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Message: 5825 - Contents - Hide Contents

Date: Fri, 03 Jan 2003 03:02:52

Subject: Minimax generator

From: Gene Ward Smith

--- 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 (|u1|^p + |u2|^p + |u3|^p)^(1/p) as p tends to infinity, in the limit we get max(|u1|,|u2|,|u3|). Minimizing this gives us the minimum maximum, or minimax--in this case, it is again 1/4 comma meantone.
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Message: 5826 - Contents - Hide Contents

Date: Sat, 04 Jan 2003 08:21:28

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

From: David C Keenan

George Secor:

>Dave Keenan:
>> 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.
Why not 7':11' if it is 11' - 7' ? But see my alternative suggestion below.
>> 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? >>
I'll just note that neither of us have answered the above yet, in case the way I edited things might have made it look like the following was answering it, which of course it is not.
>>> 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.
But this one is notating it as A#-something, not A-something. Sure in meantone you can notate 4:7 as C:A# but you'd only do it because the something happens to vanish, it's a long way up the chain of fifths and Bb \!/ is likely to be more convenient.
>> 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. True.
>> 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.
Certainly not an unusual interval, and we can already notate it. 27:28 from C is of course Db!). But I understand you mean a symbol that represents it as a modified unison. I can see that this might be useful, although I don't think its apotome complement will be of much use.
> 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.
That would be nice, but I would still want to avoid its use as much as possible.
>>> 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!) Agreed.
>> 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
When naming these commas, how about we forget about the details of whether an x:y comma is the sum or difference of x or x' and y or y' but simply parse "x:y'-comma" as "(x:y)'-comma" meaning simply the second (and less important) x:y comma. I noticed recently, an ambiguity in the spoken form of these comma names. To say "x prime comma" can be taken as merely referring to the fact that x is a prime number. Perhaps in the spoken form it would be more useful to say "small x comma" or "big x comma", except that this doesn't correspond directly to primed or unprimed because we have the 5'-comma smaller than the 5-comma while all the others have primed larger than unprimed. We could refer to the unprimed one without using "big" or "small" and use whichever of these applies for the primed one. If really necessary the unprimed could be called "normal" or "standard".
>So have I sold you on a 7' comma, 27:28?
Yes. It's acceptable because it only contains 3^3. I feel there is some sort of comma uselessness metric that increases with both the size of the comma and the number of fifths. A first guess would be to take the product of these. I wouldn't want to add any comma to the system that had this "uselessness" higher than any we've already got. The highest so far is 362 for the 11' comma (|), and a close second is the 25 comma //| at 344. The proposed 7' comma (27:28) is ok at 189, but 57344:59049 is out of the question at 507. I think we should also dump the 31" comma (65536:67797) at a uselessness of 411. Actually, I think the number of fifths should feature more strongly in uselessness, or we could have a sharp cutoff at 9 fifths. But I'm still reluctant re 27:28 because of the added complication of the 5'-comma "flag". No matter how good we can make it look, the fact remains that we haven't had to use it at all for anything else in the 15-limit plus harmonics to 31. So I think it should not be considered part of the "basic" system despite its low prime limit and low product complexity. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5827 - Contents - Hide Contents

Date: Sat, 04 Jan 2003 00:09:27

Subject: Fwd: Re: Temperament notationn

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...>wrote:

> Since the 5-limit optimal generator size depends on the optimizing > method, then it's a tossup between 31 and 50.
One rather rough and ready rule along these lines would be to pick the lastdivision which occurs as a denominator of a convergent for both the rms and minimax generator. This is rough and ready, since we can see from my examples nothing stops two opitmized values from being the same even though the rest cover a range; one could add p=1 and p=4 (both not hard to compute) or simply p=4 to the mix to help fix that. For Miracle, we get 72 in both the 7 and 11 limits, and for meantone, we get 31 in both the 5 and 7 limits (11 limit too, obviously; I didn't bother to compute it since it was clear what the result would be.) For the 7-limit schismic, we have 94, but for 5-limit, it runs all the way up to 289. I suppose 53, 118, 171 and 289, all very similar from a 5-limit schismic point of view, would look completely different when notated sagitally?
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Message: 5828 - Contents - Hide Contents

Date: Sat, 04 Jan 2003 00:09:44

Subject: Fwd: Re: Temperament notationn

From: Dave Keenan

> Any other ideas that might favor 50?
No, but it might make sense to say that for meantones that are on the 1/3-comma side of golden meantone (or some such boundary) a 50-ET-based notation may be preferable.
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Message: 5829 - Contents - Hide Contents

Date: Sat, 04 Jan 2003 01:16:55

Subject: Re: Temperament notation

From: Carl Lumma

>> >hat 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?
As many as he needs to play the music he wants to. Since learning septimal notation gives access to 300 years of Western music, the potential payoff for each new one shouldn't be grounds for complaint.
>>> 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.)
Only if the music is written in a decatonic scale, yes. There are many ways to supply the chromatic pitches outside of Miracle temperament.
>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.
I'd say there's nothing far worse about such a setup than in using the Halberstadt for 12-equal.
>>> 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,
Well, that's an important statement. Since I don't have any experience playing a generalized keyboard of any sort, I have nothing to offer. I can say that learning to read music was easier for me than learning to finger the piano. I really have no idea if this relationship would remain when learning a new keyboard/notation pair.
>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?
No! But there are other temperaments with interesting decatonic scales besides 31, 41, and 72, and all of them would get ten nominals under my pen, and I suspect the differences in accidentals would be easy for performers to learn.
>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.
Perhaps we'll have to agree to disagree.
>(Or suppose that a piece is heptatonic in one place and decatonic >in another. Do we switch notations in the middle of the page?)
Absolutely! Just like switching clefs or key signatures.
>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.
Indeed, any time we leave "diatonic" writing, the need to show scale intervals in the notation disappears. Then what I meant by 'transpositionally invariant' notation would be optimal -- a notation in which acoustic intervals always look the same. In diatonic notation it is scale intervals (2nds, 3rds, etc.) that always look the same. You mention Partch's music, which doesn't really use any fixed melodic scale. I would think transpositionally invariant notation would be optimal, for the scores at least. But Partch had the right idea... since his instruments played different scales (tonality diamond, microchromatic scales, ancient melodic scales, etc.), he notated differently for each of them.
>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?
I am genuinely interested to see how it looks. You should debut it with sample music, both original and classical.
>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.
A distinct posibility.
>> 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,
They'd work spledidly for wholetone music. It's fortunate that 12-equal supports the wholetone scale without collisions. But if you look at octatonic music, it would be much better notated with 8 nominals. I would go so far as to suggest that this held back octatonic music in the last century.
>even for Partch (since 6:7:8:9:10:11 isn't a constant structure).
The diatonic scale in 12-equal isn't a constant structure either. Actually, it's strict propriety that's important, and the violations aren't too bad here, and I think 6 nominals would be ideal. But Partch doesn't really stick to this scale, so...
>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.
They'll learn them if there's cool music written in them. Learning any notation is a tremendously difficult problem for humans, but they never cease to amaze me.
>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.
Did you see any of the virtual keyboard projector posts I've made to the main list over the past year? -Carl
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Message: 5830 - Contents - Hide Contents

Date: Sat, 04 Jan 2003 01:58:49

Subject: Fwd: Re: Temperament notationn

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...>wrote:

> Any other ideas that might favor 50?
Here's one that comes close, but instead decides that the 81-et is right for the 5-limit. Has anyone even proposed using it? For this definition, we compute the optimal rms, minimax and least fourth powers generators. We then take the smallest et which gives us a generator in the interval from the minimum to the maximum of the above values. In the case of 5-limit meantone, this is 81, though 50 comes close--very very close if we toss in the least cubes value as well, since 50 is pretty well identical to my world-famous (7+sqrt(11))/38-comma meantone (which everyone must and shall use) in practice. We could increase the size of the interval by some percentage if we wanted,of course.
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Message: 5831 - Contents - Hide Contents

Date: Sat, 04 Jan 2003 02:36:44

Subject: Fwd: Re: Temperament notationn

From: Gene Ward Smith

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

> For this definition, we compute the optimal rms, minimax and least fourthpowers generators. We then take the smallest et which gives us a generatorin the interval from the minimum to the maximum of the above values.
This method works fine as a way of choosing an et for a temperament for computer music purposes, but is a little over the top as a way to define a notation et to associate to that temperament. In the case of miracle, it gives175 for 7-limit miracle, which I've already been using for computer scores. For the 11-limit, it pulls the 401 et from out of left field, despite thefact that 72 in practice is more or less the same (the difference between 39/401 and 7/72 being all of 50/1203 = 0.04156 cents.)
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Message: 5832 - Contents - Hide Contents

Date: Sat, 04 Jan 2003 19:27:43

Subject: Re: Temperament notation

From: David C Keenan

At 12:12 AM 4/01/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith ><genewardsmith@j...>" <genewardsmith@j...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote: >
>> Since the 5-limit optimal generator size depends on the optimizing >> method, then it's a tossup between 31 and 50. >
>One rather rough and ready rule along these lines would be to pick the >last division which occurs as a denominator of a convergent for both >the rms and minimax generator.
That sounds like an excellent idea!
>For Miracle, we get 72 in both the 7 and 11 limits, and for meantone, >we get 31 in both the 5 and 7 limits (11 limit too, obviously; I >didn't bother to compute it since it was clear what the result would >be.) For the 7-limit schismic, we have 94, but for 5-limit, it runs >all the way up to 289. I suppose 53, 118, 171 and 289, all very >similar from a 5-limit schismic point of view, would look completely >different when notated sagitally?
As a matter of fact there is a sagittal notation that agrees with 118, 171 and 289 and is also compatible with 53 and 94. In the two-symbol form, it strongly resembles a linear-temperament-specific notation with 12 nominals, like the decimal notation for miracle. This is possible because the generator happens to be a fifth and the chroma happens to be the 5-comma (80;81). A chain of 60 notes looks like this. Eb\\! Bb\\! F\\! C\\! G\\! D\\! A\\! E\\! B\\! F#\\! C#\\! G#\\! Eb\! Bb\! F\! C\! G\! D\! A\! E\! B\! F#\! C#\! G#\! Eb Bb F C G D A E B F# C# G# Eb/| Bb/| F/| C/| G/| D/| A/| E/| B/| F#/| C#/| G#/| Eb//| Bb//| F//| C//| G//| D//| A//| E//| B//| F#//| C#//| G#//| The reason I say it's only "compatible" with 53 and 94 is because the standard sets for these do not use the 25-comma symbol //| I suggest this notation (or its single-symbol counterpart) should be used for all open schismics of up to 60 notes. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5833 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 08:31:25

Subject: Re: Temperament notation

From: David C Keenan

Notice that this sagittal notation for schismic depends only on the 
temperament's mapping from generators to primes and not on any particular 
range of generator sizes. The fact that it only uses 5-limit comma symbols 
(if //| is taken as a double 5-comma rather than a 25 comma) means that it 
is valid for schismic at all odd limits.

Eb\\! Bb\\! F\\! C\\! G\\! D\\! A\\! E\\! B\\! F#\\! C#\\! G#\\!
Eb\!  Bb\!  F\!  C\!  G\!  D\!  A\!  E\!  B\!  F#\!  C#\!  G#\!
Eb    Bb    F    C    G    D    A    E    B    F#    C#    G#
Eb/|  Bb/|  F/|  C/|  G/|  D/|  A/|  E/|  B/|  F#/|  C#/|  G#/|
Eb//| Bb//| F//| C//| G//| D//| A//| E//| B//| F#//| C#//| G#//|

This points the way to similar multi-symbol sagittal notations for other 
linear temperaments (LTs). As well as being as independent of generator 
size as possible and hence independent of any particular ET (which may not 
be fully acheivable when the generator is not a fifth) it will make the 
sagittal notation for an LT resemble as closely as possible the ideal 
notation where more than 7 nominals are allowed.

You first decide how many nominals there should be. Call this N. Then 
examine the available symbol commas in order of popularity, applying the 
LT's primes-to-generators mapping, until a symbol is found that corresponds 
to a chain of N generators (the chroma), also possibly 2N, 3N etc.

Naming the notes of the central chain is another matter which I haven't 
worked out in general, except by using an ET.

For example, in the case of Miracle, we would want 10 nominals and we find 
that |) as the 7-comma 63;64 corresponds to the chroma.

C     C#/|  D|)   Eb/|\ E||)  F||\  G     G#/|  A|)   Bb/|\
C!)   C#\!  D     Eb|\  E|)   F/|\  G!)   G#\!  A     Bb/|
C!!)  C#\!/ D!)   Eb!/  E     F/|   G!!)  G#\!/ A!)   Bb\!

The following is perhaps a more natural sagittal notation (based on 72-ET), 
but makes no attempt to look like it has 10 nominals.

C     Db/|  D|)   E\!/  F!)   F#\!  G     Ab/|  A|)   B\!/
C!)   C#\!  D     Eb|\  E|)   F/|\  G!)   G#\!  A     Bb/|
B|)   C/|\  D!)   D#!/  E     F/|   Gb|)  G/|\  A!)   A#\!
B
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)


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Message: 5834 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 21:55:33

Subject: Some improved poptimal generators

From: Gene Ward Smith

I wrote code for least-cubes optimization for linear temperaments, and
used it to improve a few of the values. I was surprised by how often
the least cubes value allowed me to slightly enlarge the interval of
known poptimal generators, but for the most part this made no
difference. It did, however, in three instances; and I was especially
happy to find that the 94-et is poptimal for schismic.

Quartaminorthirds [9, 5, -3, -21, 30, -13]

[1, 16/247], ==> [1, 9/139]


Supermajor seconds [3, 12, -1, -36, 10, 12]

[1, 59/305], ==> [1, 53/274]


Schismic [1, -8, -14, -10, 25, -15]

[1, 134/229], ==> [1, 55/94]


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Message: 5835 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 08:40:46

Subject: Re: Temperament notation

From: David C Keenan

At 02:32 AM 4/01/2003 +0000, wallyesterpaulrus 
<wallyesterpaulrus@xxxxx.xxx> wrote:
>dave, > >MAD means "Mean Absolute Deviation" > >which is another error criterion yet. > >so MA would breed confusion.
OK. I admit defeat on the acronym. But I'd still prefer to call it max-absolute rather than minimax.
>--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" ><d.keenan@u...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" >> >>
>>> 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.
-- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5836 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 15:42:45

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Here's my latest suggestion regarding symbolising the 5'-comma (5-schisma) 
up and down:

Make them separate symbols. Like an accent mark on a character but placed 
beside the associated arrow symbol, not above or below it. To which side? I 
haven't decided, but currently favour the left, at least in scores (as 
opposed to in text).

See Yahoo groups: /tuning-math/files/Dave/5Schisma... * [with cont.] 
for some examples.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)


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Message: 5837 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 05:46:19

Subject: Poptimal generators

From: Gene Ward Smith

"Poptimal" is short for "p-optimal". The p here is a real variable
p>=2, which is what analysts normally use when discussing these Holder
type normed linear spaces.

A pair of generators [1/n, x] for a linear temperament is *poptimal* if there is some p, 2 <= p <=
infinity, such that x is precisely optimal according to the
sum-of-absolute-values-of-pth-powers definition of optimal, and where
p=infinity is shorthand for minimax.

We can show a generator is poptimal by using a continuity argument;
the sum of pth powers is a sum

SUM abs(ai/n + bi*x - l2(consonant interval))^p

over the consonances of the tuning. It is twice continuously
differentiable with respect to x, and the second derviative is
positive, since not all of the error terms can be zero. Hence it has a
unique minimum, which is the p-optimized tuning value.

If we denote by F(x, p) the above derivative with respect to x, we can
check that it is continuously differentiable with respect to both x
and p (it might look as if the p derivative blows up, but
|x|^n log|x| is dominated by x^n near zero for n>0.) We also have that
the partial is never zero, since the partial with respect to x,
by the above, is positive. Hence the conditions of the implicit
function theorem obtain, and we can not only find x as a continuous
function of p, we have p as a continuous function of x. The set of
poptimal generators is therefore convex--if x1 and x2 are poptimal, so
is any x such that x1<x<x2. 

Since p=2 can be computed by least squares, p=infinity by setting it
up as a linear programming problem, and p=4 by solving a cubic
polynomial with a single real root, they are easy to compute. If x2,
x4, xinf are the three values we obtain, we can find the interval
[xa, xb] where xa = min(x2,x4,xinf) and xb = max(x2, x4, xinf).
Usually, this interval will be more than a single point, and so we can
find a rational poptimal generator. This is a generator which can
reasonably claim to be *precisely* optimal, since it is optimal by
some reasonable definition of "optimal", and yet it is an interval of
an equal temperament.

In some cases, a rational poptimal generator does not exist, since
there is only one unique poptimal generator. This can happen for "funky" temperaments where two of the consonances are identified.
I don't know if it can happen otherwise, and might try to prove that
impossible.

Below are poptimal generators for 44 7-limit linear temperaments. In
many cases they will be the rational poptimal generator of smallest
height, but this is not assured, since they were calculated by the x2,
x4, xinf method.


"Dominant seventh" [1, 4, -2, -16, 6, 4] [1, 24/41]

"Diminished" [4, 4, 4, -2, 5, -3] [1/4, 1/16]

"Blackwood" [0, 5, 0, -14, 0, 8] [1/5, 3/25]

"Augmented" [3, 0, 6, 14, -1, -7] [1/3, 8/33]

"Pajara" [2, -4, -4, 2, 12, -11] [1/2, 5/56]

"Hexadecimal" [1, -3, 5, 20, -5, -7] [1, 23/41]

"Tertiathirds" [4, -3, 2, 13, 8, -14] [1, 5/48]

"Kleismic" [6, 5, 3, -7, 12, -6] [1, 14/53]

"Tripletone" [3, 0, -6, -14, 18, -7] [1/3, 2/27]

"Hemifourth" [2, 8, 1, -20, 4, 8] [1, 4/19]

"Meantone" [1, 4, 10, 12, -13, 4] [1, 119/205]

"Injera" [2, 8, 8, -4, -7, 8] [1/2, 1/13]

"Double wide" [8, 6, 6, -3, 13, -9] [1/2, 7/26]

"Porcupine" [3, 5, -6, -28, 18, 1] [1, 8/59]

"Superpythagorean" [1, 9, -2, -30, 6, 12] [1, 45/76]

"Muggles" [5, 1, -7, -19, 25, -10] [1, 62/197]

"Beatles" [2, -9, -4, 16, 12, -19] [1, 19/64]

"Flattone" [1, 4, -9, -32, 17, 4] [1, 63/109]

"Magic" [5, 1, 12, 25, -5, -10] [1, 13/41]

"Nonkleismic" [10, 9, 7, -9, 17, -9] [1, 23/89]

"Semisixths" [7, 9, 13, 5, -1, -2] [1, 44/119]

"Orwell" [7, -3, 8, 27, 7, -21] [1, 26/115]

"Miracle" [6, -7, -2, 15, 20, -25] [1, 17/175]

"Quartaminorthirds" [9, 5, -3, -21, 30, -13] [1, 16/247]

"Supermajor seconds" [3, 12, -1, -36, 10, 12] [1, 59/305]

"Schismic" [1, -8, -14, -10, 25, -15] [1, 134/229]

"Superkleismic" [9, 10, -3, -35, 30, -5] [1, 37/138]

"Squares" [4, 16, 9, -24, -3, 16] [1, 93/262]

"Semififth" [2, 8, -11, -48, 23, 8] [1, 119/410]

"Diaschismic" [2, -4, -16, -26, 31, -11] [1/2, 9/104]

"Octafifths" [8, 18, 11, -25, 5, 10] [1, 13/177]

"Tritonic" [5, -11, -12, 3, 33, -29] [1, 163/337]

"Supersupermajor" [3, 17, -1, -50, 10, 20] [1, 25/128]

"Shrutar" [4, -8, 14, 55, -11, -22] [1/2, 4/91]

"Catakleismic" [6, 5, 22, 37, -18, -6] [1, 52/197]

"Hemiwuerschmidt" [16, 2, 5, 6, 37, -34] [1, 37/229]

"Hemikleismic" [12, 10, -9, -49, 48, -12] [1, 25/189]

"Hemithird" [15, -2, -5, -6, 50, -38] [1, 24/149]

"Wizard" [12, -2, 20, 52, 2, -31] [1/2, 23/72]

"Duodecimal" [0, 12, 24, 22, -38, 19] [1/12, 1/76]

"Slender" [13, -10, 6, 42, 27, -46] [1, 5/156]

"Amity" [5, 13, -17, -76, 41, 9] [1, 155/548]

"Hemififth" [2, 25, 13, -40, -15, 35] [1, 70/239]

"Ennealimmal" [18, 27, 18, -34, 22, 1] [1/9, 43/612]


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Message: 5838 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 06:03:22

Subject: Re: Poptimal generators

From: Dave Keenan

I think I actually followed most of that, and what's more, found it
very interesting. Thanks. 

But with regard to sagittal notation for these temperaments, some of
those results looked pretty wild, such as 205-ET for 7-limit Meantone.

What do you mean by rational generator of minimum height? What is
"height" here?


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Message: 5839 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 06:06:01

Subject: Re: Poptimal generators

From: Gene Ward Smith

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

> "Injera" [2, 8, 8, -4, -7, 8] [1/2, 1/13] > "Duodecimal" [0, 12, 24, 22, -38, 19] [1/12, 1/76]
Maple, which I used to print this out, automatically reduces fractions to their lowest terms, so you need to take the lcm of the denomiators to find the et--for Injera, it would be 26, for Duodecimal, 228.
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Message: 5840 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 06:09:26

Subject: Re: Poptimal generators

From: Gene Ward Smith

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

> But with regard to sagittal notation for these temperaments, some of > those results looked pretty wild, such as 205-ET for 7-limit Meantone.
I agree, that is way out there, but I think these are worth considering. They certainly make sense for computer music, at least, if not for notational purposes.
> What do you mean by rational generator of minimum height? What is > "height" here?
Size of denominator, or numerator times denomiator, etc etc. It doesn't really matter which one you choose. "Height" is what number theorists call a function which measures the complexity of a rational (or algebraic) number.
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Message: 5841 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 06:24:29

Subject: Re: Poptimal generators

From: Gene Ward Smith

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

> But with regard to sagittal notation for these temperaments, some of > those results looked pretty wild, such as 205-ET for 7-limit Meantone.
Incidentally, the convergents for the log base 2 of the fourth root of five, which is the 1/4-comma meantone fifth, are as follows: 1, 1/2, 3/5, 4/7, 7/12, 11/19, 18/31, 101/174, 119/205, 458/789, 1493/2572, 1951/3361, 5395/9294, 7346/12655, 56817/97879 ... Whether this justifies using 119/205 for notating meantone may be doubted, but better it than than 56817/97879, also poptimal.
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Message: 5842 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 06:56:43

Subject: Re: Poptimal generators

From: Gene Ward Smith

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

> Whether this justifies using 119/205 for notating meantone may be doubted, but better it than than 56817/97879, also poptimal.
I used my r3 value of (7+sqrt(11))/38-comma meantone, and found that 47/81 is poptimal for 7-limit meantone. Since it was also what I got for a poptimal 5-limit meantone, it is looking good. Forget (7+sqrt(11))/38-comma meantone; I think I'll opt for this as my personal meantone. It's about 5/19-comma, or 1/3.8 comma if you prefer, meantone. You could reasonably use it to notate meantone if you wanted. Of course the next theory I expect to hear is that we *should* be looking at weighted poptimal generators, which would probably be enough to bring 31-et back into the fold.
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Message: 5843 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 07:19:30

Subject: Re: Poptimal generators

From: Gene Ward Smith

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

> I used my r3 value of (7+sqrt(11))/38-comma meantone, and found that > 47/81 is poptimal for 7-limit meantone. Since it was also what I got for a poptimal 5-limit meantone, it is looking good. Forget > (7+sqrt(11))/38-comma meantone; I think I'll opt for this as my personal meantone. It's about 5/19-comma, or 1/3.8 comma if you prefer, meantone.
The same r3 value allows us to determine that Golden Meantone is in the rather small range of values which are poptimal in both the 5 and 7 limits. Notating it is a whole other story, of course; I presume one would use ordinary sharps and flats--and just keep on using them.
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Message: 5844 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 10:13:26

Subject: Re: Poptimal generators

From: Gene Ward Smith

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

I
think I need to get some sleep; neither 81 nor, sadly, Golden Meantone
can be added in the 7-limit by using r3. The 205-et I gave for 7-limit
Meantone may be the best that can be done. Of course, I could get some
rather different ones by moving on up to the 9-limit.


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Message: 5845 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 10:17:03

Subject: Re: Poptimal generators

From: Carl Lumma

>"Height" is what number theorists call a function >which measures the complexity of a rational (or >algebraic) number.
What common feature would all such functions share? IOW, how would they define "complexity"? -C.
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Message: 5846 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 10:31:59

Subject: Re: Poptimal generators

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> What common feature would all such functions share?
Roughly, the bigger the coefficients of the defining polynomial of the algebraic number, the greater the height; but of course there are different ways to measure it. Common is the sum of absolute values of the coefficients, but there are height functions with nifty properties like Mahler height or Silverman height. I've never thought about musical applications, but we don't usually worry about algebraic numbers anyway.
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Message: 5847 - Contents - Hide Contents

Date: Sun, 05 Jan 2003 10:32:50

Subject: Scale question

From: Gene Ward Smith

By the way, Carl, did you see my question about scale theory resources?


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Message: 5848 - Contents - Hide Contents

Date: Mon, 06 Jan 2003 19:38:53

Subject: Improved generators for 7-limit linear temperaments

From: Gene Ward Smith

Since 9-odd-limit and 7-odd-limit involve the same set of primes, we
can justify using either. If we regard them both as important, we can
justify using anything between the ranges of the poptimal generators
even when these are disjoint. To be sure, when 3 has high complexity
we might be interested in 7 but not 9, but I think this is pretty much
of a quibble. I calculated a 9-limit poptimal, (the second listed
below) and then found the best which can be had by putting 7 and 9
together. The "univeral" generators listed below are where all the p
values give the same result; since this didn't happen for both 7 and 9
I still got an et.

At this point I would say things are generally looking good. We now
have 72 for Miracle, 84 for Orwell, 12 for Diminished and Dominant
Seventh, 22 for Porcupine, 31 for Semififths and Supermajor Seconds, 
41 for Superkleismic and 46 for Semisixths. The 112-et for Meantone is
still a little over the top, but is at least worth considering; the
tendency for Meantone to want a generator in this very small interval
is rather striking.


            pops:=[
            "Decimal",
            [4, 2, 2, -1, 8, -6], [[2, 0, 3, 4], [0, 2, 1, 1]],

          
            [1/2 l2(4/3)), "universal"], [1/2, 5/24], [1/2, 5/24],




            "Dominant seventh",
            [1, 4, -2, -16, 6, 4], [[1, 0, -4, 6], [0, 1, 4, -2]],

            [1, 24/41], [1, 7/12], [1, 7/12],


            "Diminished",
            [4, 4, 4, -2, 5, -3], [[4, 0, 3, 5], [0, 1, 1, 1]],

            [1/4, 1/16], [1/4, l2(3 sqrt(2)/4), "universal"], [1/4,
1/12],



            "Blackwood",
            [0, 5, 0, -14, 0, 8], [[5, 8, 0, 14], [0, 0, 1, 0]],

           [1/5, 3/25], [1/5, 1/15], [1/5, 1/15],


            "Augmented",
            [3, 0, 6, 14, -1, -7], [[3, 0, 7, -1], [0, 1, 0, 2]],

            [1/3, 8/33], [1/3, 11/45], [1/3, 8/33],



            "Pajara",
            [2, -4, -4, 2, 12, -11], [[2, 0, 11, 12], [0, 1, -2, -2]],

            
            [1/2, 5/56], [1/2, 5/56], [1/2, 5/56],


            "Hexadecimal",
            [1, -3, 5, 20, -5, -7], [[1, 0, 7, -5], [0, 1, -3, 5]], 

            [1, 23/41], [1, 41/73], [1, 23/41],


            "Tertiathirds",
            [4, -3, 2, 13, 8, -14], [[1, 2, 2, 3], [0, 4, -3, 2]],

            [1, 5/48], [1, 5/48], [1, 5/48],


            "Kleismic",
            [6, 5, 3, -7, 12, -6], [[1, 0, 1, 2], [0, 6, 5, 3]],

            
            [1, 14/53], [1, 14/53], [1, 14/53],


            "Tripletone",
            [3, 0, -6, -14, 18, -7], [[3, 0, 7, 18], [0, 1, 0, -2]],

          

            [1/3, 2/27], [1/3, 3/33], [1/3, 1/12],


            "Hemifourths",
            [2, 8, 1, -20, 4, 8], [[1, 0, -4, 2], [0, 2, 8, 1]],

           

            [1, 4/19], [1, 4/19], [1, 4/19],


            "Meantone",
            [1, 4, 10, 12, -13, 4], [[1, 0, -4, -13], [0, 1, 4, 10]],

          
            [1, 119/205], [1, 65/112], [1, 65/112],


            "Injera",
            [2, 8, 8, -4, -7, 8], [[2, 0, -8, -7], [0, 1, 4, 4]],

           

            [1/2, 2/26], [1/2, 2/26], [1/2, 2/26],


            "Double wide",
            [8, 6, 6, -3, 13, -9], [[2, 1, 3, 4], [0, 4, 3, 3]],

           
            [1/2, 7/26], [1/2, 13/48], [1/2, 7/26],


            "Porcupine",
            [3, 5, -6, -28, 18, 1], [[1, 2, 3, 2], [0, 3, 5, -6]],

          

            [1, 8/59], [1, 3/22], [1, 3/22],


            "Superpythagorean",
            [1, 9, -2, -30, 6, 12], [[1, 0, -12, 6], [0, 1, 9, -2]],

          
            [1, 45/76], [1, 29/49], [1, 29/49],


            "Muggles",
            [5, 1, -7, -19, 25, -10], [[1, 0, 2, 5], [0, 5, 1, -7]],

          

            [1, 62/197], [1, 53/168], [1, 17/54],


            "Beatles",
            [2, -9, -4, 16, 12, -19], [[1, 1, 5, 4], [0, 2, -9, -4]],

           

            [1, 19/64], [1, 8/27], [1, 8/27],


            "Flattone",
            [1, 4, -9, -32, 17, 4], [[1, 0, -4, 17], [0, 1, 4, -9]],

          

            [1, 63/109], [1, 37/64], [1, 37/64],


            "Magic",
            [5, 1, 12, 25, -5, -10], [[1, 0, 2, -1], [0, 5, 1, 12]],

          

            [1, 13/41], [1, 71/224], [1, 13/41],


            "Nonkleismic",
            [10, 9, 7, -9, 17, -9], [[1, 9, 9, 8], [0, 10, 9, 7]],


            [1, 23/89], [1, 38/147], [1, 23/89],


            "Semisixths",
            [7, 9, 13, 5, -1, -2], [[1, 6, 8, 11], [0, 7, 9, 13]],

            

            [1, 44/119], [1, 17/46], [1, 17/46], 


            "Orwell",
            [7, -3, 8, 27, 7, -21], [[1, 0, 3, 1], [0, 7, -3, 8]],

          
            [1, 26/115], [1, 43/190], [1, 19/84],


            "Miracle",
            [6, -7, -2, 15, 20, -25], [[1, 1, 3, 3], [0, 6, -7, -2]],

          

            [1, 17/175], [1, 32/329], [1, 7/72],


            "Quartaminorthirds",
            [9, 5, -3, -21, 30, -13], [[1, 1, 2, 3], [0, 9, 5, -3]],

           

            [1, 9/139], [1, 17/262], [1, 7/108],


            "Supermajor seconds",
            [3, 12, -1, -36, 10, 12], [[1, 1, 0, 3], [0, 3, 12, -1]],

           
            [1, 53/274], [1, 6/31], [1, 6/31],


            "Schismic",
            [1, -8, -14, -10, 25, -15], [[1, 0, 15, 25], [0, 1, -8,
-14]],

            
            [1, 55/94], [1, 55/94], [1, 55/94],


            "Superkleismic",
            [9, 10, -3, -35, 30, -5], [[1, 4, 5, 2], [0, 9, 10, -3]],

            

            [1, 37/138], [1, 106/395], [1, 11/41],


            "Squares",
            [4, 16, 9, -24, -3, 16], [[1, 3, 8, 6], [0, -4, -16, -9]],

          
            [1, 93/262], [1, 104/293], [1, 93/262],


            "Semififths",
            [2, 8, -11, -48, 23, 8], [[1, 1, 0, 6], [0, 2, 8, -11]],

            

            [1, 119/410], [1, 9/31], [1, 9/31],


            "Diaschismic",
            [2, -4, -16, -26, 31, -11], [[2, 0, 11, 31], [0, 1, -2,
-8]],

          
            [1/2, 9/104], [1/2, 9/104], [1/2, 9/104],


            "Octafifths",
            [8, 18, 11, -25, 5, 10], [[1, 1, 1, 2], [0, 8, 18, 11]],

          

            [1, 13/177], [1, 8/109], [1, 8/109], 


            "Tritonic",
            [5, -11, -12, 3, 33, -29], [[1, 4, -3, -3], [0, 5, -11,
-12]],

         

            [1, 163/337], [1, 191/395], [1, 59/122],


            "Supersupermajor",
            [3, 17, -1, -50, 10, 20], [[1, 1, -1, 3], [0, 3, 17, -1]],

           

            [1, 25/128], [1, 25/128], [1, 25/128],


            "Shrutar",
            [4, -8, 14, 55, -11, -22], [[2, 1, 9, -2], [0, 2, -4, 7]],

           
            [1/2, 8/182], [1/2, 9/205], [1/2, 8/182],


            "Catakleismic",
            [6, 5, 22, 37, -18, -6], [[1, 0, 1, -3], [0, 6, 5, 22]],

         

            [1, 52/197], [1, 52/197], [1, 52/197],


            "Hemiwuerschmidt",
            [16, 2, 5, 6, 37, -34], [[1, 15, 4, 7], [0, 16, 2, 5]],

           

            [1, 37/229], [1, 37/229], [1, 37/229],


            "Hemikleismic",
            [12, 10, -9, -49, 48, -12], [[1, 0, 1, 4], [0, 12, 10,
-9]],

           

            [1, 25/189], [1, 62/469], [1, 16/121],


            "Hemithirds",
            [15, -2, -5, -6, 50, -38], [[1, 4, 2, 2], [0, -15, 2, 5]],

          

            [1, 24/149], [1, 43/267], [1, 24/149],


            "Wizard",
            [12, -2, 20, 52, 2, -31], [[2, 1, 5, 2], [0, 6, -1, 10]],

          

            [1/2, 23/72], [1/2, 99/310], [1/2, 23/72],


            "Duodecimal",
            [0, 12, 24, 22, -38, 19], [[12, 19, 0, -22], [0, 0, 1,
2]],

           
            [1/12, 3/228], [1/12, 3/228], [1/12, 3/228],


            "Slender",
            [13, -10, 6, 42, 27, -46], [[1, 2, 2, 3], [0, 13, -10,
6]],

         

            [1, 5/156], [1, 4/125], [1, 4/125],


            "Amity",
            [5, 13, -17, -76, 41, 9], [[1, 3, 6, -2], [0, 5, 13,
-17]],

           

            [1, 155/548], [1, 99/350], [1, 99/350],


            "Hemififths",
            [2, 25, 13, -40, -15, 35], [[1, 1, -5, -1], [0, 2, 25,
13]],

         
            [1, 70/239], [1, 70/239], [1, 70/239],


            "Ennealimmal",
            [18, 27, 18, -34, 22, 1], [[9, 1, 1, 12], [0, 2, 3, 2]],

          

            [1/9, 43/612]], [1/9, 227/3231], [1/9, 43/612]]:


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Message: 5849 - Contents - Hide Contents

Date: Mon, 06 Jan 2003 13:40:01

Subject: Re: A common notation for JI and ETs

From: David C Keenan

In case anyone has already looked at the .bmp for my latest suggestion 
regarding symbolising the 5'-comma (5-schisma) up and down:

It was riddled with vertical alignment errors so I've had another go at it.

See Yahoo groups: /tuning-math/files/Dave/5Schisma... * [with cont.] 
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)


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