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Message: 5775 - Contents - Hide Contents

Date: Fri, 20 Dec 2002 18:02:29

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> 
wrote:
> Welcome to this discussion Margo. > > Here's my suggestion for notating Peppermint-24. > > I will use "L" for the sagittal 7-comma-down symbol which has previously > been ASCIIfied as "!)", and "^" for the sagittal 11-diesis-up symbol which > has previously been ASCIIfied as "/|\". > > The lower chain of 704.1 cent fifths is notated > F C G D A E B F# C# G# D# A# > The upper chain, a 58.7 cent quasi-diesis above it, is notated > F^ C^ G^ D^ A^ FL CL GL DL AL EL BL > with FL and CL optionally being notated as > E^ B^
This essentially agrees with what I suggested as a "quasi-217 mapping," and it is good that this can be more firmly established by identifying a proper ET mapping.
> This is based on 121-ET, so other "enharmonic" spellings are possible, and > the two chains may be extended in both directions well beyond 12 notes (or > other chains added) without two different notes having the same name.
Upon examining 121, I found that ratios of 13 aren't compatible with ratios of 7 for the notation (as often occurs with ETs). For Margo's benefit I will explain: /| is 3deg and |) is 2deg, so /|) would be 5deg by adding the flags, but the 13 comma should be 6deg121. So one must make a choice between the 7 and 13-comma symbols for the symbol set. Even though 13 has less error in 121-ET than either 7 or 11, 7 is preferred over 13 because 1) it's a lower prime, and 2) the 11 and 11' commas are the same number of degrees as the 13 and 13' commas, respectively, so the symbols for a lower prime, 11, can be used for both 11 and 13 (which is appropriate, since 351:352 vanishes in both 121-ET and Peppermint). The one thing that differs from what I suggested is this: Should Margo would want to respell E (~14/11 relative to C) as F-something, then the symbol I suggested, using the 7:11 comma -- Fb(| or F)!!~ -- is not in the 121 standard set that Dave has proposed (below). His set has the 5:11 comma, which gives Fb(|( or F~!!(. Either one of these is valid for 5deg121, but since Peppermint lacks ratios of 5, Margo might prefer the 7:11 comma symbol. Something to consider when we're mapping something to an ET is that the standard symbol set might not be the "best," i.e., the most meaningful or useful one.
> I couldn't find anywhere that we agreed on sagittal notation for 121-ET. So > here's my proposal for its single shaft symbols. > > 121: )| |) /| (|( (|~ /|\ (|)
As you have probably concluded by now, I agree with this for the standard set. With rational complements this becomes: 121: )| |) /| (|( (|~ /|\ (|) )|| ~||( ||\ ||) (||~ /||\ I will add this ET to Table 4 in the paper. For Peppermint my suggested modification is: 121: )| |) /| (| (|~ /|\ (|) )|| )||~ ||\ ||) (||~ /||\ --George
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Message: 5776 - Contents - Hide Contents

Date: Sun, 22 Dec 2002 09:04:44

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 10:04 AM 20/12/2002 -0800, George Secor wrote:
>The one thing that differs from what I suggested is this: Should Margo >would want to respell E (~14/11 relative to C) as F-something,
Why would she want to do that? Do you want to do that, Margo?
>then the >symbol I suggested, using the 7:11 comma -- Fb(| or F)!!~ -- is not in >the 121 standard set that Dave has proposed (below). His set has the >5:11 comma, which gives Fb(|( or F~!!(. Either one of these is valid >for 5deg121, but since Peppermint lacks ratios of 5, Margo might prefer >the 7:11 comma symbol. Something to consider when we're mapping >something to an ET is that the standard symbol set might not be the >"best," i.e., the most meaningful or useful one.
Might not be, but in this case I think it is, simply because there will never be any call for the 4deg121 symbol in notating Peppermint unless the chains are long enough that there _are_ ratios of 5. But even if Margo did want to do as you suggest and notate the approx 11:14 as a kind of diminished fourth rather than a kind of major third, I don't believe that your proposed use of (| or )||~ above are valid in the sagittal system. Read on.
>> I couldn't find anywhere that we agreed on sagittal notation for >121-ET. So
>> here's my proposal for its single shaft symbols. >> >> 121: )| |) /| (|( (|~ /|\ (|) >
>As you have probably concluded by now, I agree with this for the >standard set. With rational complements this becomes: > >121: )| |) /| (|( (|~ /|\ (|) )|| ~||( ||\ ||) (||~ /||\ Agreed. >I will add this ET to Table 4 in the paper. >For Peppermint my suggested modification is: > >121: )| |) /| (| (|~ /|\ (|) )|| )||~ ||\ ||) (||~ /||\
By my calculations, (| as the 7:11' comma (45056:45927) is 5deg121, not 4deg121 as you have it above. 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). We had better check all existing uses of )|( for agreement with this. I don't think )|( appears in the article. But I guess we should check for other useful 15-limit dual-prime commas we may have missed. Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5777 - Contents - Hide Contents

Date: Mon, 23 Dec 2002 12:00:16

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Manuel Op de Coul has kindly written a program that went thru the entire 
Scala archive to tell us what are the most popular rational pitches and how 
many times each occurs.

Actually the information we need for this notation effort is a little  more 
complicated than that. Factors of 2 and 3 are irrelevant, and so are 
inversions, in deciding what commas we should have symbols for. For example 
5/4, 8/5, 5/3, 6/5, 10/9, 9/5, 27/20, 40/27 etc will all use the same 
sagittal symbol (possibly in combination with sharps and flats) and 7/5, 
10/7, 28/15, 15/14, 21/20, 40/21 etc, (but not 35/32 or 64/35) are use 
another. So after taking these into account we have the following "top 40" 
in order of popularity, shown with the cumulative percentage of pitches 
covered.

If we only notated the first 14 of these (up to and including 25/7) we'd be 
doing pretty well, since this would let us notate 80% of the rational 
_pitch_instances_ in the Scala archive. This would probably enable us to 
notate significantly more than 80% of the rational _scales_ in the archive 
and possibly more than 90% of the scales both rational and otherwise.

1/1   25.9%
5/1   44.2%
7/1   54.5%
25/1  59.9%
7/5   64.4%
11/1  67.8%
35/1  70.8%
125/1 72.5%
49/1  74.0%
13/1  75.6%
11/5  76.7%
11/7  77.8%
17/1  78.9%
25/7  80.0%
49/5  80.8%
13/5  81.5%
175/1 82.1%
19/1  82.6%
245/1 83.2%
13/7  83.7%
625/1 84.2%
23/1  84.6%
49/25 85.1%
55/1  85.5%
77/1  85.9%
17/5  86.2%
19/5  86.6%
35/11 86.9%
13/11 87.2%
31/1  87.5%
343/1 87.7%
29/1  87.9%
125/7 88.1%
55/7  88.3%
17/11 88.5%
77/5  88.7%
19/7  88.9%
385/1 89.1%
55/49 89.2%
17/7  89.4%
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)


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Message: 5778 - Contents - Hide Contents

Date: Mon, 23 Dec 2002 22:19:44

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> 
wrote:
> At 10:04 AM 20/12/2002 -0800, George Secor wrote:
>> The one thing that differs from what I suggested is this: Should Margo >> would want to respell E (~14/11 relative to C) as F-something, >
> Why would she want to do that? Do you want to do that, Margo?
I was just looking for some basis for respelling notes in the first chain. But now that you ask the question, that wouldn't seem to be necessary.
>> then the >> symbol I suggested, using the 7:11 comma -- Fb(| or F)!!~ -- is not in >> the 121 standard set that Dave has proposed (below). His set has the >> 5:11 comma, which gives Fb(|( or F~!!(. Either one of these is valid >> for 5deg121, but since Peppermint lacks ratios of 5, Margo might prefer >> the 7:11 comma symbol. Something to consider when we're mapping >> something to an ET is that the standard symbol set might not be the >> "best," i.e., the most meaningful or useful one. >
> Might not be, but in this case I think it is, simply because there will > never be any call for the 4deg121 symbol in notating Peppermint unless the > chains are long enough that there _are_ ratios of 5. But even if Margo did > want to do as you suggest and notate the approx 11:14 as a kind of > diminished fourth rather than a kind of major third, I don't believe that > your proposed use of (| or )||~ above are valid in the sagittal system. > Read on. >
>>> I couldn't find anywhere that we agreed on sagittal notation
for 121-ET. So
>>> here's my proposal for its single shaft symbols. >>> >>> 121: )| |) /| (|( (|~ /|\ (|) >>
>> As you have probably concluded by now, I agree with this for the >> standard set. With rational complements this becomes: >> >> 121: )| |) /| (|( (|~ /|\ (|) )|| ~||( ||\ ||) (||~ /||\ > > Agreed. >
>> I will add this ET to Table 4 in the paper. >> For Peppermint my suggested modification is: >> >> 121: )| |) /| (| (|~ /|\ (|) )|| )||~ ||\ ||) (||~ /||\ >
> By my calculations, (| as the 7:11' comma (45056:45927) is 5deg121, not > 4deg121 as you have it above.
Sorry, my mistake.
> 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.
> We had better check all existing uses of )|( for agreement with this. I > don't think )|( appears in the article. But I guess we should check for > other useful 15-limit dual-prime commas we may have missed.
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) Expressed another way: 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 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 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 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 Your 5' and -5' flag idea continues to be a winner! If you agree with the three new symbols I have proposed above, then all of the 3- flag symbol symbols to date will be ones in which the intended comma is exactly the sum of its parts. In other words, we would be keeping the complicated symbols as simple as possible. Except for the new symbols proposed, nothing is valid in 217, but )|( is the only one that isn't valid in 494. I am being forced to rethink the notion that a JI mapping to 494 isn't practical; a 15- limit <0.5-cent maximum error can make a lot of things work, and if someone is going to be very demanding with regard to precision, then we might as well have a JI option that will satisfy them. (In other words, If there's an application for something, then it's practical.) --George
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Message: 5779 - Contents - Hide Contents

Date: Mon, 23 Dec 2002 21:13:24

Subject: Re: A common notation for JI and ETs

From: M. Schulter

Please let me thank both George and Dave for your most helpful and
engrossing comments, and also take note of my possible wisdom in
writing a first response after George's initial reply, but waiting a
bit to follow the further dialogue before posting.

This process has helped me both to evaluate which parts of my first
draft might be of most interest and relevance here, and to confirm
some of my initial impressions: for example, that 121-ET notation
seemed an attractive choice for Peppermint, albeit with one possibly
not so relevant "glitch" of 121-ET itself.

At the outset, I should respond to your most gracious welcome, George:

> I'll have to express my apologies to just about anyone else who > might be reading this (and who hasn't seen the paper), that this > probably isn't going to mean a whole lot to you, and hopefully I'll > have a presentation on this soon. Margo asked me a question in a > private communication about whether the new notation would provide a > way to do such-and-such, and my explanation wouldn't have been very > meaningful without sending along the latest draft of the paper to > her. An apology is also extended to you, Margo, since I haven't yet > responded to a message from last Thursday -- after looking at what > you had to say, I started getting bogged down in the mathematics and > couldn't find a quick answer.
Please let me apologize and take full responsibility for possibly showing more enthusiasm than prudence in excitedly leaping into this discussion without fully considering the distinction between the public discussion here so far and our private communications. While I hope that no harm has been done, I apologize for any confusion or puzzlement on the part of other participants -- my responsibility, not yours. Also, I would like to emphasize your generosity and grace in responding to my e-mail questions while you are at an important point in this process -- which hopefully I won't complicate too much with my suggested "refinements" <grin>. The reply in which you welcomed me is one example of your generosity, and of your gifts as a teacher. Thank you for helping me get oriented with excellent points such as the drawbacks of attempting to notate a tempered system in "quasi-JI" style, and the frequent advantages of notating JI using ET symbols. Some of the commas you used for certain equivalent spellings also give me an opportunity to become acquainted with more symbols, and I might pursue some questions in further posts or via e-mail. However, recalling the length of my first post here, I'll try to be a bit more concise this time by focusing mainly on the question of choosing an ET notation for Peppermint 24 -- or for a larger tuning set based on the same parameters of two chains of fifths in the Wilson/Pepper temperament (~704.096 cents) at ~58.680 cents apart, yielding some pure ratios of 6:7. [As you'll see, I was not so successful in keeping this brief -- but possibly somewhat briefer than if I had tried to address more topics.] Since the general focus here is "tuning-math," I have an excuse to describe one method for looking for such an ET approximation through a Fibonacci series -- hopefully without wandering too far from our main concern in this thread with sagittal notation. Where this digressions leads is my own practical conclusion that 121-ET notation could be fine for Peppermint 24 or even Peppermint 34 (two 17-note chains), despite a minor inconsistency or nonuniqueness of 121-ET itself. Here I invite the opinions of more experienced people. In the Wilson/Pepper tuning, the ratio of logarithmic sizes (e.g. in cents) between the whole-tone and chromatic semitone is equal to the Golden Section, or Phi (~1.618034), and the ratio between whole-tone and the smaller diatonic semitone equal to Phi + 1 (~2.618034). We can use a Fibonacci series of equal temperaments -- an actual Fibonacci series, as opposed to the "Fibonacci-like" series of Yasser's theory of harmonic evolution (5,7,12,19,31...) -- to produce successively closer approximations of the Wilson/Pepper temperament. Here I use the abbreviations of (L) for limma or diatonic semitone; (A) for apotome or chromatic semitone; and (T) for whole-tone. Easley Blackwood's R is defined as the ratio between the logarithmic sizes of whole-tone and limma. The generator is the fifth in the tuning closest to the Wilson/Pepper fifth of ~704.096 cents, while the quasi-diesis (Q-D) is equal to the difference between two of these generators less an octave and the best approximation of 6:7. ET L A T R Generator Steps Q-D Steps ----------------------------------------------------------------------- 12 1 1 2 2.000 700.000 7 100.000 1 17 1 2 3 3.000 ~705.882 10 ~70.588 1 29 2 3 5 2.500 ~703.448 17 ~41.379 1 46 3 5 8 ~2.667 ~704.348 27 ~52.174 2 75 5 8 13 2.600 704.000 44 64.000 4 121 8 13 21 2.625 ~704.132 71 ~59.504 6 196 13 21 34 ~2.615 ~704.082 115 ~61.224 10 317 21 34 55 ~2.619 ~704.101 186 ~60.568 16 513 34 55 89 ~2.6176 ~704.094 301 ~58.480 25 -------------------------------------------------------------------- Wilson/Pepper ~2.6180 ~704.096 ~58.680 -------------------------------------------------------------------- A complication here is that while successive equal temperaments in this Fibonacci series yield closer and closer approximations of the Wilson/Pepper generator, they do not necessarily yield closer approximations of the quasi-diesis, or of the just 6:7 around which this interval in Peppermint is defined. Indeed 121 does considerably better than 196 or 317, with 513 as the first ET in the series coming really close (quasi-diesis ~0.200 cents narrow of Peppermint, and best approximation of 6:7 at ~266.667 cents, ~0.204 cents narrow). Returning to our main theme of sagittal notation, the question before us is whether 121 is quite accurate enough for our purposes -- or whether the precise properties of 121-ET might not be so important, as long as the symbol set can accurately represent Peppermint. First, let me quote the most attractive symbol set for Peppermint proposed in this thread by George, to which I am really drawn, with the number of 121-notation steps shown: 19 35 69 90 106 C/|||\ E(!) F/|||\ G/|||\ B(!) C/|\ D/|\ E/|\ F/|\ G/|\ A/|\ B/|\ C/|\ 6 27 48 56 77 98 119 127 -------------------------------------------------------------------- 13 29 63 84 100 C/||\ E\!!/ F/||\ G/||\ B\!!/ C D E F G A B C 0 21 42 50 71 92 113 121 Now for the arguable glitch in 121-ET, which I raise in order to seek advice from more experienced hands as to whether it is actually a problem as long as the 121-symbols show what is happening in Peppermint itself. One feature of Peppermint is that a fourth less the Archytas comma or 7 comma (2deg121) -- e.g. 4F-4A/|\ -- yields an approximation of 16:21 at ~475.062 cents, or ~4.282 cents narrow, the same absolute deviation as with 8:9 in the wide direction. This could alternately be spelled as 4F-4B!!!) using your notational equivalents, George. In a 34-note version of Peppermint with two chains of 17 notes, the next MOS size for Wilson/Pepper after 12, we would also have an approximation for 13:17 -- G\!!/-A/||\ or G(!)-A/|||\ -- formed from a chain of 16 regular fifths up at ~465.530 cents, or ~1.102 cents wide. In 121-ET, the ~16:21 of Peppermint would map to 48 steps at ~476.033 cents, and the ~13:17 to 47 steps at ~466.116 cents. We can derive the first interval as regular major third plus quasi-diesis (42 + 6) or fourth less 7 comma (50 - 2); and the second as a major third plus the regular diesis or "Pythagorean-like comma" of 5 steps (42 + 5). Unfortunately, in 121-ET as opposed to Peppermint, the Archytan comma or 7 comma at 2 steps is only ~19.835 cents rather than ~20.842 cents, so that the 48deg121 interval of ~476.033 cents is ~5.252 cents wide, actually further from 16:21 than 47deg121 (~4.665 cents narrow). One way of putting the problem is to say that in 121-ET, in contrast to Peppermint, the best approximation of 4:7 less the best approximation of 3:4 does not yield the best approximation of 16:21. Another approach is to say that in 121-ET, the 272:273 comma (~6.353 cents) defining the distinction between 16:21 and 13:17 is tempered out, while in Peppermint this distinction is observed. Now for the practical issue: given that the 121-notation seems neatly to fit what actually happens in Peppermint, where F-A/|\ or F-B!!!) is indeed the best approximation of 16:21, and G\!!/-A/||\ of 13:17, is the "inconsistency" or "nonuniqueness" of 121-ET really relevant here? For example, if we are simply seeking a convenient symbol set for Peppermint 34 itself, the following sagittal spellings give the best representations for the isoharmonic chords 9:13:17:21 and 11:16:21:26, the latter also available in Peppermint 24: 5D/|\ 5F/|\ 4A/|||\ 5C/|||\ 4G(!) 4A 4C 4D/|\ ~9:13:17:21 ~11:16:21:26 Incidentally, speaking of equivalents, if we choose some note on the upper keyboard as the 1/1, the second sonority could also be written: 5F 5C/||\ 4A\!/ 4D Having digested the first lesson that sagittal symbols should be kept as simple as possible, I would be inclined to go with the 121-notation for Peppermint 24. Even with Peppermint 34, I might guess that this notation could present a problem only if someone draws the inference that 16:21 and 13:17 as notated map accurately in 121-ET -- unless there are other "glitches" to be found. If we really want an ET yet more closely modelling Peppermint, then I might look at 513-ET. Now to some points in your responses, Dave and George. First, Dave, you gave this alternative ASCII version of a 121-notation:
> The lower chain of 704.1 cent fifths is notated > F C G D A E B F# C# G# D# A# > The upper chain, a 58.7 cent quasi-diesis above it, is notated > F^ C^ G^ D^ A^ FL CL GL DL AL EL BL > with FL and CL optionally being notated as > E^ B^
Here I see that ^ saves some typing in comparison with /|\, although I like the sagittal style of the latter -- a bias possibly related to my custom of using ^ in nonsagittal notation to show a note raised by about a Pythagorean comma or Archytan (63:64) comma, as George has aptly named the second ratio.
> Upon examining 121, I found that ratios of 13 aren't compatible with > ratios of 7 for the notation (as often occurs with ETs). For > Margo's benefit I will explain: /| is 3deg and |) is 2deg, so /|) > would be 5deg by adding the flags, but the 13 comma should be > 6deg121.
Let's be sure that I'm following this. In Peppermint, or generally in 121-ET, the Archytan comma |) maps to 2deg121, and the Didymic comma /| to 3deg121, with the 13 comma usually the sum of these or /|) -- with a schisma in JI of 4095:4096. Here, however, it's actually equal to the quasi-diesis of 6deg121, the difference between a regular minor sixth and a ~8:13, e.g. 3B-4G and 3B-4G/|\ (~7:11 and ~8:13).
> So one must make a choice between the 7 and 13-comma symbols for the > symbol set. Even though 13 has less error in 121-ET than either 7 > or 11, 7 is preferred over 13 because 1) it's a lower prime, and 2) > the 11 and 11' commas are the same number of degrees as the 13 and > 13' commas, respectively, so the symbols for a lower prime, 11, can > be used for both 11 and 13 (which is appropriate, since 351:352 > vanishes in both 121-ET and Peppermint).
Why don't I first confirm my own intution that the 7-comma symbols are indeed best, and offer the additional argument that the one just interval in Peppermint other than the 1:2 octave is the 6:7 or 7:12. An interesting point is that while "11" or "13" often suggests to me a family of intervals sharing a factor (e.g. 11:14, 11:12, 9:11; or 11:13, 8:13, 7:13, 9:13), here we are referring to a single ratio or comma, thus: 3rd harmonic 2:3 +~2.141 cents 7th harmonic 4:7 +~2.141 cents 9th harmonic 8:9 +~4.282 cents 11th harmonic 8:11 +~3.266 cents 13th harmonic 8:13 +~1.770 cents As discussed in the paper, this approach permits comparing these deviations to calculate the accuracy of other ratios -- for example 9:11 at (~3.266c - ~4.282c) or ~1.015 cents narrow in Peppermint 24. Also, I get the point that the 11-diesis (regular fourth vs. ~8:11) is identical to the 13-diesis for ~8:13, thus 4D-4G/|\ and 4E-5C/|\ respectively. Thus 351:352 is tempered out -- as is also 891:896, with the 11-diesis or ~32:33 also representing ~27:28, the difference between a regular major second and a 6:7 third, e.g. 3B\!!/-4C/|\ or 3B\!!/-4D!!!).
> The one thing that differs from what I suggested is this: Should > Margo would want to respell E (~14/11 relative to C) as F-something, > then the symbol I suggested, using the 7:11 comma -- Fb(| or F)!!~ > -- is not in the 121 standard set that Dave has proposed (below). > His set has the 5:11 comma, which gives Fb(|( or F~!!(. Either one > of these is valid for 5deg121, but since Peppermint lacks ratios of > 5, Margo might prefer the 7:11 comma symbol. Something to consider > when we're mapping something to an ET is that the standard symbol > set might not be the "best," i.e., the most meaningful or useful > one.
Here I can quickly understand that the modification upward of F\!!/ in F)!!~ is here equivalent to the usual diesis or "Pythagorean-like" comma of 5deg121. To understand this, I consider the 7:11 comma as the difference between (|) and |). If I define (!) as the difference between a regular major third C-E and a ~9:11 at C-E(!), this gives 7deg121, which compared to the Archytan comma of 2deg121 places the 7:11 comma at 5deg121. I see the advantage that these relationships involve intervals actually present in Peppermint 24 -- unlike the 5-comma. [Dave:]
> I couldn't find anywhere that we agreed on sagittal notation for > 121-ET. So here's my proposal for its single shaft symbols. > 121: )| |) /| (|( (|~ /|\ (|) [George:] > As you have probably concluded by now, I agree with this for the > standard set. With rational complements this becomes: > 121: )| |) /| (|( (|~ /|\ (|) )|| ~||( ||\ ||) (||~ /||\ > I will add this ET to Table 4 in the paper. > For Peppermint my suggested modification is: > 121: )| |) /| (| (|~ /|\ (|) )|| )||~ ||\ ||) (||~ /||\
Why don't I take up some of the symbols less familiar to me in another post, since this one is already longer than I intended -- but concluding for now by saying that I'm really excited about this symbol set, and will try some examples to test my usage at this point and get your feedback. Most appreciatively, Margo mschulter@xxxxx.xxx
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Message: 5780 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 13:54:22

Subject: Re: Ultimate 5-limit comma list

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> grasping, yelping for an answer . . .
I guess I gotta do it, but recall that last time you wanted the limit upped, you told me what I did was all you wanted. :)
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Message: 5781 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 14:15:32

Subject: Re: Ultimate 5-limit comma list

From: wallyesterpaulrus

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>> grasping, yelping for an answer . . . >
> I guess I gotta do it, but recall that last time you wanted the >limit upped, you told me what I did was all you wanted. :)
well, you're kinda too late -- see the tuning list post that should have just appeared . . .
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Message: 5782 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 06:13:39

Subject: Re: Ultimate 5-limit comma list

From: wallyesterpaulrus

grasping, yelping for an answer . . .

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus 
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> still hoping for an answer to this . . . > > --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" > <wallyesterpaulrus@y...> wrote:
>> what if we pushed up the badness limit until ampersand made it in? >> the 5-limit comma does have a name, so it must be of some
use . . .
>> >> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> > wrote:
>>> Not that any list is really ultimate, but with rms error < 40,
>> geometric complexity < 500, and badness < 3500, it covers a lot of >> ground. >>>
>>> 27/25 3.739252 35.60924 1861.731473 >>> >>> 135/128 4.132031 18.077734 1275.36536 >>> >>> 256/243 5.493061 12.759741 2114.877638 >>> >>> 25/24 3.025593 28.851897 799.108711 >>> >>> 648/625 6.437752 11.06006 2950.938432 >>> >>> 16875/16384 8.17255 5.942563 3243.743713 >>> >>> 250/243 5.948286 7.975801 1678.609846 >>> >>> 128/125 4.828314 9.677666 1089.323984 >>> >>> 3125/3072 7.741412 4.569472 2119.95499 >>> >>> 20000/19683 9.785568 2.504205 2346.540676 >>> >>> 531441/524288 13.183347 1.382394 3167.444999 >>> >>> 81/80 4.132031 4.217731 297.556531 >>> >>> 2048/2025 6.271199 2.612822 644.408867 >>> >>> 67108864/66430125 15.510107 .905187 3377.402314 >>> >>> 78732/78125 12.192182 1.157498 2097.802867 >>> >>> 393216/390625 12.543123 1.07195 2115.395301 >>> >>> 2109375/2097152 12.772341 .80041 1667.723301 >>> >>> 4294967296/4271484375 18.573955 .483108 3095.692488 >>> >>> 15625/15552 9.338935 1.029625 838.631548 >>> >>> 1600000/1594323 13.7942 .383104 1005.555381 >>> >>> (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263 >>> >>> (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204 >>> >>> (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306 >>> >>> (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873 >>> >>> (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562 >>> >>> (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615 >>> >>> (3)^8*(5)/(2)^15 9.459948 .161693 136.885775 >>> >>> (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497 >>> >>> (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642 >>> >>> (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207 >>> >>> (2)*(5)^18/(3)^27 33.653272 .025593 975.428947 >>> >>> (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149 >>> >>> (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379 >>> >>> (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233 >>> >>> (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284 >>> >>> (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255 >>> >>> (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661 >>> >>> (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786 >>> >>> (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237 >>> >>> (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721 >>> >>> (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642 >>> >>> (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514 >>> >>> (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003 >>> >>> (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867 >>> >>> (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668 >>> >>> (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625 >>> >>> (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527 >>> >>> (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421 >>> >>> (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113 >>> >>> (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823 >>> >>> (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684
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Message: 5783 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 14:36:33

Subject: Re: Ultimate 5-limit comma list

From: wallyesterpaulrus

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

> well, you're kinda too late -- see the tuning list post that should > have just appeared . . .
well, it didn't . . . but i've done these: 404 Not Found * [with cont.] Search for http://www.stretch-music.com/_wsn/page2.html in Wayback Machine Yahoo groups: /tuning/database? * [with cont.] method=reportRows&tbl=10&sortBy=9&sortDir=up
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Message: 5784 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 00:04:55

Subject: Re: A common notation for JI and ETs

From: M. Schulter

Hello, there, Dave and George, and now that I've read the latest
digest, I can share my own puzzlement about the 7:11 comma in 121-ET
or Peppermint, and check my understanding of a few other less familiar
(to me) symbols also.

> At 10:04 AM 20/12/2002 -0800, George Secor wrote:
>> The one thing that differs from what I suggested is this: Should Margo >> would want to respell E (~14/11 relative to C) as F-something,
> Why would she want to do that? Do you want to do that, Margo?
Really, I'm not sure -- it's not something that would obviously occur to me. The way I would read it is defining E (~14/11) as F (~4/3) down by a ~21:22, the same interval as the regular limma of 8deg121. The JI notation in the paper presents this as 2187:2048 or the usual apotome down and a 7:11 comma up at 45056:45927 (~33.148 cents), the latter equal to the difference between 64:81 and 9:11, or 27:32 and 22:27. As a beginner, maybe I should ask an open question: are there reasons I might want to use this kind of equivalence, either in certain musical contexts or else to validate the consistency of a given mapping, for example?
>> then the symbol I suggested, using the 7:11 comma -- Fb(| or F)!!~ >> -- is not in >the 121 standard set that Dave has proposed (below). >> His set has the 5:11 comma, which gives Fb(|( or F~!!(. Either >> one of these is valid for 5deg121, but since Peppermint lacks >> ratios of 5, Margo might prefer >the 7:11 comma symbol. Something >> to consider when we're mapping >something to an ET is that the >> standard symbol set might not be the "best," i.e., the most >> meaningful or useful one.
An aside to George: this point about the best symbol set for a given musical application is something I much like, with an example I might offer stemming from your earlier mention of a possible 46-notation for Peppermint. Why don't I take this up in due course? [Dave:]
>>> here's my proposal for its single shaft symbols. >>> >>> 121: )| |) /| (|( (|~ /|\ (|) [George:]
>> As you have probably concluded by now, I agree with this for the >> standard set. With rational complements this becomes:
>121: )| |) /| (|( (|~ /|\ (|) )|| ~||( ||\ ||) (||~ /||\ [Dave:] > Agreed. [George:]
>> I will add this ET to Table 4 in the paper. >> For Peppermint my suggested modification is: >> >> 121: )| |) /| (| (|~ /|\ (|) )|| )||~ ||\ ||) (||~ /||\ [Dave:]
> By my calculations, (| as the 7:11' comma (45056:45927) is 5deg121, > not 4deg121 as you have it above.
This is what I also wanted to query, with some beginner's caution since I could simply be puzzled by unfamiliar symbols. As I understand, the 7:11' comma is the difference between 704:729 (regular major third vs. ~9:11) and the 7 or Archytas comma at 63:64. In 121-ET, the 704:729 is 42deg121 (~11:14) less 35deg121 (~9:11), or 7d121, and the 63:64 is 2deg121, with 5deg121 as the difference -- the same as the usual limma. By the way, I take the 5:11 comma (|( at 44:45 as the difference between 704:729 and the 5 or Didymus comma at 80:81, so that the 5:11 would define the difference between ~4:5 and ~9:11. In 121-ET, I have this (|) at 7deg121 less /| at 3deg121, or 4deg121. I can see why it's a natural choice for the standard 121-ET set, but not of obvious relevance to Peppermint 24. The ~5:6 and ~4:5 in Pepper/Wilson are represented by 20 fifths down (~318.088 cents) and 21 fifths up (~386.008 cents), so the 5 comma comes up in larger sets. Let me also seek confirmation that the 11:19 diesis (|~ is equal to the 11 diesis at 32:33 less the 19 comma at 512:513, or the difference between ~16:19 and ~9:11. In 121-ET, the 512:513 is 1 step, and the 32:33 is 6 steps, so that indeed yields 5 steps for this diesis at 171:176. That explains the 5-step symbol in both versions of the symbol set for 121.
> 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).
That's how I might describe a temperament like Peppermint -- an "eventone" at close to 1/4-891:896 to produce a regular ~11:14 major third, analogous to meantone (which tempers out the 80:81) or to 22-ET (which tempers out the 63:64). The symbol )|( appeals to me, since 891:896 is almost exactly twice 351:352 -- actually 891:896 is realized in one 17-note JI system I use as 351:352 (~4.925c) plus 363:364 (~4.763c), with a small schisma or harmonisma of 10647:10648 (~0.163c), another example of a schisma which could be disregarded in sagittal notation. George, recalling an earlier post of yours, I also realize that adopting an 891:896 symbol doesn't necessarily mean that I should use it every time I write 11:14 in a JI context -- you mentioned that it can be a good idea at times to map JI to an ET symbol set as long as the symbols are consistently defined and adequately explained. Getting back to Peppermint for a moment, I've noticed that we might also say that 891:896 is tempered out in that the same quasi-diesis (mapped to 6deg121) represents both 32:33 and 27:28. On the theme of comparing prime factors, I also notice that if we take the deviations for 4:7 and 8:11 at ~2.14c and ~3.26c wide, the difference is the amount by which 11:14 is narrow or 7:11 wide, ~1.13c.
> We had better check all existing uses of )|( for agreement with > this. I don't think )|( appears in the article. But I guess we > should check for other useful 15-limit dual-prime commas we may have > missed.
There are some ratios about which I have questions, but I'm not sure which ones would fall in this category. Why don't I mention some of these in another post? Most appreciatively, Margo mschulter@xxxxx.xxx
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Message: 5785 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 19:15:54

Subject: Re: Ultimate 5-limit comma list

From: Carl Lumma

>> >ell, you're kinda too late -- see the tuning list post that >> should have just appeared . . . >
>well, it didn't . . . but i've done these: > >404 Not Found * [with cont.] Search for http://www.stretch-music.com/_wsn/page2.html in Wayback Machine Good, good. >Yahoo groups: /tuning/database? * [with cont.] >method=reportRows&tbl=10&sortBy=9&sortDir=up
Aha! Now all we need is RMS optimum generators, and a blurb on what the heck heuristic error and complexity are. -C.
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Message: 5786 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 13:14:18

Subject: Re: A common notation for JI and ETs

From: M. Schulter

Hello, everyone, and please let me quickly correct what I might
describe as an incomplete definition at the end of this paragraph:

> Really, I'm not sure -- it's not something that would obviously > occur to me. The way I would read it is defining E (~14/11) as F > (~4/3) down by a ~21:22, the same interval as the regular limma of > 8deg121. The JI notation in the paper presents this as 2187:2048 or > the usual apotome down and a 7:11 comma up at 45056:45927 (~33.148 > cents), the latter equal to the difference between 64:81 and 9:11, > or 27:32 and 22:27.
Here I should have written the last sentence something like this, possibly more readable if broken into two sentences: The JI notation in the paper presents this as 2187:2048 or the usual apotome down and a 7:11 comma up at 45056:45927 (~33.148 cents). The 7:11 comma is equal to the difference between the 7 or Archytas comma at 63:64 (27.264 cents), and the 11' diesis at 704:729 (~60.412 cents) defined as the difference between 64:81 and 9:11, or 27:32 and 22:27. My mistake, obvious on rereading, was to give the correct ratio for the 7:11 comma, but to define it as 64:81 vs. 9:11 or 27:32 vs. 22:27 -- actually the definition of the 704:729, which less a 7 comma yields the 7:11 comma. Most appreciatively, Margo Schulter mschulter@xxxxx.xxx
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Message: 5787 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 13:17:54

Subject: Re: A common notation for JI and ETs

From: M. Schulter

Hello, everyone, and here's an omission repaired: a promised Scala scale
file for Peppermint 24 to go with my examples of sagittal notation posted
today:

! peprmint.scl
!
Peppermint 24: Wilson/Pepper apotome/limma=Phi, 2 chains spaced for pure 7:6
 24
!
 58.679693 cents
 128.669246 cents
 187.348938 cents
 208.191213 cents
 7/6
 287.713180 cents
 346.392873 cents
 416.382426 cents
 475.062119 cents
 495.904393 cents
 554.584086 cents
 624.573639 cents
 683.253332 cents
 704.095607 cents
 762.775299 cents
 832.764852 cents
 891.444545 cents
 912.286820 cents
 970.966512 cents
 991.808787 cents
 1050.488479 cents
 1120.478033 cents
 1179.157725 cents
 2/1

Most appreciatively,

Margo


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Message: 5788 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 12:51:53

Subject: Re: A common notation for JI and ETs

From: M. Schulter

Hello, there, George and Dave and everyone, and Happy Holidays!

Now that we have a symbol set for Peppermint, or at least for the
steps and intervals that I'm likely to use in Peppermint 24, I don't
want to miss the point that the idea of this notation is to put it
into practice.

Of course, the discussion about the fine points of developing a symbol
set have been an opportunity for me to learn some sagittal philosophy,
and also to get more understanding by joining in on the process of
problem-solving with more experienced users. However, this festive day
seems an apt occasion to write down a few progressions and see how
musically understandable they are to others.

In part, this is a fun exercise for me as a beginner, and like a
counterpoint student in one of the classic dialogues I wouldn't be
surprised if I commit a few errors and oversights, warmly inviting
your corrections. At the same time, it's a kind of trying out of the
notation, also, assuming that I can write it correctly -- how does
this progression or passage look to someone else?

To provide a bit of a reference on interval sizes, I've included a
Scala file for Peppermint 24 at the end of this post.

First, here's a scale I find quite beautiful in Peppermint, which I
give in two spellings, and placing octave numbers before note names
(C4 = middle C):

       3D     3E    3E/|\   3G    3A    3B    3B/|\    4D

       3D     3E    3F!)    3G    3A    3B    4C!)     4D

Next is a progression which presents a kind of problem in creative
tempering of intervals for which Peppermint provides an interesting
solution:

          4G/||\  4G/|\ 4B\!!/ 4B(!)
          4F/|\   4G/|\ ------ 4F/|\
          4E\!!/  4F    ------ 4F/|\
          3B(!)   4C/|\ ------ 3B(!)

Here's a different progression using some of the same intervals, and
one I much like -- can anyone give a 13-prime rational approximation
for the opening sonority?

           4B(!)    4B
           4F/|||\  4F/||\
           4E       4F/||\
           3B/|\    3B

Now we come to a fingering exercise for my right hand in an historical
scale, originally given in rational form, with the tempered intervals
quite close to the theoretical values. First I'll notate, as above, in
a "keyboard tablature" style taking the lower chain of fifths (on the
lower keyboard) as the reference:

            4G/|\   4F/|\  4E/|\   4F/|\  4G/|\
            4C/|\   4D     4E/|\   4D     4C/|\

Here's a version with the "1/1" or lowest note of the pentachord as
the reference:

            4G   4F     4E   4F    4G
            4C   4D\|/  4E   4D\|/ 4C

By the way, the JI intervals in one rendition:

            4G   4F     4E)|(  4F    4G
            4C   4D\|/  4E)|(  4D\|/ 4C

Given some recent "popularity polling" of rational intervals used in
Scala tunings, I can't help contributing this example, featuring a
tempered version of an isoharmonic chord with all intervals within one
cent of just (the resolving sonority is less accurate):

            5G/|||\ 5F/|||\
            5E\!!/  5C/|||\
            4F/|\   4F/|||\

Finally, here's a kind of procedure I often use when improvising in
Peppermint, with the following example only a sketch inviting lots of
elaboration (mainly the meandering of "florid" but rather stately
melody in a note-against-note texture, given the nature of the
keyboard technique and my modest accomplishments therein), and some
more cadences with descending semitones (quasi-steps which could be
called small thirdtones or possibly large quartertones) as in the
opening phrase:

     4G  -----  4B\!!/ 4A     4G     4F     4G
     4D  -----  4F     4E     4D     4C     4D
     3G  3A/|\  4C/|\  3B/|\  3A/|\  3G/|\  3G

     4G     4A     5C     5C/|\
     4D     4E     4G     4G/|\
     3A/|\  3B/|\  4D/|\  4C/|\

     4B\!!/  4A     4G     4F     4F/|\
     4F      4E     4D     4C     4C/|\
     4C/|\   3B/|\  3A/|\  3G/|\  3F/|\

     4F     4G     4A     4A/|\
     4C     4D     4E     4E/|\
     3G/|\  3A/|\  3B/|\  3A/|\

     4G     4F     4E\!!/  4D     4D/|\
     4D     4C     3B\!!/  3A     3A/|\
     3A/|\  3G/|\  3F/|\   3E/|\  3D/|\
     
Again, Happy Holidays to all!

Most appreciatively,

Margo
mschulter@xxxxx.xxx


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Message: 5789 - Contents - Hide Contents

Date: Tue, 24 Dec 2002 22:31:05

Subject: Re: Ultimate 5-limit comma list

From: wallyesterpaulrus

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" 
<clumma@y...> wrote:
>>> well, you're kinda too late -- see the tuning list post that >>> should have just appeared . . . >>
>> well, it didn't . . . but i've done these: >> >> 404 Not Found * [with cont.] Search for http://www.stretch-music.com/_wsn/page2.html in Wayback Machine > > Good, good. > >> Yahoo groups: /tuning/database? * [with cont.] >> method=reportRows&tbl=10&sortBy=9&sortDir=up >
> Aha! Now all we need is RMS optimum generators, and a blurb > on what the heck heuristic error and complexity are. > > -C.
as you know, error and complexity can both be defined in various ways -- rms/minimax/mad, weightedthisway/weightedthatway/unweighted, euclidean/taxicab, etc. what i've come up with are heuristics for error and complexity, which are approximately correct no matter what definition you use. they're calculated as follows: d = odd limit of comma ratio; s = numerator minus denominator ln(d) = heuristic complexity |s|/(d*ln(d)) = heuristic error (actually i multiplied by 1200/ln(2)?) what i would like to figure out is what natural choice of metric, weighting, loss function, etc. makes these heuristics *exactly* correct. then report *those* optimal generators . . .
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Message: 5790 - Contents - Hide Contents

Date: Wed, 25 Dec 2002 00:01:16

Subject: Temperament notation

From: Gene Ward Smith

This Peppermint notation business raises again the question of how, in
general, you two are proposing to deal with notating temperaments. Is
is to be a cut down version of JI, or do you relate it to ets
supporting the temperament, or something else?


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Message: 5791 - Contents - Hide Contents

Date: Wed, 25 Dec 2002 03:23:16

Subject: Re: Ultimate 5-limit comma list

From: Carl Lumma

>>> >ttp://groups.yahoo.com/group/tuning/database? >>> method=reportRows&tbl=10&sortBy=9&sortDir=up >>
>> Aha! Now all we need is RMS optimum generators, and a blurb >> on what the heck heuristic error and complexity are. >> >> -C. >
>as you know, error and complexity can both be defined in >various ways -- rms/minimax/mad, >weightedthisway/weightedthatway/unweighted, >euclidean/taxicab, etc. > >what i've come up with are heuristics for error and complexity, >which are approximately correct no matter what definition you >use.
Sounds like a good idea.
>they're calculated as follows: > >d = odd limit of comma ratio; s = numerator minus denominator >ln(d) = heuristic complexity >|s|/(d*ln(d)) = heuristic error >(actually i multiplied by 1200/ln(2)?) > >what i would like to figure out is what natural choice of metric, >weighting, loss function, etc. makes these heuristics *exactly* >correct. then report *those* optimal generators . . .
Why would you assume the heuristics are more ideal than any of the current formulations they approximate? Do they require fewer assumptions? Can you plot unweighted Graham complexity against heuristic complexity for me? -Carl
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Message: 5792 - Contents - Hide Contents

Date: Thu, 26 Dec 2002 01:31:25

Subject: Re: Ultimate 5-limit comma list

From: wallyesterpaulrus

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" 
<clumma@y...> wrote:
>>>> Yahoo groups: /tuning/database? * [with cont.] >>>> method=reportRows&tbl=10&sortBy=9&sortDir=up >>>
>>> Aha! Now all we need is RMS optimum generators, and a blurb >>> on what the heck heuristic error and complexity are. >>> >>> -C. >>
>> as you know, error and complexity can both be defined in >> various ways -- rms/minimax/mad, >> weightedthisway/weightedthatway/unweighted, >> euclidean/taxicab, etc. >> >> what i've come up with are heuristics for error and complexity, >> which are approximately correct no matter what definition you >> use. >
> Sounds like a good idea. >
>> they're calculated as follows: >> >> d = odd limit of comma ratio; s = numerator minus denominator >> ln(d) = heuristic complexity >> |s|/(d*ln(d)) = heuristic error >> (actually i multiplied by 1200/ln(2)?) >> >> what i would like to figure out is what natural choice of metric, >> weighting, loss function, etc. makes these heuristics *exactly* >> correct. then report *those* optimal generators . . . >
> Why would you assume the heuristics are more ideal than any > of the current formulations they approximate?
a bit of "numerology" one might say . . .
> Do they require > fewer assumptions?
hmm . . . you might say that, since the formulas are simpler. especially if you're given the ratios and have to factorize them yourself.
> Can you plot unweighted Graham complexity > against heuristic complexity for me?
the only thing i can do on this computer is surf yahoo and a few other websites. i'm sitting in a convenience store . . .
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Message: 5793 - Contents - Hide Contents

Date: Thu, 26 Dec 2002 17:41:15

Subject: Re: Temperament notation

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> This Peppermint notation business raises again the question of how,
in general, you two are proposing to deal with notating temperaments. Is is to be a cut down version of JI, or do you relate it to ets supporting the temperament, or something else? It looks as if the method will be: 1) Identify an ET that closely resembles the temperament; 2) Use the ET notation for that temperament; and 3) Identify any instances in which symbols apart from that ET's standard set might be more harmonically meaningful, and make appropriate substitutions. For example, if you want to notate the meantone temperament extended to include ratios of 7, you could use the 31-ET notation. But the standard 31-ET symbol set uses the 11-comma (32:33) symbol for a single degree, and you're not interested in ratios of 11. So you would probably want to use the 7-comma symbol (63:64) instead. But if you're using the 11 limit in extended meantone, you could even use both symbols (in different places) if you wanted to indicate harmonic function (as long as it didn't confuse the player). This brings up the question of whether it would be advisable to do this in an ET as well. We have gone out of our way to define standard symbol sets that would require the minimum amount of memorization and confusion and maximum utility, so our policy is to discourage departure from standard sets. But we're not going to be the notation police, either, so a composer will be free to experiment with things like this to see what's helpful and what isn't. --George
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Message: 5794 - Contents - Hide Contents

Date: Thu, 26 Dec 2002 19:45:32

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@xxxxxxxxxxx.xxxx "M. Schulter" <mschulter@m...> 
wrote:
> ... Now for the arguable glitch in 121-ET, which I raise in order to seek > advice from more experienced hands as to whether it is actually a > problem as long as the 121-symbols show what is happening in > Peppermint itself. > > One feature of Peppermint is that a fourth less the Archytas comma or > 7 comma (2deg121) -- e.g. 4F-4A/|\ -- yields an approximation of 16:21 > at ~475.062 cents, or ~4.282 cents narrow, the same absolute deviation > as with 8:9 in the wide direction. This could alternately be spelled > as 4F-4B!!!) using your notational equivalents, George. > > In a 34-note version of Peppermint with two chains of 17 notes, the > next MOS size for Wilson/Pepper after 12, we would also have an > approximation for 13:17 -- G\!!/-A/||\ or G(!)-A/|||\ -- formed from a > chain of 16 regular fifths up at ~465.530 cents, or ~1.102 cents wide. > > In 121-ET, the ~16:21 of Peppermint would map to 48 steps at ~476.033 > cents, and the ~13:17 to 47 steps at ~466.116 cents. We can derive the > first interval as regular major third plus quasi-diesis (42 + 6) or > fourth less 7 comma (50 - 2); and the second as a major third plus the > regular diesis or "Pythagorean-like comma" of 5 steps (42 + 5). > > Unfortunately, in 121-ET as opposed to Peppermint, the Archytan comma > or 7 comma at 2 steps is only ~19.835 cents rather than ~20.842 cents, > so that the 48deg121 interval of ~476.033 cents is ~5.252 cents wide, > actually further from 16:21 than 47deg121 (~4.665 cents narrow). > > One way of putting the problem is to say that in 121-ET, in contrast > to Peppermint, the best approximation of 4:7 less the best > approximation of 3:4 does not yield the best approximation of 16:21.
A concise way of stating this is to say that 121-ET is not 1,3,7,21- consistent.
> Another approach is to say that in 121-ET, the 272:273 comma (~6.353 > cents) defining the distinction between 16:21 and 13:17 is tempered > out, while in Peppermint this distinction is observed.
Since there are two ways of arriving at 16:21 in 121-ET, I wouldn't put it that way, especially since the one you're interested in maintains the distinction.
> Now for the practical issue: given that the 121-notation seems neatly > to fit what actually happens in Peppermint, where F-A/|\ or F-B!!!) is > indeed the best approximation of 16:21, and G\!!/-A/||\ of 13:17, is > the "inconsistency" or "nonuniqueness" of 121-ET really relevant here?
Not as far as I can tell. The place where you might start to experience a problem is if a tone at one end of one chain of fifths is the same number of degrees as a tone at the other end of the other chain of fifths, and you've reached the point where you've gotten very close -- the 34th tone in a downward chain of fifths is 6deg121 (or go 17 fifths in opposite directions from the middle of each chain). But you're out of the woods if you happen to use different nominals (or letter names) to spell the two tones.
> ... If we really want an ET yet > more closely modelling Peppermint, then I might look at 513-ET.
That's possible; 513 isn't 13-limit consistent, but you could get around that by taking 13 as 360 instead of 359 degrees -- you just wouldn't be using the best representation of 13.
> Now to some points in your responses, Dave and George. First, Dave, > you gave this alternative ASCII version of a 121-notation: >
>> The lower chain of 704.1 cent fifths is notated >> F C G D A E B F# C# G# D# A# >> The upper chain, a 58.7 cent quasi-diesis above it, is notated >> F^ C^ G^ D^ A^ FL CL GL DL AL EL BL >> with FL and CL optionally being notated as >> E^ B^ >
> Here I see that ^ saves some typing in comparison with /|\, although > I like the sagittal style of the latter -- a bias possibly related to > my custom of using ^ in nonsagittal notation to show a note raised by > about a Pythagorean comma or Archytan (63:64) comma, as George has > aptly named the second ratio. >
>> Upon examining 121, I found that ratios of 13 aren't compatible with >> ratios of 7 for the notation (as often occurs with ETs). For >> Margo's benefit I will explain: /| is 3deg and |) is 2deg, so /|) >> would be 5deg by adding the flags, but the 13 comma should be >> 6deg121. >
> Let's be sure that I'm following this. In Peppermint, or generally in > 121-ET, the Archytan comma |) maps to 2deg121, and the Didymic comma > /| to 3deg121, with the 13 comma usually the sum of these or /|) -- > with a schisma in JI of 4095:4096. Here, however, it's actually equal > to the quasi-diesis of 6deg121, the difference between a regular minor > sixth and a ~8:13, e.g. 3B-4G and 3B-4G/|\ (~7:11 and ~8:13).
You've got it.
>> So one must make a choice between the 7 and 13-comma symbols for the >> symbol set. Even though 13 has less error in 121-ET than either 7 >> or 11, 7 is preferred over 13 because 1) it's a lower prime, and 2) >> the 11 and 11' commas are the same number of degrees as the 13 and >> 13' commas, respectively, so the symbols for a lower prime, 11, can >> be used for both 11 and 13 (which is appropriate, since 351:352 >> vanishes in both 121-ET and Peppermint). >
> Why don't I first confirm my own intution that the 7-comma symbols are > indeed best, and offer the additional argument that the one just > interval in Peppermint other than the 1:2 octave is the 6:7 or 7:12. > An interesting point is that while "11" or "13" often suggests to me a > family of intervals sharing a factor (e.g. 11:14, 11:12, 9:11; or > 11:13, 8:13, 7:13, 9:13), here we are referring to a single ratio or > comma, thus: > > 3rd harmonic 2:3 +~2.141 cents > 7th harmonic 4:7 +~2.141 cents > 9th harmonic 8:9 +~4.282 cents > 11th harmonic 8:11 +~3.266 cents > 13th harmonic 8:13 +~1.770 cents > > As discussed in the paper, this approach permits comparing these > deviations to calculate the accuracy of other ratios -- for example > 9:11 at (~3.266c - ~4.282c) or ~1.015 cents narrow in Peppermint 24. Correct. > Also, I get the point that the 11-diesis (regular fourth vs. ~8:11) is > identical to the 13-diesis for ~8:13, thus 4D-4G/|\ and 4E-5C/|\ > respectively. Thus 351:352 is tempered out -- as is also 891:896, with > the 11-diesis or ~32:33 also representing ~27:28, the difference > between a regular major second and a 6:7 third, e.g. 3B\!!/-4C/|\ or > 3B\!!/-4D!!!).
Since you need the symbol for the 7 comma, but since the difference between the 11 and 13 commas vanishes in 121, you choose the 11-comma symbol, because the 13-comma symbol is inconsistent with the 7-comma symbol in 121-ET.
>> The one thing that differs from what I suggested is this: Should >> Margo would want to respell E (~14/11 relative to C) as F- something,
Forget about this. --George
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Message: 5795 - Contents - Hide Contents

Date: Thu, 26 Dec 2002 19:53:13

Subject: Re: A common notation for JI and ETs

From: M. Schulter

Hello, George and Dave and everyone, and please let me express my
excitement upon actually starting to use the sagittal system for
notating pieces of music by hand, where the actual graphic symbols can
apply.

Very quickly I realized that the apotome symbols very graphically
indicate the frequent tendency of a flat to descend by a limma, and of
a sharp to ascend by a limma. Since the actual sign for the apotome
has changed -- with the medieval European symbols corresponding to
both the modern natural and sharp playing this role -- the sagittal
symbols could be taken as a new chapter of this tradition.

With Peppermint 24, I find the 121-ET notation very intuitive in
practice as well as theory, as writing down some examples at the
keyboard has shown me. Counting in approximate 1/21-tones can help in
checking some points of notation, as I'll illustrate below.

To celebrate my excitement in this festive season, why don't I give
two examples: first, a short complete piece; and secondly, a sketch or
"harmonic plan" for a passage in three alternative versions.

The first item, a complete piece, shows the "conventional" side of the
notation where only diatonic notes and apotome signs are required:
this could be played in a 17-tone equal or well-tempered or just
system, or in Peppermint.

For convenience in ASCII notation, I have followed the convention that
a note is sustained in a given part until another note appears, or a
rest as shown by the symbol "r," or the end of a piece or section. 
Octave numbers appear before the note names and sagittal signs (with
C4 = middle C).  

                     Cantilena for George Secor

A
1     2      | 1       2    |  1     2      |  1    +     2      ||
5C    4B               4A      5C    4B\!!/         4A    4B\!!/
4A    4G/||\           4A      4A    4G                   4F
4F             4E      4D      4F              4E         4F
4D             4C/||\  4D      4D              4C         3B\!!/

B                                 B                              DC
1       2    | 1     +    2    |  1       2    |  1    +     2   ||
4B\!!/  4A           4G   4A      4B\!!/  4A           4G    4A
4G      4F                4E      4G      4F                 4E
4E\!!/         4D         4E      4E\!!/          4D         4E
4C             3B\!!/     3A      4C              3B\!!/     3A

Here the form is ABBAA, with the identical repetition of the
two-measure B section written out, a pattern rather like the
14th-century French virelai or Italian ballata. The miniature design,
however, is inspired especially by the 13th-century rondeaux in three
voices of Adam de la Halle -- with some curious suggestions, at least
to my ears, of certain touches of 20th-century "popular music,"
possibly because of the minor seventh sonorities.

This is a little piece that I did some months ago, but take special
pleasure in sharing in the sagittal notation.

Now we come to the harmonic sketch for a passage in Peppermint, which
I'll write in three alternative versions to practice some equivalents
and show how the approximate 21-fold division of the tone in the
121-ET notational model can be helpful in converting between these
equivalents.

First, what I call a "keyboard tablature" style of sagittal notation
which takes the lower manual or chain of fifths as the reference. 
Since this is a very rough conceptual sketch, I won't try to show any
specific rhythm, but only to suggest how the simultaneities line up,
again with each note in a part sustained until another note or rest
(r) occurs, or until the end of the piece or section. Here barlines
simply show structural ideas or units:

5D/|\  5D     5C/|\ | 5F/|\  5E(!)  5E\!!/  5F | 5F  5E\!!/  5D    5D/|\
4A/|\  4A     4F/|\   5C/|\  4B(!)  4B\!!/  5C   5C  4B\!!/  4A    4A/|\
4D/|\  4E/|\  4F/|\   4F/|\                 4F   4F          4E/|\ 4D/|\  

This "tablature-style" notation show which keys are pressed based on a
fixed layout, in effect a sagittal equivalent of a keyboard diagram.
The opening progression is a three-voice harmonization of a two-voice
example given by Marchettus of Padua in his _Lucidarium_ (1318), with
this Peppermint rendition possibly not too far from an idiomatic
14th-century intonation in the style he chronicles.

In a second version, we take the sonority 4D/|\-4A/|\-5D/|\ as the
"1/1," representing it simply as "4D-4A-5D." This leads to a keyboard
mapping as follows:

       C/||\     E\!!/            F/||\     G/||\   B\!!/
  C           D         E     F         G         A         B    C
---------------------------------------------------------------------
        C(|)     E\!!!/            F(|)      G(|)   B\!!!/
  C\|/        D\|/      E\|/  F\|/      G\|/      A\|/      B\|/ C\|/

Note how the approximate 21-fold division of the tone helps in keeping
track of some of these symbols. For example, the F/||\ key on the
upper manual is 13-steps (the usual apotome) above F --and the
accidental key on the lower manual corresponding to this is 6 steps
lower (the quasi-diesis between the two chains), placing it at 7 steps
above F, or F(|).

5D   5D\|/ 5C | 5F  5E\!!/  5E\!!!/  5F\!/ | 5F\!/  5E\!!!/ 5D\!/ 5D
4A   4A\|/ 4F   5C  4B\!!/  4B\!!!/  5C\!/   5C\!/  4B\!!!/ 4A\!/ 4A
4D   4E    4F   4F                   4F\!/   4F\!/          4E    4D  

A third version likewise treats the upper keyboard as the reference
for the "1/1," but additionally shows some septimal equivalents making
explicit the 2-3-7-9 basis for the harmony and the shift of an
Archytas of 7 comma (63:64) in the last part of the example:

5D   5C|||) 5C | 5F  5E\!!/  5D|)  5E|) | 5E|)  5D|)  5C|||)  5D
4A   4G|||) 4F   5C  4B\!!/  4A|)  4B|)   4B|)  4A|)  4G|||)  4A
4D   4E     4F   4F                4E|)   4E|)        4E      4D  

In this version, the difference between the 8-step limma E-F and the
6-step quasi-diesis F\!/-F is equivalent to the 2-step Archytas comma,
thus 4F\!/=4E|) and so forth.

Anyway, I'm impressed with how nice the sagittal notation can be for
"conventional" uses as well as intonational fine points.

Happy Holidays to All,

Margo Schulter
mschulter@xxxxx.xxx


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Message: 5796 - Contents - Hide Contents

Date: Sat, 28 Dec 2002 11:07:33

Subject: Re: Temperament notation

From: David C Keenan

At 08:43 AM 26/12/2002 -0800, George Secor wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith ><genewardsmith@j...>" <genewardsmith@j...> wrote:
>> This Peppermint notation business raises again the question of how,
>in general, you two are proposing to deal with notating temperaments. >Is is to be a cut down version of JI, or do you relate it to ets >supporting the temperament, or something else? > >It looks as if the method will be: > >1) Identify an ET that closely resembles the temperament;
This is the tricky bit. There will generally be more than one such ET. Which one should we use. For an octave-based linear temperament, if we know the exact size of the generator and period then we can be a bit more specific. (a) The cardinality of the ET must be (octave/period times) the denominator of a convergent of generator/period, and (b) The best fifth (approx 2:3) in the ET must be the same as the fifth calculated by applying the temperament's mapping to the best approximation of the generator (and period) in that ET, and (c) The cardinality of the ET must not be so small that the desired length of generator chains wrap around or run into each other. So the cardinality of the ET must be greater (and may need to be a lot greater) than the cardinality of the finite scale being used in the temperament, and (d) The cardinality of the ET must not be so large that it cannot be notated in sagittal notation, or cannot be notated without resorting to more unusual or complex sagittal symbols than necessary. (e) The cardinality of the ET must not be so large that its validity according to criterion (a) above is overly sensitive to the exact value of the generator. Now criteria (a) (b) and (c) above are completely defined, but (d) and (e) are fuzzy. One way around this is to replace (d) and (e) with (f) The ET must be the smallest one that satisfies (a) (b) and (c). The only problem with this is that different size scales within a temperament may end up with completely different notations. This could be turned on its head and instead a popular temperament might be used as an argument for determining the standard sagittal symbol set for some ET, so that the central parts of the generator chains agree between the various convergent ETs. e.g. for meantone 12, 19, 31, 50 miracle 31, 41, 72 But it isn't possible to make this work with all temperaments and their ETs.
>2) Use the ET notation for that temperament; and > >3) Identify any instances in which symbols apart from that ET's >standard set might be more harmonically meaningful, and make >appropriate substitutions. > >For example, if you want to notate the meantone temperament extended to >include ratios of 7, you could use the 31-ET notation. But the >standard 31-ET symbol set uses the 11-comma (32:33) symbol for a single >degree, and you're not interested in ratios of 11. So you would >probably want to use the 7-comma symbol (63:64) instead.
I really don't see why it should change from the standard set of the ET. Someone using 31-ET without reference to any temperament may also not be interested in ratios of 11 and yet should probably just accept that /|\ is the symbol for both the 11-diesis and the 7-comma in 31-ET, because of its stronger association as a half-apotome symbol.
>But if you're using the 11 limit in extended meantone, you could even >use both symbols (in different places) if you wanted to indicate >harmonic function (as long as it didn't confuse the player). This >brings up the question of whether it would be advisable to do this in >an ET as well. We have gone out of our way to define standard symbol >sets that would require the minimum amount of memorization and >confusion and maximum utility, so our policy is to discourage departure >from standard sets. But we're not going to be the notation police, >either, so a composer will be free to experiment with things like this >to see what's helpful and what isn't.
Agreed. I'm just saying that I severely doubt it will be helpful. One major reason for having standards in the first place, is so we can communicate with each other. If everyone invents their own sagittal notation for an ET, they may find it personally "helpful" right up until they want someone else to read it. Whether the standards proposed by we two, will be thought acceptable by a wider community is another question, but there will only be a very brief period (starting now) when it will be possible to negotiate a change to any standard. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5797 - Contents - Hide Contents

Date: Sat, 28 Dec 2002 01:49:45

Subject: Re: Temperament notation

From: Gene Ward Smith

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

> Whether the standards proposed by we two, will be thought acceptable by a > wider community is another question, but there will only be a very brief > period (starting now) when it will be possible to negotiate a change to any > standard.
Whatever you do, I suggest you keep in mind that notating linear temperaments is just as important as notating ets, if not more so. 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.
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Message: 5798 - Contents - Hide Contents

Date: Sat, 28 Dec 2002 21:18:48

Subject: Re: A common notation for JI and ETs

From: M. Schulter

Hello, George and everyone, and thank you for your suggestions on
consistency and related topics. Here I'll start by quoting a bit from
my post to which you were responding, to give a bit of context:

>> Unfortunately, in 121-ET as opposed to Peppermint, the Archytan >> comma or 7 comma at 2 steps is only ~19.835 cents rather than >> ~20.842 cents, so that the 48deg121 interval of ~476.033 cents is >> ~5.252 cents wide, actually further from 16:21 than 47deg121 >> (~4.665 cents narrow). >> One way of putting the problem is to say that in 121-ET, in >> contrast to Peppermint, the best approximation of 4:7 less the best >> approximation of 3:4 does not yield the best approximation of >> 16:21.
> A concise way of stating this is to say that 121-ET is not 1,3,7,21- > consistent.
Thanks for this notational tip; I see that we can list the applicable odd factors, and this is very neat and compact.
>> Another approach is to say that in 121-ET, the 272:273 comma >> (~6.353 cents) defining the distinction between 16:21 and 13:17 is >> tempered > out, while in Peppermint this distinction is observed.
> Since there are two ways of arriving at 16:21 in 121-ET, I wouldn't > put it that way, especially since the one you're interested in > maintains the distinction.
Yes, I see your point that while "the best approximation of 13:17 in 121-ET also happens to be the best approximation of 16:21," the two derivations are indeed distinct, and in Peppermint yield the desired results, with a fourth less 2d121 as the best ~16:21.
>> Now for the practical issue: given that the 121-notation seems >> neatly to fit what actually happens in Peppermint, where F-A/|\ or >> F-B!!!) is indeed the best approximation of 16:21, and G\!!/-A/||\ >> of 13:17, is the "inconsistency" or "nonuniqueness" of 121-ET >> really relevant here?
> Not as far as I can tell. The place where you might start to > experience a problem is if a tone at one end of one chain of fifths > is the same number of degrees as a tone at the other end of the > other chain of fifths, and you've reached the point where you've > gotten very close -- the 34th tone in a downward chain of fifths is > 6deg121 (or go 17 fifths in opposite directions from the middle of > each chain). But you're out of the woods if you happen to use > different nominals (or letter names) to spell the two tones.
First, please let me confirm that I love the 121-ET notation, and find the 21-fold division of the tone very intuitive. The equivalence in 121-ET of the quasi-diesis at 6deg121 and 34 fifths down is something that I hadn't considered, but easy to see once you point it out: this quasi-diesis is equal to two "17-fifth" commas at 3deg121 each, the difference between the limma at 8deg121 and the natural diesis at 5deg121. Using different nominals, if chains this long come up, could be a reasonable solution -- I really like 121 as a symbol set.
>> ... If we really want an ET yet >> more closely modelling Peppermint, then I might look at 513-ET.
> That's possible; 513 isn't 13-limit consistent, but you could get > around that by taking 13 as 360 instead of 359 degrees -- you just > wouldn't be using the best representation of 13.
Paul Erlich has sometimes commented that consistency is most important for smaller ET's, because with large ones, an inaccuracy of one step isn't necessarily that significant. By the way, I understand that someone -- is it Carl Lumma -- has developed a routine or program to test consistency for specified factors in a given tuning. My own inclination, if "seeking near-perfection," would be to pick something that's 2-3-7-11-13 consistent and also closely approximates both the Peppermint generator and the quasi-diesis -- but that could mean a large ET with lots of sagittal symbols. In practice, 121 looks fine to me. Happy Holidays, Margo
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Message: 5799 - Contents - Hide Contents

Date: Sat, 28 Dec 2002 22:52:13

Subject: Re: A common notation for JI and ETs

From: M. Schulter

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 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 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\!!/ Happy Holidays, Margo
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