwise keep it as close to 9 as possible,
Dave's right, Gene. Where did you get 26 from?
>more than 12 is probably useless. I think that even 4 would be >better than 13 or more, if such a choice were available.
I'd hate to compare like this. 4 is seriously too small. If we're allowed to write music that allows the listener to subset melodies, then 13 would be far better. If we're forced to write tone rows then probably 13 would be worse.
>But as you say, this is not what George and I are trying to do. >When one learns one such temperament-specific notation there is >almost nothing one can carry over to an unrelated temperament, >particularly if it involves a different number of nominals.
I disagree (see above). I imagine that once the fingerings are learned, the mind could transform them to scale degrees and back fairly easily, and that much of the ability to extract scale degree motion from one 'diatonic' notation would work on other 'diatonic' notations (it's quite graphic, after all). Again, I'd take very seriously any input from George on this matter.
>Nor is there anything much one can use from one's knowledge of >conventional notation.
I would think that learning some new fingerings would be easier than keeping track of all the bizare accidental motion if you force decatonic music (say) onto 7 nominals.
>We want notations where C:G is always the temperament's >approximation of a 1:3 (if it has one) and C:E* is always it's >approximation of a 1:5, where * stands for a single saggital >symbol or none at all, and so on up the primes.
Easy! You just use 7-et and slap on as many accidentals as needed.
>But we want more than this. We want it so the notation for any >reasonable-sized MOS in the temperament can be arranged in pitch >order with monotonic letters, e.g. one is never forced to have >any C* higher in pitch than any D*,
? You mean 7 monotonic letters? If you're forcing 7 nominals, you'll have to give up the idea that accidentals represent chromatic uvs.
>and one is not forced to use more than one symbol with any letter.
Well, you can always get that by just adding more flags. Not sure if such flag profusion is any better or worse that stacking a single flag... in standard notation, double-sharp has a dedicated flag, while double-flat does not... IOW, flags are independent of accidentals. -Carl
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Message: 5802 - Contents - Hide Contents

Date: Sun, 29 Dec 2002 04:02:58

Subject: Re: Temperament notation

From: Gene Ward Smith

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

> We should try to develop rules for notating them which do not depend > on predetermined ET notations. Ideally the notation would depend only > on the LT's mapping from gens and periods to primes, and not on any > particular value for the generator.
I don't know how you expect to do that. The most obvious approach seems to me to pick nominals for a MOS with 26 steps or fewer, pick the generator which has the lowest height in the correct p-limit, and use something like sharps and flats. That is completely at odds with what you are doing. If I understand how Graham's decimal notation works, it would be a generalization of that.
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Message: 5803 - Contents - Hide Contents

Date: Sun, 29 Dec 2002 23:29:01

Subject: Re: A common notation for JI and ETs

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "M. Schulter" <mschulter@m...> wrote:
> Hello, everyone, and here are two "tests" of the JI symbols, or > possibly rather of my imperfect understanding of them as a beginner. > > There are some ratios which I'm not sure how best to express -- an > exercise which could help in seeing how best to use the present > symbols or add new ones -- but I've chosen two examples where the > symbols on hand seem sufficient, provided that I've used and > interpreted them correctly. > > First, here's a 12-note JI system with a 7-note Pythagorean chain at > F-B plus some others ratios for the accidentals. As this example > illustrates, I often am looking to write 13:14 as a chromatic > semitone, or apotome at 2048:2187 plus 28672:28431 (~14.613 cents), > with the 17' comma ~|( at 4096:4131 (~14.730 cents) as a neat > solution, as also noted in a recent post from George. > >> deg217 deg494
>> C# 2048:2187 ~113.685c 21 47 >> vs. 14/13 ~128.298c 23 53 >> 28431:28672 ~14.613c 2 6 >> 17' comma ~|( ~14.730c 3 6 >
> 14/13 7/6 21/16 21/13 7/4 > C~|||( E!!!) F!) G~|||( B!!!) > C D E F G A B C > 1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1 Hi Margo,
I don't read multishaft sagittal very well, particularly in ASCII, so permit me to rewrite it in single-shaft (dual symbol). 14/13 7/6 21/16 21/13 7/4 C#~|( Eb!) F!) G#~|( Bb!) C D E F G A B C 1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1 Yes, thats quite correct. C#~|( and G#~|( could also be notated Db(|( and Ab(|( but of course these involve a larger comma (larger by a Pythagorean comma) and so there seems little point in doing so. In multishaft those would be D~!!( and A~!!(
> Here's an example of a diatonic scale in a 17-note JI tuning I use, > showing the 351:352 and 891:896 symbols -- very intuitive for me, > since 351:352 is very close to half of 891:896. > > B\!!/ C|( D)|( E\!!/ F G|( A)|( B\!!/ > 1/1 44/39 14/11 4/3 3/2 22/13 21/11 2/1
Yes, quite correct. It does work out nicely doesn't it.
> Of course, the precise sagittal symbols wouldn't always be necessary: > from a user's viewpoint, this is simply a "justly tempered" diatonic > scale with some fifths pure and others wide by around a 351:352 or > about 5 cents. > > Anyway, there might be two points to this post: I find these symbols > useful, and also intuitive, especially the 351:352 and 891:896 > symbols. Of course, I realize that they're not valid for certain ET's, > but if I'm using them, it suggests a precise kind of JI notation > bringing out the "rational mapping apart from an ET" style. > > By the way, speaking of ET's and standard symbol sets, I should offer a > bit of reassurance that when notating in 72-ET, I would write > > 4A/| 4B\!!/ > 4G 4F > 4E/| 4F > 4C 3B\!!/ > > rather than > > 4A)|( 4B\!!/ > 4G 4F > 4E)|( 4F > 4C 3B\!!/ Right.
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Message: 5804 - Contents - Hide Contents

Date: Sun, 29 Dec 2002 06:18:56

Subject: Re: Temperament notation

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote: >
>> We should try to develop rules for notating them which do not depend >> on predetermined ET notations. Ideally the notation would depend only >> on the LT's mapping from gens and periods to primes, and not on any >> particular value for the generator. >
> I don't know how you expect to do that. The most obvious approach
seems to me to pick nominals for a MOS with 26 steps or fewer, pick the generator which has the lowest height in the correct p-limit, and use something like sharps and flats. That is completely at odds with what you are doing. If I understand how Graham's decimal notation works, it would be a generalization of that. Yes, it is completely at odds. Yes, it is a generalisation of multi-sharp/flat meantone notation and Graham's decimal notation for miracle temperament, and the 4, 7 and 8 natural notations for kleismic described in my "Chain of minor thirds" article. I have nothing against these temperament-specific notations. Very early in the "Common notation ..." thread I said the same, and one could certainly use sagittal symbols consistently for this purpose. By the way, although there are 26 letters in our alphabet, there are good reasons, relating to human cognition, why one should aim to have between 5 and 9 naturals if such a proper MOS exists for the temperament, and otherwise keep it as close to 9 as possible, more than 12 is probably useless. I think that even 4 would be better than 13 or more, if such a choice were available. But as you say, this is not what George and I are trying to do. When one learns one such temperament-specific notation there is almost nothing one can carry over to an unrelated temperament, particularly if it involves a different number of nominals. Nor is there anything much one can use from one's knowledge of conventional notation. George and I are attempting to design an _evolution_ for those who are unlikely to be interested in such a _revolution_, and we'd love your help. We want notations where C:G is always the temperament's approximation of a 1:3 (if it has one) and C:E* is always it's approximation of a 1:5, where * stands for a single saggital symbol or none at all, and so on up the primes. But we want more than this. We want it so the notation for any reasonable-sized MOS in the temperament can be arranged in pitch order with monotonic letters, e.g. one is never forced to have any C* higher in pitch than any D*, and one is not forced to use more than one symbol with any letter. As one extends the chains of generators in any temperament, there eventually comes a proper MOS which is so close to an equal temperament that there is little point in adding more notes. There seems no disagreement that this happens at 72 for miracle, but for meantone it is unclear to me whether it is 31 or 50. It seems to be 50 in the 5-limit optimised case, and 31 for 7 and 11 limit. For schismic is it 41 or 53? For kleismic is it 53 or 72? etc.
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Message: 5805 - Contents - Hide Contents

Date: Mon, 30 Dec 2002 18:04:35

Subject: Scale theory resources

From: Gene Ward Smith

Does anyone know of a good source for these, especially on-line? I was
browsing around and found that Clampitt has been using Ramsey theory,
which sounds as if people are wading out past the shallow end of the
pond, and I'd like to catch up.


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Message: 5806 - Contents - Hide Contents

Date: Mon, 30 Dec 2002 23:10:12

Subject: Re: A common notation for JI and ETs

From: M. Schulter

Hello, Dave and everyone, and thank you for your response to my
examples of JI notation.

As I've read the draft of the article for Xenharmonikon 18, and
followed some of the discussions about reviewing and possibly adding
to the list of commas and dieses, I've noticed that the available
symbols cover many of my favorite ratios, but leave a few questions.

The following JI system, which I came up with earlier this year, is
based in part on an arithmetic or subharmonic series a la Kathleen
Schlesinger of 28-27-26-24-23-22-21, with the ratio of 22:28 or 11:14
divided into whole-tone steps of 39:44 and 242:273. The first ratio in
this division is wider than 8:9 by 351:352, the second by 363:364.

The tuning, as it happens, is generally similar to the 17-note
triaphonic system of John Chalmers. He and Erv Wilson, it turns out,
got the idea of combining Schlesinger's _harmoniai_ or arithmetic
divisions with a tetrachord (3:4) structure some years before I did.

The general philosophy of my Just Octachord Tuning (JOT-17) -- with an
"octachord" or eight inclusive steps for each 3:4 tetrachord -- is
that of a JI system offering many of the features of a 17-note
well-temperament, but with some quirks making it "a bit different"
than a closed circle.

Like a 17-note well-temperament, JOT-17 has at each position some kind
of diatonic "thirdtone" (1deg17) ranging in size from 88:91 to 21:22;
each whole-tone (3deg17) is divided into three such steps. Minor
thirds (4deg17) range from Pythagorean to septimal, with major thirds
(6deg17) having a similar range, and neutral thirds (5deg17) from
52:63 to 21:26, etc.

However, the presence of some pure septimal ratios and sonorities such
as 12:14:18:21 or 14:18:21:24 is associated with two fifths (10deg17)
wide by a full 63:64 (~27.264 cents) and 729:736 (~16.544 cents). Thus
a sagittal notation, at least, might seek to modify a usual 17-ET
notation for these special intervals in the interest of "least
astonishment," as well as to reflect the availability of such
sonorities as a just 16:21:24:28 in one position.

With JOT-17 we might therefore ask two questions: how _can_ we
precisely notate the diverse intervals in an approach showing
modifications of Pythagorean tuning (following the sagittal convention
of disregarding small schismas such as 10647:10648)?; and how _should_
we notate such a system to best effect, at once following many of the
norms of a 17-ET notation and duly representing divergences of
interest to performers?

Here is a Scala file for JOT-17, presented for simplicity in an
arrangement where the "1/1" is B\!!/ or Bb, the lowest note of the
first 3:4 tetrachord or "octachord":

! jot17a.scl
!
Just octachord tuning -- 4:3-9:8-4:3 division, 17 steps (7 + 3 + 7), Bb-Bb
 17
!
 28/27
 14/13
 44/39
 7/6
 28/23
 28/22
 4/3
 112/81
 56/39
 3/2
 14/9
 21/13
 22/13
 7/4
 42/23
 21/11
 2/1

Here's a Scala "show scale" data file:

|
Just octachord tuning -- 4:3-9:8-4:3 division, 17 steps (7 + 3 + 7), Bb-Bb
  0:          1/1            0.000000 unison, perfect prime
  1:         28/27           62.96093 1/3-tone   
  2:         14/13           128.2983 2/3-tone
  3:         44/39           208.8353
  4:          7/6            266.8710 septimal minor third
  5:         28/23           340.5516
  6:         14/11           417.5081 undecimal diminished fourth
  7:          4/3            498.0452 perfect fourth
  8:        112/81           561.0061
  9:         56/39           626.3435
 10:          3/2            701.9553 perfect fifth
 11:         14/9            764.9162 septimal minor sixth
 12:         21/13           830.2536
 13:         22/13           910.7907
 14:          7/4            968.8264 harmonic seventh
 15:         42/23           1042.507
 16:         21/11           1119.463
 17:          2/1            1200.000 octave

In my 17-ET notation, I'd really like to consider the 1/1 as B\!!/,
placing the 21:32 between the steps A!!!) and E\!!/ -- however, for
this try, why don't I instead consider the 1/1 as C, to simplify a
certain point I'll explain.

Here are the steps I can readily notate, and some on which I'm unsure:

  1/1            C           C
 28/27           D!!!)       Db!)
 14/13           C~|||(      C#~|(
 44/39           D|(         D|(
  7/6            E!!!)       Eb!)
 28/23           This is Pythagorean D/||\ + 452709:458752 (~22.957c)
 14/11           E)|(        E)|(
  4/3            F           F
112/81           G!!!(       Gb!)
 56/39           F~|||(      F#~|(
  3/2            G           G
 14/9            A!!!)       Ab!)
 21/13           G~|||(      G#~|(       
 22/13           A|(         A|(
  7/4            B!!!)       Bb!)
 42/23           This is Pythagorean A/||\ + 452709:458752 (~22.957c)
 21/11           B)|(        B)|(
  2/1            C           C 

Interestingly, the 23:28 or 23:42 is very close to a Pythagorean
augmented second or sixth plus a Pythagorean comma -- so if there's a
sign for a Pythagorean comma, that might do (within ~0.5 cents of the
actual ratio).

Now for the complication. When I took B\!!/ for the 1/1, I realized
than using 17-ET conventions, A/||\ would be a thirdtone higher, here
a just 27:28 -- but in Pythagorean, this is only a Pythagorean comma,
so the modifications become unclear to me as a beginner.

In practice, I would guess that keeping track of the 351:352
adjustments and the like would be unnecessary, much as in a 17-tone
well-temperament. However, three fifths do seem to invite some
explicit sagittal modifications. Taking 1/1 as B\!!/ or Bb, these
would be in conventional notation F#-C# (69:104, wide by 207:208 or
~8.343 cents); G#-D# (243:368, wide by 729:736 or ~16.544 cents); and
Ab-Eb (21:32, wide by 63:64 or ~27.264 cents). 

Anyway, the problem of notating the 23:28 as a modification of a
Pythagorean augmented second might be another reason for a Pythagorean
comma sign, which as I recall has been proposed by adding at least one
flag to one of the basic symbols.

Most appreciatively,

Margo
mschulter@xxxxx.xxx


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Message: 5807 - Contents - Hide Contents

Date: Tue, 31 Dec 2002 19:03:08

Subject: Re: Temperament notation

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" 
<d.keenan@u...> wrote:
> ... > As one extends the chains of generators in any temperament, there > eventually comes a proper MOS which is so close to an equal > temperament that there is little point in adding more notes. There > seems no disagreement that this happens at 72 for miracle, but for > meantone it is unclear to me whether it is 31 or 50. It seems to be 50 > in the 5-limit optimised case, and 31 for 7 and 11 limit.
It depends on what method you're using to optimize. The 5-limit minimax generator is a 1/4-comma fifth, which would also favor 31. In order to get the slow-beating minor third of 50-ET you also get a slow-beating major third, plus a faster-beating fifth that is not acceptable to some. So I always thought that 31 was the clear choice for meantone.
> For schismic is it 41 or 53?
How about 94?
> For kleismic is it 53 or 72?
I'll have to taken a better look at this one.
> etc. etc. --George
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Message: 5808 - Contents - Hide Contents

Date: Wed, 01 Jan 2003 20:37:15

Subject: Re: A common notation for JI and ETs

From: David C Keenan

In tuning-math@xxxxxxxxxxx.xxxx "M. Schulter" <mschulter@m...> wrote:

>The following JI system, which I came up with earlier this year, is >based in part on an arithmetic or subharmonic series a la Kathleen >Schlesinger of 28-27-26-24-23-22-21, with the ratio of 22:28 or 11:14 >divided into whole-tone steps of 39:44 and 242:273. The first ratio in >this division is wider than 8:9 by 351:352, the second by 363:364.
Oh dear! You're really testing us aren't you? _SUB_harmonic series, ratios of 23, and tempering (albeit by ratios).
>Here is a Scala file for JOT-17, presented for simplicity in an >arrangement where the "1/1" is B\!!/ or Bb, the lowest note of the >first 3:4 tetrachord or "octachord": > >! jot17a.scl >! >Just octachord tuning -- 4:3-9:8-4:3 division, 17 steps (7 + 3 + 7), Bb-Bb > 17 >! > 28/27 > 14/13 > 44/39 > 7/6 > 28/23 > 28/22 > 4/3 > 112/81 > 56/39 > 3/2 > 14/9 > 21/13 > 22/13 > 7/4 > 42/23 > 21/11 > 2/1 ... >In my 17-ET notation, I'd really like to consider the 1/1 as B\!!/, >placing the 21:32 between the steps A!!!) and E\!!/ -- however, for >this try, why don't I instead consider the 1/1 as C, to simplify a >certain point I'll explain. > >Here are the steps I can readily notate, and some on which I'm unsure: > > 1/1 C C > 28/27 D!!!) Db!) > 14/13 C~|||( C#~|( > 44/39 D|( D|( > 7/6 E!!!) Eb!) > 28/23 This is Pythagorean D/||\ + 452709:458752 (~22.957c) > 14/11 E)|( E)|( > 4/3 F F >112/81 G!!!( Gb!) > 56/39 F~|||( F#~|( > 3/2 G G > 14/9 A!!!) Ab!) > 21/13 G~|||( G#~|( > 22/13 A|( A|( > 7/4 B!!!) Bb!) > 42/23 This is Pythagorean A/||\ + 452709:458752 (~22.957c) > 21/11 B)|( B)|( > 2/1 C C > >Interestingly, the 23:28 or 23:42 is very close to a Pythagorean >augmented second or sixth plus a Pythagorean comma -- so if there's a >sign for a Pythagorean comma, that might do (within ~0.5 cents of the >actual ratio).
The above notation is quite correct. You _could_ use a Pythagorean comma symbol for 28/23 and 42/23, but I don't think we've agreed on that symbol, and some of the candidates are rather complicated 3-flaggers, and in any case I have a much simpler suggestion. Use the 5-comma symbol /| . 28/23 D/||| D#/| 42/23 A/||| A#/| You will find that this does not imply any actual ratios of 5 in this tuning and happens to be consistent with 212-ET, which models it rather well. Proposal 212-ET: |( )|( ~|( /| |) (| (|( //| /|\ (/| (|) I'm not saying that /| is always appropriate to represent a 7:23 comma, but in this tuning I do not believe these two pitches are justly intoned relative to any combination of other pitches in the tuning (except each other) and therefore I feel that the 1.5 cent notational schisma it involves is insignificant. We can just as easily decide to read an Archytas comma symbol as a 7:23-comma, as we can a Pythagorean comma symbol.
>Now for the complication. When I took B\!!/ for the 1/1, I realized >than using 17-ET conventions, A/||\ would be a thirdtone higher, here >a just 27:28 -- but in Pythagorean, this is only a Pythagorean comma, >so the modifications become unclear to me as a beginner.
There is no way, in the rational sagittal notation, to notate a 27:28 above a Bb, as a variety of A#. Nor do I think there ever should be. That's equivalent to wanting to notate a 4:7 above C as a variety of Gx. With 1/1 as Bb, 28/27 would be Cb!) or C !!!) and 112/81 would be Fb!) or F!!!). You could invoke the 212-ET notation and call them A#(|( and D#(|( but this seems wrong to me because it actually makes use of the slight difference between the 212-ET fifth and the just fifth, which we are otherwise ignoring.
>In practice, I would guess that keeping track of the 351:352 >adjustments and the like would be unnecessary, much as in a 17-tone >well-temperament. However, three fifths do seem to invite some >explicit sagittal modifications. Taking 1/1 as B\!!/ or Bb, these >would be in conventional notation F#-C# (69:104, wide by 207:208 or >~8.343 cents); G#-D# (243:368, wide by 729:736 or ~16.544 cents); and >Ab-Eb (21:32, wide by 63:64 or ~27.264 cents).
What's wrong with 1/1 as C and then the wolves are G#~|( to D#/| 8 cents wide, A#/| to Gb!) 17 cents wide, Bb!) to F 27 cents wide. Surely you would want A#-Gb to be a wolf? I'm afraid I don't understand the advantage of the Bb 1/1. We also have 5c wide fifths G to D|( A|( to E)|( B)|( to F#~|(
>Anyway, the problem of notating the 23:28 as a modification of a >Pythagorean augmented second might be another reason for a Pythagorean >comma sign, which as I recall has been proposed by adding at least one >flag to one of the basic symbols.
Yes one proposal is to add a very tiny straight right flag (or some other tiny graphical addition) to the 5-comma symbol /| . Using Pythagorean rather than Archtus certainly would make the distinction between the 5 cent wide and the 8 cent wide fifths, which are not distinguished in a 212-ET mapping. There's another angle to this which I won't go into since you seem happy with the way it is. That is: How would you notate it if you did not want to have C-something being a higher pitch than D-something etc.; what I refer to as having monotonic letters? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5809 - Contents - Hide Contents

Date: Wed, 01 Jan 2003 21:25:27

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 01:21 PM 23/12/2002 -0800, George Secor wrote:
>> The 7:11 comma that is relevant to Peppermint is 891:896 which I >don't
>> believe we have considered before in regard to the sagittal notation. >The
>> appropriate symbol for it would be )|(, however it vanishes in >Peppermint >> (and 121-ET). >
>The symbol )|( is not valid as 891:896 in either 217 or 494, but that >shouldn't stop anyone from using it for JI, unless we can figure out >something else. Well, here's something else: The (19'-17)-5' comma >~)|' would come within 0.3 cents and would be valid in both 217 and 494 >(plus 224, 270, 282, 311, 342, 364, 388, 400, 525, and 612, to name >more than a few). The only thing I can say against it is that it seems >rather contrived and not at all intuitive, but it works in more places >than I would have expected.
I'd prefer to go with )|( as the 7:11 comma since it only involves a 0.55 cent schisma. I feel that a 3 flag symbol for something under 10 cents could not be justified when a 2 flagger is within 0.98 cents. It seems 891:896 )|( should be called the 7:11 comma while the comma represented by (| is called the 7:11'-comma. Are there any ETs in which we should now prefer )|( over some other symbol given that it now has such a low prime-limit or low product complexity?
>They are all 7-related. In a 13-limit heptad (8:9:10:11:12:13:14) it >is 7 that introduces scale impropriety; e.g., the fifth 5:7 is smaller >than the fourth 7:10. Replace 14 with 15 in the heptad and I believe >the scale is proper. So it would not be surprising that someone might >want to respell the intervals involving 7 -- 4:7 as a sixth, 5:7 as a >fourth, 6:7 as a second, 7:9 as a fourth, 11:14 as a third, and 13:14 >as an altered unison. > >So we would want to notate the following ratios of 7 using these >commas: > > deg217 deg494 > ------ ------ >A# 32768:59049 ~1019.550c 185 420 >vs. 7/4 ~968.826c 175 399 >57344:59049 ~50.724c 10 21 >(apotome complement of 27:28 - this could be called the 7' comma) >11:19 comma (|~ ~49.895c 9 21 >But a new symbol /|)` would represent it exactly >(if the flags are added up separately 5+7+5' comma)
I really don't think it is necessary or desirable to notate this 7'-comma. It is larger than the standard 7-comma and it involves a longer chain of fifths than _any_ other comma we've ever used. I think we should only accept the need for a _larger_ alternative comma for some prime (or ratio of primes) if it involves a _shorter_ chain of fifths.
>Expressed another way:
I don't see the following quote as expressing the above quote another way. It is a completely different 7-comma. With this comma a 4:7 above C would be a kind of A, not A#. A 16:27 vs. 4:7
>F 3:4 ~498.045c 90 205 >vs. 9/7 ~435.084c 79 179 >27:28 ~62.961c 11 26 >symbol )|| 12 26 >But a new symbol (|\' would represent it exactly
It is very large, and the absolute value of its power of 3 is still larger than that of the standard 7 comma, although only by 1. I'm not convinced there's any need for it.
>F# 512:729 ~611.730c 111 252 >vs. 7/5 ~582.512c 105 240 >3584:3645 ~29.218c 6 12 >This is the 5:7' comma, or 7+5' comma, or 7'-5 comma >A new symbol |)` would represent it exactly
This contains 3^6 while the standard 5:7-comma has 3^-6 so I think there could be some demand for this one. I think the proposed symbol is good, being only 2 flags, however I'd like it even better if we could come up with some way that the 5'-comma (ordinary schisma) could be notated as a modification of the shaft rather than as a flag, or if the two flags were not on the same side. But in any case, it seems we should avoid using it if possible because of its containing that very unfamiliar flag. It's kind of strange if we should need to use this obscurte new flag as low as the 7-limit. You should leave it out of the XH18 paper.
>E 64:81 ~407.820c 74 168 >vs. 14/11 ~417.508c 75 172 >891:896 ~9.688c 1 4 >5:7+19 comma )|( ~9.136c 2 3 Agreed. >C# 2048:2187 ~113.685c 21 47 >vs. 14/13 ~128.298c 23 53 >28431:28672 ~14.613c 2 6 >17' comma ~|( ~14.730c 3 6
Agreed. I though we already had that one. I believe we called this the 7:13 comma while (|( is the 7:13' comma. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5810 - Contents - Hide Contents

Date: Thu, 02 Jan 2003 16:17:53

Subject: Re: Temperament notation

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" 
<gdsecor@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" > <d.keenan@u...> wrote: >> ...
>> As one extends the chains of generators in any temperament, there >> eventually comes a proper MOS which is so close to an equal >> temperament that there is little point in adding more notes. There >> seems no disagreement that this happens at 72 for miracle, but for >> meantone it is unclear to me whether it is 31 or 50. It seems to be > 50
>> in the 5-limit optimised case, and 31 for 7 and 11 limit. >
> It depends on what method you're using to optimize. The 5-limit > minimax generator is a 1/4-comma fifth, which would also favor 31. > In order to get the slow-beating minor third of 50-ET you also get a > slow-beating major third, plus a faster-beating fifth that is not > acceptable to some. So I always thought that 31 was the clear choice > for meantone. >
>> For schismic is it 41 or 53? >
> How about 94?
Now that I've more time to look at this, I would say definitely 94, for two reasons: 1) If you're including the 7th harmonic, then you might as well take this to the 11 limit (since the minimax generator for both is the same). The 11th harmonic occurs in the series of fifths in 41, 53, and 94 in the +23 position, but in 41 it is also closer -- in the -18 position, which is not typical for the schismic family of temperaments. So I eliminate 41 as my choice. 2) The 7 and 11-limit minimax generator is ~702.193c (7:9 being exact) giving a maximum error of ~4.331c (for 5:7 and 5:9). The 53- ET fifth is ~701.887c (max. error ~12.681c), but the 94-ET fifth is much closer to the ideal: ~702.178c (max. error ~4.722c), and the same may be said for the 13 and 15-limit minimax generator (~702.109c, 13:14 being exact). So I eliminate 53, leaving 94 as my choice.
>> For kleismic is it 53 or 72? >
> I'll have to taken a better look at this one.
I've done that and I conclude that, unless you are sticking with a 5 limit, the choice is clearly 72 over 53. It is best to put the figures in a table to show this: Generator Size Max. error Exact 5-minimax ~316.993c ~1.351c 2:3 53-ET ~316.981c ~1.408c 72-ET ~316.667c ~2.980c 125-ET ~316.800c ~2.314c 7,9-minimax ~316.765c ~2.732c 4:7 53-ET ~316.981c ~6.167c 72-ET ~316.667c ~3.910c 125-ET ~316.800c ~3.088c 11-minimax ~316.745c ~2.976c 9:11 53-ET ~316.981c ~12.681c 72-ET ~316.667c ~3.910c 125-ET ~316.800c ~4.892c I threw 125 in there also, since it does slightly better than 72 at the 7 and 9 limit (and also at the 13 and 15 limit, which has the same minimax generator as for the 7 and 9 limit). But since the 11 limit is the highest you can go while keeping the max error under 4 cents, that's the limit I would use, and 72 has the advantage. Something else I noticed about the choice of 72 as the notation for both the Miracle and kleismic temperaments: the progression of sagittal symbols for a 72-ET panchromatic scale (one passing through all the tones) is the same as that for a sequence of tones differing by the generating interval in both temperaments. To illustrate: 72-ET: C C\! C!) C\!/ B|) B/| B Miracle: C B\! Bb!) A\!/ G|) F#/| F kleismic: C A\! F#!) Eb\!/ B|) G#/| F The pattern then repeats. I believe that this is a useful property that provides a further justification for basing the kleismic notation on 72. --George
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Message: 5811 - Contents - Hide Contents

Date: Thu, 02 Jan 2003 18:36:37

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> 
wrote:
> In tuning-math@xxxxxxxxxxx.xxxx "M. Schulter" <mschulter@m...> wrote: >
>> The following JI system, which I came up with earlier this year, is >> based in part on an arithmetic or subharmonic series a la Kathleen >> Schlesinger of 28-27-26-24-23-22-21, with the ratio of 22:28 or 11:14 >> divided into whole-tone steps of 39:44 and 242:273. The first ratio in >> this division is wider than 8:9 by 351:352, the second by 363:364. >
> Oh dear! You're really testing us aren't you? _SUB_harmonic series, ratios > of 23, and tempering (albeit by ratios). > ... > You _could_ use a Pythagorean comma symbol for 28/23 and 42/23, but I don't > think we've agreed on that symbol, and some of the candidates are rather > complicated 3-flaggers,
I think by now that we would agree that it would be the 5 comma plus 5' comma (or "schisma") symbol, but that we have yet to agree on the best way to symbolize the 5' comma.
> and in any case I have a much simpler suggestion. > Use the 5-comma symbol /| . > > 28/23 D/||| D#/| > 42/23 A/||| A#/| > > You will find that this does not imply any actual ratios of 5 in this > tuning and happens to be consistent with 212-ET, which models it rather well. > > Proposal > 212-ET: |( )|( ~|( /| |) (| (|( //| /|\ (/| (|)
Filling out the rational complementation for a complete apotome this would be: 212a: |( )|( ~|( /| |) (| (|( //| /|\ (/| (|) ~|| ~||( ) ||~ ||) ||\ (||( ||~) /||) /||\ (DK) I have only one question: Since the 17th harmonic is so far off in 212-ET as to be almost midway between tones (and inconsistent besides), whereas the 23rd is almost exact, would it be more appropriate to substitute the 23 comma for the 17' comma symbol? That would give: 212b: |( )|( |~ /| |) (| (|( //| /|\ (/| (|) ~|| ~||( ) ||~ ||) ||\ ~||) ||~) /||) /||\ (GS) But if you still prefer 212a for the standard set, then at least Margo could use 212b as a modification, since 23 is present in her tuning. --George
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Message: 5812 - Contents - Hide Contents

Date: Thu, 02 Jan 2003 21:30:03

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> 
wrote:
> At 01:21 PM 23/12/2002 -0800, George Secor wrote:
>>> The 7:11 comma that is relevant to Peppermint is 891:896 which I don't >>> believe we have considered before in regard to the sagittal notation. The >>> appropriate symbol for it would be )|(, however it vanishes in Peppermint >>> (and 121-ET). >>
>> The symbol )|( is not valid as 891:896 in either 217 or 494, but that >> shouldn't stop anyone from using it for JI, unless we can figure out >> something else. Well, here's something else: The (19'-17)-5' comma >> ~)|' would come within 0.3 cents and would be valid in both 217 and 494 >> (plus 224, 270, 282, 311, 342, 364, 388, 400, 525, and 612, to name >> more than a few). The only thing I can say against it is that it seems >> rather contrived and not at all intuitive, but it works in more places >> than I would have expected. >
> I'd prefer to go with )|( as the 7:11 comma since it only involves a 0.55 > cent schisma. I feel that a 3 flag symbol for something under 10 cents > could not be justified when a 2 flagger is within 0.98 cents.
Agreed, but I would call it the 7':11 comma for reasons given below.
> It seems 891:896 )|( should be called the 7:11 comma while the comma > represented by (| is called the 7:11'-comma.
Before we used the colon designation for these two-prime commas, we were expressing them as the sum or difference of two single-prime commas, e.g., the 5:7 comma was the 7-5 comma. How would you do that for 891:896 other than as the 11'-7' comma? (However, since you don't like what I have for the 7' comma, see below.) Since this is the comma that is arrived at by invoking a new (7') comma, I think that this should be called the 7':11 comma.
> Are there any ETs in which we should now prefer )|( over some other symbol > given that it now has such a low prime-limit or low product complexity? >
>> They are all 7-related. In a 13-limit heptad (8:9:10:11:12:13:14) it >> is 7 that introduces scale impropriety; e.g., the fifth 5:7 is smaller >> than the fourth 7:10. Replace 14 with 15 in the heptad and I believe >> the scale is proper. So it would not be surprising that someone might >> want to respell the intervals involving 7 -- 4:7 as a sixth, 5:7 as a >> fourth, 6:7 as a second, 7:9 as a fourth, 11:14 as a third, and 13:14 >> as an altered unison. >> >> So we would want to notate the following ratios of 7 using these >> commas: >> >> deg217 deg494 >> ------ ------ >> A# 32768:59049 ~1019.550c 185 420 >> vs. 7/4 ~968.826c 175 399 >> 57344:59049 ~50.724c 10 21 >> (apotome complement of 27:28 - this could be called the 7' comma) >> 11:19 comma (|~ ~49.895c 9 21 >> But a new symbol /|)` would represent it exactly >> (if the flags are added up separately 5+7+5' comma) >
> I really don't think it is necessary or desirable to notate this 7'- comma. > It is larger than the standard 7-comma and it involves a longer chain of > fifths than _any_ other comma we've ever used.
I think it's a matter of waiting to see if we'll have to, because I don't think that wanting to notate 7/4 relative to C as A-something would be unusual or weird.
> I think we should only accept the need for a _larger_ alternative comma for > some prime (or ratio of primes) if it involves a _shorter_ chain of fifths. >
>> Expressed another way: >
> I don't see the following quote as expressing the above quote another way. > It is a completely different 7-comma. With this comma a 4:7 above C would > be a kind of A, not A#.
Okay, then call its apotome complement, 27:28, the 7' comma and use the 13'-5' symbol (|\' to represent it (replacing ' with whatever we eventually agree on for the 5' comma). I notice that this is the next symbol I proposed:
> A 16:27 > vs. 4:7 >
>> F 3:4 ~498.045c 90 205 >> vs. 9/7 ~435.084c 79 179 >> 27:28 ~62.961c 11 26 >> symbol )|| 12 26 >> But a new symbol (|\' would represent it exactly >
> It is very large,
But it's still smaller than the 13' comma, ~65.3c, so this doesn't take us outside our upper boundary for single-shaft symbols.
> and the absolute value of its power of 3 is still larger > than that of the standard 7 comma, although only by 1. I'm not convinced > there's any need for it.
As I said, let's wait and see. The nice thing about this is that it doesn't require any new flags other than the 5' (for which it offers further justification for having that new flag or whatever) and that it's exact. Come to think of it, I seem to recall that Margo wrote me a couple of weeks ago that she wanted a JI symbol for 27:28 -- you must admit that this is not a weird or unusual interval. If there is any problem with this, I think it is that we need to be able to represent the 5' comma in such a way that the symbols in which it is used don't look weird.
>> F# 512:729 ~611.730c 111 252 >> vs. 7/5 ~582.512c 105 240 >> 3584:3645 ~29.218c 6 12 >> This is the 5:7' comma, or 7+5' comma, or 7'-5 comma >> A new symbol |)` would represent it exactly >
> This contains 3^6 while the standard 5:7-comma has 3^-6 so I think > there could be some demand for this one. I think the proposed symbol is > good, being only 2 flags, however I'd like it even better if we could come > up with some way that the 5'-comma (ordinary schisma) could be
notated as a
> modification of the shaft rather than as a flag, or if the two flags were > not on the same side.
We need to find a good way to represent *both* the 5' and -5' alterations that involves something other than a flag -- something laterally aligned with the shaft, if not a modification to the shaft itself. (So back to the drawing board!)
> But in any case, it seems we should avoid using it if possible because of > its containing that very unfamiliar flag. It's kind of strange if we should > need to use this obscurte new flag as low as the 7-limit. You should leave > it out of the XH18 paper.
I have a feeling that the 5' comma is going to be useful for notating all sorts of things regardless of the prime limit (we have already proposed incorporating it into the diaschisma, Pythagorean comma, and 5-diesis symbols), particularly if it will indicate intervals exactly, so I wouldn't call this an obscure flag -- just a very small one. And the idea of using something other than a lateral flag to symbolize it strikes me as highly appropriate -- just so long as it looks good (and therein lies the problem).
>> E 64:81 ~407.820c 74 168 >> vs. 14/11 ~417.508c 75 172 >> 891:896 ~9.688c 1 4 >> 5:7+19 comma )|( ~9.136c 2 3 > > Agreed. >
>> C# 2048:2187 ~113.685c 21 47 >> vs. 14/13 ~128.298c 23 53 >> 28431:28672 ~14.613c 2 6 >> 17' comma ~|( ~14.730c 3 6 >
> Agreed. I though we already had that one. I believe we called this the 7:13 > comma while (|( is the 7:13' comma.
I have been calling (|( the 7:13 comma, since it is the 13'-7 comma; however 28431:28672 isn't the 13-7 comma -- it's the 7'-13 comma (if 27:28 is now the 7' comma), so I would then also call it the 7':13 comma. If 27:28 is the 7' comma, then I would also have to rename the following in what I gave above: 3584:3645 as the 5:7' comma or 7+5' comma (but now not the 7'-5 comma) 891:896 as the 7':11 comma or 7'-11 comma So have I sold you on a 7' comma, 27:28? --George
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Message: 5813 - Contents - Hide Contents

Date: Thu, 02 Jan 2003 22:48:02

Subject: Re: Temperament notation

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" 
<clumma@y...> wrote:
>>> I don't know how you expect to do that. The most obvious approach >>> seems to me to pick nominals for a MOS with 26 steps or fewer, >>> pick the generator which has the lowest height in the correct >>> p-limit, and use something like sharps and flats. That is >>> completely at odds with what you are doing. If I understand how >>> Graham's decimal notation works, it would be a generalization of >>> that. >>
>> Yes, it is completely at odds. Yes, it is a generalisation of >> multi-sharp/flat meantone notation and Graham's decimal notation >> for miracle temperament, and the 4, 7 and 8 natural notations for >> kleismic described in my "Chain of minor thirds" article. >
> I believe that the simplest way to notate 'diatonic' music is to > put the transposition in the fingers and have the scale degrees > make sense on paper. Handing the mind scale degrees is > indispensible pre-processing for working with 'diatonic' music. > The generalized keyboard would reduce the number of fingerings for > each scale -- the musician could learn twelve tunings, reading > from 'diatonic' notation in each case, with the same amount of > effort needed to learn to read standard notation on the piano. > > But George's thoughts go a long way with me... could it be that > for a strict performer, who had to cover lots of tunings, > 'transpositionally invarient' notation is the way to go? >
>> By the way, although there are 26 letters in our alphabet, there >> are good reasons, relating to human cognition, why one should aim >> to have between 5 and 9 naturals if such a proper MOS exists for >> the temperament, and otherwise keep it as close to 9 as possible, >
> Dave's right, Gene. Where did you get 26 from? >
>> more than 12 is probably useless. I think that even 4 would be >> better than 13 or more, if such a choice were available. >
> I'd hate to compare like this. 4 is seriously too small. If we're > allowed to write music that allows the listener to subset melodies, > then 13 would be far better. If we're forced to write tone rows > then probably 13 would be worse. >
>> But as you say, this is not what George and I are trying to do. >> When one learns one such temperament-specific notation there is >> almost nothing one can carry over to an unrelated temperament, >> particularly if it involves a different number of nominals. >
> I disagree (see above). I imagine that once the fingerings are > learned, the mind could transform them to scale degrees and back > fairly easily, and that much of the ability to extract scale degree > motion from one 'diatonic' notation would work on other 'diatonic' > notations (it's quite graphic, after all). Again, I'd take very > seriously any input from George on this matter.
I've been thinking this over while in the process of answering other things. I'm not familiar with the problems of notating some of the esoteric ETs, for example, those for which Blackwood wrote etudes (though I have a recording), so my reply is going to be from a somewhat limited perspective. So take it for what it's worth. Let me offer an example from my viewpoint that may shed some light on some of the issues involved. Carl, you seem to be looking at the problem of notation from a keyboardist's viewpoint (from your mention of fingerings), but one's understanding and framework for a notation must be broader than and/or independent of that. But even if we stick to a keyboard application for the moment, let me give an example of how a "transpositionally invariant" notation (if I understand the term correctly) could pose more problems than it would solve. Let's consider the Miracle tuning notated using Graham's decimal notation, which I presume you would consider transpositionally invariant. Suppose you write something for Blackjack or Canasta in decimal notation, and I want to play it on my Scalatron -- with a Bosanquet generalized rather than a decimal keyboard. My only option is to map Canasta into a 31-tone octave (41 would also be possible if I had that many program boards on the instrument, but that's a more difficult fingering pattern and another issue entirely). If I am doing Blackjack, then I have that as a subset -- so far, so good. Now I already know how to read 31-ET in Fokker's notation or sagittal notation -- both with 7 nominals and the same semantics, differing only in the cosmetics. And I can also read 72-ET sagittal notation and can easily convert it at sight to 31-ET -- the rules are simple: the 5 comma vanishes and the 7 comma is translated to an 11 comma. So I would have no problem reading Miracle mapped to either 31 or 72 sagittal. But what value is the decimal notation to me, and what incentive would I have to learn it without a decimal keyboard? Now reverse the circumstances. Give me a decimal keyboard and sufficient time to learn it, and then give me something in Miracle to read in sagittal notation. I should do just fine with it, because I can translate the notes into pitches and the pitches into keyboard locations. Now give me the same decimal keyboard with Partch's 11-limit JI mapped onto it (observe that this was the reason that I originally came up with the layout). Again, I should do just fine with 72-ET sagittal notation, assuming that I am proficient with the keyboard. So what is the point of learning a decimal notation, if it will have a slight advantage only with a specific keyboard and/or a specific tuning geometry? If I decide to use the Miracle temperament instead of 11-limit JI for Partch's music, is there any advantage in using the decimal notation with 10 nominals for this purpose over a 7- nominal sagittal notation? We're all familiar with a 7-nominal / heptatonic / diatonic notation, and that's what Dave and I have been building on to produce what I call a "generalized" notation -- one that is semantically independent of any particular tonal geometry or division of the octave. It may not be the best notation for everything, but it will do a lot of different things and will do many of them extremely well. --George
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Message: 5815 - Contents - Hide Contents

Date: Fri, 3 Jan 2003 14:17:59

Subject: Re: Minimax generator

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Minimax generators can also be calculated with Scala by the
"calculate/minimax" command. It shows the least squares optimum
at the same time, so you don't need to enter everything twice
if you want that also.

The next version still to come will have a new dialog to support 
the easy calculation of equal beating temperaments, in the Tools
menu.

Manuel


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Message: 5816 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 18:23:34

Subject: Fwd: Re: A common notation for JI and ETss

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> 
wrote:
> George Secor: >
>> Filling out the rational complementation for a complete apotome this >> would be: >> >> 212a: |( )|( ~|( /| |) (| (|( //| /|\ (/| (|) ~|| ~||
( )||~ ||) ||\ (||( ||~) /||) /||\ (DK)
>> >> I have only one question: Since the 17th harmonic is so far off in >> 212-ET as to be almost midway between tones (and inconsistent >> besides), whereas the 23rd is almost exact, would it be more >> appropriate to substitute the 23 comma for the 17' comma symbol? >> That would give: >> >> 212b: |( )|( |~ /| |) (| (|( //| /|\ (/| (|) ~|| ~||
( )||~ ||) ||\ ~||) ||~) /||) /||\ (GS)
>> >> But if you still prefer 212a for the standard set, then at least >> Margo could use 212b as a modification, since 23 is present in her >> tuning. >
> 23 is present in the tuning, but not in those pitches that might be notated > with the 3deg212 symbol. Margo and I both chose ~|( because of its > interpretation as a 7:13 comma, not a 17 comma. So whatever we might decide > for 3deg212, ~|( seems like the right symbol for Margo's tuning.
Okay, now I get it. (Sorry, I didn't read through everything as carefully as I should have.) So you're getting two commas represented here for the price of one.
> In determining what is best for 3deg212 I agree that 17 commas should be > avoided because of the inconsistency and inaccuracy, but should the primary > interpretation of ~|( be the 17' comma or the 7:13 comma? The only > popularity stats we have, say that ignoring powers of 2 and 3, 17/1 is > twice as popular as 13/7, so 17' should be the primary interpretation. > However these same stats say that 13/7 is slightly more popular than 23/1, > so perhaps we should use ~|( for that reason. 212-ET is at least > 1,3,9,17-consistent. > > Is there any other advantage conferred by using |~ instead of ~|( ?
Not that I can see. It seems that the dual usage for ~|( would argue in its favor, i.e., you should compare the combined usefulness or popularity of 13/7 and 17 against that of 23. So I agree with 212a as the standard symbol set. --George
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Message: 5817 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 18:38:49

Subject: Fwd: Re: Temperament notationn

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> 
wrote:
> George Secor: >
>>> It depends on what method you're using to optimize. The 5-limit >>> minimax generator is a 1/4-comma fifth, which would also favor 31. >>> In order to get the slow-beating minor third of 50-ET you also get >> a
>>> slow-beating major third, plus a faster-beating fifth that is not >>> acceptable to some. So I always thought that 31 was the clear >> choice >>> for meantone. >
> Me too. But some prefer RMS error, and some minimax beating, both of which > favour something closer to 50-ET in the 5-limit.
Since the 5-limit optimal generator size depends on the optimizing method, then it's a tossup between 31 and 50. I believe that the choice of the division that most closely approximates the characterstics of meantone should then be determined by the one that does the 7 limit best. The 7-limit minimax generator is also a fifth narrow by 1/4-comma, and 31-ET wins hands down no matter which method you use.
> Perhaps what we should do is give upper and lower bounds on the size of > generator for which a particular ET notation is valid. Perhaps we > should propose standard notations that fully cover the spectrum of > generator sizes from say 50 cents to 600 cents. This would be for single > chain (octave period) temperaments. We'd have to repeat this for 2,
3, 4, 5
> chain. > > On second thoughts maybe the most popular temperaments would be enough.
If it comes down to a popularity contest, I would say that 31 takes it on that basis, too. Any other ideas that might favor 50? --George
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Message: 5818 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 08:32:44

Subject: Re: Fwd: Re: A common notation for JI and ETss

From: David C Keenan

George Secor:

>Filling out the rational complementation for a complete apotome this >would be: > >212a: |( )|( ~|( /| |) (| (|( //| /|\ (/| (|) ~|| ~||( ) >||~ ||) ||\ (||( ||~) /||) /||\ (DK) > >I have only one question: Since the 17th harmonic is so far off in >212-ET as to be almost midway between tones (and inconsistent >besides), whereas the 23rd is almost exact, would it be more >appropriate to substitute the 23 comma for the 17' comma symbol? >That would give: > >212b: |( )|( |~ /| |) (| (|( //| /|\ (/| (|) ~|| ~||( ) >||~ ||) ||\ ~||) ||~) /||) /||\ (GS) > >But if you still prefer 212a for the standard set, then at least >Margo could use 212b as a modification, since 23 is present in her >tuning.
23 is present in the tuning, but not in those pitches that might be notated with the 3deg212 symbol. Margo and I both chose ~|( because of its interpretation as a 7:13 comma, not a 17 comma. So whatever we might decide for 3deg212, ~|( seems like the right symbol for Margo's tuning. In determining what is best for 3deg212 I agree that 17 commas should be avoided because of the inconsistency and inaccuracy, but should the primary interpretation of ~|( be the 17' comma or the 7:13 comma? The only popularity stats we have, say that ignoring powers of 2 and 3, 17/1 is twice as popular as 13/7, so 17' should be the primary interpretation. However these same stats say that 13/7 is slightly more popular than 23/1, so perhaps we should use ~|( for that reason. 212-ET is at least 1,3,9,17-consistent. Is there any other advantage conferred by using |~ instead of ~|( ? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5819 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 18:56:43

Subject: Re: Minimax generator

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote: >>
>> Would someone explain "minimax" generator (I understand rms generator) >
> Let's take meantone for an example. If we define the three linear functions > > u1 = x - l2(3) > > u2 = 4x - 4 - l2(5) > > u3 = 3x -4 - l2(5/3) > > using "l2" to mean log base 2, then the rms generator can be found > by minimizing u1^2+u2^2+u3^2. On the other hand, we can minimize > |u1| + |u2| + |u3| instead (which gives 1/4 comma meantone.) In fact, > for any p>1, we can minimize |u1|^p + |u2|^p + |u3|^p; if for instance > p is 4, this gives us essentially 1/3 comma meantone (.33365 comma.) > If we consider
It turns out this is so high because of round off error. I redid the calculation using 100 digits of accuracy, and got: p=2 7/26 comma meantone p=4 7/26 comma meantone again p=6 .26295498 comma meantone, about 5/19 comma p=8 .25942006 comma meantone, about 7/27 comma p=10 .25739442 comma meantone, about 9/35 comma p=1 and p=infinity, 1/4 comma meantone
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Message: 5820 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 08:44:29

Subject: Re: Fwd: Re: Temperament notationn

From: David C Keenan

George Secor:

>> It depends on what method you're using to optimize. The 5-limit >> minimax generator is a 1/4-comma fifth, which would also favor 31. >> In order to get the slow-beating minor third of 50-ET you also get >a
>> slow-beating major third, plus a faster-beating fifth that is not >> acceptable to some. So I always thought that 31 was the clear >choice >> for meantone.
Me too. But some prefer RMS error, and some minimax beating, both of which favour something closer to 50-ET in the 5-limit. Perhaps what we should do is give upper and lower bounds on the size of generator for which a particular ET notation is valid. Perhaps we should propose standard notations that fully cover the spectrum of generator sizes from say 50 cents to 600 cents. This would be for single chain (octave period) temperaments. We'd have to repeat this for 2, 3, 4, 5 chain. On second thoughts maybe the most popular temperaments would be enough.
>>
>>> For schismic is it 41 or 53? >>
>> How about 94? >
>Now that I've more time to look at this, I would say definitely 94, >for two reasons: > >1) If you're including the 7th harmonic, then you might as well take >this to the 11 limit (since the minimax generator for both is the >same). The 11th harmonic occurs in the series of fifths in 41, 53, >and 94 in the +23 position, but in 41 it is also closer -- in the -18 >position, which is not typical for the schismic family of >temperaments. So I eliminate 41 as my choice. > >2) The 7 and 11-limit minimax generator is ~702.193c (7:9 being >exact) giving a maximum error of ~4.331c (for 5:7 and 5:9). The 53- >ET fifth is ~701.887c (max. error ~12.681c), but the 94-ET fifth is >much closer to the ideal: ~702.178c (max. error ~4.722c), and the >same may be said for the 13 and 15-limit minimax generator >(~702.109c, 13:14 being exact). So I eliminate 53, leaving 94 as my >choice.
OK. 94 sounds good.
>>> For kleismic is it 53 or 72? >>
>> I'll have to taken a better look at this one. >
>I've done that and I conclude that, unless you are sticking with a 5 >limit, the choice is clearly 72 over 53. It is best to put the >figures in a table to show this: > >Generator Size Max. error Exact > >5-minimax ~316.993c ~1.351c 2:3 >53-ET ~316.981c ~1.408c >72-ET ~316.667c ~2.980c >125-ET ~316.800c ~2.314c > >7,9-minimax ~316.765c ~2.732c 4:7 >53-ET ~316.981c ~6.167c >72-ET ~316.667c ~3.910c >125-ET ~316.800c ~3.088c > >11-minimax ~316.745c ~2.976c 9:11 >53-ET ~316.981c ~12.681c >72-ET ~316.667c ~3.910c >125-ET ~316.800c ~4.892c > >I threw 125 in there also, since it does slightly better than 72 at >the 7 and 9 limit (and also at the 13 and 15 limit, which has the >same minimax generator as for the 7 and 9 limit). But since the 11 >limit is the highest you can go while keeping the max error under 4 >cents, that's the limit I would use, and 72 has the advantage. > >Something else I noticed about the choice of 72 as the notation for >both the Miracle and kleismic temperaments: the progression of >sagittal symbols for a 72-ET panchromatic scale (one passing through >all the tones) is the same as that for a sequence of tones differing >by the generating interval in both temperaments. To illustrate: > >72-ET: C C\! C!) C\!/ B|) B/| B >Miracle: C B\! Bb!) A\!/ G|) F#/| F >kleismic: C A\! F#!) Eb\!/ B|) G#/| F > >The pattern then repeats. I believe that this is a useful property >that provides a further justification for basing the kleismic >notation on 72.
OK. 72 is good. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5821 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 19:45:37

Subject: Re: Minimax generator

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:

> p=2 7/26 comma meantone > > p=4 7/26 comma meantone again > > p=6 .26295498 comma meantone, about 5/19 comma > > p=8 .25942006 comma meantone, about 7/27 comma > > p=10 .25739442 comma meantone, about 9/35 comma > > p=1 and p=infinity, 1/4 comma meantone
To this you may add p=3 (7+sqrt(11))/38 comma meantone, .271490126 comma, a little flatter than the rest of them, but still far from 1/3 comma. If you want an irrational meantone, this seems to be at least as well motivated as golden meantone,and more so that Lucy tuning. Maybe I should claim it and pester everyone to adopt it.
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Message: 5822 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 21:13:09

Subject: Re: Temperament notation

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" 
<clumma@y...> wrote:
> Hi George, >
>> Let me offer an example from my viewpoint that may shed some >> light on some of the issues involved. Carl, you seem to be >> looking at the problem of notation from a keyboardist's >> viewpoint (from your mention of fingerings), but one's >> understanding and framework for a notation must be broader than >> and/or independent of that. >
> Everything I said should apply to any instrument. I used > keyboarding as a 'worst-case' example. >
>> But even if we stick to a keyboard application for the moment, >> let me give an example of how a "transpositionally invariant" >> notation (if I understand the term correctly) could pose more >> problems than it would solve. >
> That was a bad choice of terminology. The issue that Gene and > I are raising is: should notation be based on the melodic scale > being used, or should they be based on the 7-tone meantone > diatonic scale musicians are already familiar with?
The issue that I raise is: how many different notations can you expect a person to learn?
> The issue sort-of assumes the notation will be used to write > what I've been calling "diatonic music" -- music that takes > melodies and primary harmonies from the same small scale, as > 90% of Western music does. >
>> Let's consider the Miracle tuning notated using Graham's decimal >> notation, which I presume you would consider transpositionally >> invariant. >
> Let's nix that term, but yes, Graham's decimal notation is what > I'd advocate for Miracle. Of course.
>> But what value is the decimal notation to me, and what incentive >> would I have to learn it without a decimal keyboard? >
> () It gives you an invaluable tool for understanding the music.
Okay, but only as long as the music is written in the Miracle temperament, yes? (More about this below.)
> () The re-learning won't be as bad as you fear on a decimal > keyboard.
Oh, I have no problem with the idea of learning a new keyboard. But I don't expect that I will ever have a decimal keyboard (since it's *getting* a new keyboard that is the biggest problem, and I've already been through that once). And even if I did, I expect that most of the things that I would play on it wouldn't even be in the Miracle temperament, in which case I pose the question: should everything playable on this keyboard be in decimal notation? (More about this below.)
> () Since your meantone generalized keyboard is at root a planar > hexagonal tiling, many mappings exist to make it more 'decimal'. > It may be that none reflect the secor:octave cycles in the > correct way (as far as the distance of the keys from the player, > etc.), but there should be a way to get the ten nominals of the > decimal scale under the fingers.
The tones would be on a diagonal row of keys that (ascending in pitch) would go off the near edge of the keyboard; but they could be picked up at the far edge, so yes, it can be done without extraordinary effort.
>> Now give me the same decimal keyboard with Partch's 11-limit JI >> mapped onto it (observe that this was the reason that I originally >> came up with the layout). Again, I should do just fine with 72-ET >> sagittal notation, assuming that I am proficient with the >> keyboard. >
> You seem to be saying that it's easier to learn to find pitches > on a keyboard than it is to learn to find pitches in a notation... > For me, it's the opposite.
Actually I do find it easier to perceive the pitch relationships on a generalized keyboard (of whatever sort) than from a notation, but that's not what I was trying to say. The broader point that I was trying to make seems to have gotten lost in all of the details of the discussion. I was trying to show that there is no particular advantage in using decimal notation to notate music that is *not* based on the Miracle geometry (e.g., Partch's music, which is better understood in reference to an 11-limit tonality diamond), but for which the tones may still be very suitably mapped onto a decimal keyboard. The advantage of decimal notation comes into effect only when and if you are using the Miracle temperament itself, i.e., exploiting the tonal relationships that are unique to Miracle. Likewise, if I play something in 31, 41, or 72-ET on a decimal keyboard that was composed by someone utterly ignorant of Miracle as an organizing principle for tonality (as I believe *all* of us were up until a couple of years ago -- myself included), is the decimal notation going to benefit me in any way if the composition which I am playing was not conceived as being decatonic? I think not. These three divisions can be treated as *either* heptatonic *or* decatonic, and I could even show you a unidecatonic MOS subset of 31-ET (with a very useful tetrad that occurs in 5 places). But it would be unrealistic to expect anyone to learn three different notations for one tonal system, according to which tonal relationships are exploited in a given piece. (Or suppose that a piece is heptatonic in one place and decatonic in another. Do we switch notations in the middle of the page?) My point is that alternate tunings often do not tie us down to specific tonal organizations, so the choice of a tuning is often not enough to determine how many nominals would be "best" for its notation. Since we are already acquainted with a notation that uses 7 nominals, and if that works reasonably well for many alternative tunings, then why not have a generalized notation that builds on that? So I would therefore require a microtonal musician to learn no more than one new notation. A composer may wish to do otherwise when composing, but a translation would be provided for the player.
>> If I decide to use the Miracle temperament instead of 11-limit JI >> for Partch's music, is there any advantage in using the decimal >> notation with 10 nominals for this purpose over a 7-nominal >> sagittal notation? >
> Let's ask it this way: take the well-tempered clavier and re- > write it with 6 nominals. Is that only a slight disadvantage?
I was about to say no, but only because 6 nominals will hardly work well with anything, even for Partch (since 6:7:8:9:10:11 isn't a constant structure). But if you can think in terms of a 6+6 keyboard, maybe it isn't that bad after all (until you decide that maybe sharps and flats should be different in pitch). I'm not really arguing against specialized notations with other than 7 nominals, but I don't think that we can expect very many players to learn them.
> Yes, my way means more work for the person interested in learning > multiple scales. But: > > () If you use a generalized keyboard, you could learn 12 scales > with the same amount of work as people spend on learning the > diatonic scale on the piano, excluding the work it takes to > find the pitches in a tuning from blobs on paper... which I seem > to think is easier than you do...
The notes on the paper give you a two-dimensional visual orientation - - placement on the staff (first dimension, but with non-uniform steps) plus altering symbols (second dimension). A generalized keyboard gives you two visual dimensions with uniform steps, plus sound, plus a tactile or kinesthetic experience. But the two together are even better, or lacking a real keyboard, a keyboard diagram labeled with the notation is still good. (I often refer to the decimal keyboard diagram labeled with ratios and 72-ET sagittal notation that I put here in the files section.) I have no doubt that the decimal keyboard and notation together would be easy and even fun to learn, but I wonder whether very many persons would ever have the opportunity to do it.
>> We're all familiar with a 7-nominal / heptatonic / diatonic >> notation, and that's what Dave and I have been building on to >> produce what I call a "generalized" notation -- one that is >> semantically independent of any particular tonal geometry or >> division of the octave. It may not be the best notation for >> everything, but it will do a lot of different things and will >> do many of them extremely well. >
> Definitely a worthwhile project.
I have been wondering whether I should temporarily put the portion of my XH18 article that explains the basics of the sagittal notation in the tuning-math files section so that some of those who have been reading these discussions can take a look and offer comments -- maybe even *use* it. --George
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Message: 5823 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 02:56:09

Subject: Re: Temperament notation

From: Carl Lumma

Hi George,

>Let me offer an example from my viewpoint that may shed some >light on some of the issues involved. Carl, you seem to be >looking at the problem of notation from a keyboardist's >viewpoint (from your mention of fingerings), but one's >understanding and framework for a notation must be broader than >and/or independent of that.
Everything I said should apply to any instrument. I used keyboarding as a 'worst-case' example.
>But even if we stick to a keyboard application for the moment, >let me give an example of how a "transpositionally invariant" >notation (if I understand the term correctly) could pose more >problems than it would solve.
That was a bad choice of terminology. The issue that Gene and I are raising is: should notation be based on the melodic scale being used, or should they be based on the 7-tone meantone diatonic scale musicians are already familiar with? The issue sort-of assumes the notation will be used to write what I've been calling "diatonic music" -- music that takes melodies and primary harmonies from the same small scale, as 90% of Western music does.
>Let's consider the Miracle tuning notated using Graham's decimal >notation, which I presume you would consider transpositionally >invariant.
Let's nix that term, but yes, Graham's decimal notation is what I'd advocate for Miracle.
>But what value is the decimal notation to me, and what incentive >would I have to learn it without a decimal keyboard?
() It gives you an invaluable tool for understanding the music. () The re-learning won't be as bad as you fear on a decimal keyboard. () Since your meantone generalized keyboard is at root a planar hexagonal tiling, many mappings exist to make it more 'decimal'. It may be that none reflect the secor:octave cycles in the correct way (as far as the distance of the keys from the player, etc.), but there should be a way to get the ten nominals of the decimal scale under the fingers.
>Now give me the same decimal keyboard with Partch's 11-limit JI >mapped onto it (observe that this was the reason that I originally >came up with the layout). Again, I should do just fine with 72-ET >sagittal notation, assuming that I am proficient with the >keyboard.
You seem to be saying that it's easier to learn to find pitches on a keyboard than it is to learn to find pitches in a notation... For me, it's the opposite.
>If I decide to use the Miracle temperament instead of 11-limit JI >for Partch's music, is there any advantage in using the decimal >notation with 10 nominals for this purpose over a 7-nominal >sagittal notation?
Let's ask it this way: take the well-tempered clavier and re- write it with 6 nominals. Is that only a slight disadvantage? Yes, my way means more work for the person interested in learning multiple scales. But: () If you use a generalized keyboard, you could learn 12 scales with the same amount of work as people spend on learning the diatonic scale on the piano, excluding the work it takes to find the pitches in a tuning from blobs on paper... which I seem to think is easier than you do...
>We're all familiar with a 7-nominal / heptatonic / diatonic >notation, and that's what Dave and I have been building on to >produce what I call a "generalized" notation -- one that is >semantically independent of any particular tonal geometry or >division of the octave. It may not be the best notation for >everything, but it will do a lot of different things and will >do many of them extremely well.
Definitely a worthwhile project. -Carl
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Message: 5824 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 23:55:54

Subject: Re: Temperament notation

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> > Would someone explain "minimax" generator (I understand rms generator) > > Thanks
Frankly I think "minimax" is a silly term. I prefer to call it the max-absolute (MA) generator. Obviously we're trying to minimise the error measure in both cases, but we don't say "miniRMS". In one case we minimise (I'd prefer to say "optimise") the Root of the Mean of the Squares of the errors and in the other it is the Maximum of the Absolute-values of the errors. RMS and MA.
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