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Message: 6300 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 13:26:10
Subject: Re: A common notation for JI and ETs
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:
> Monz George and Dave
> --------------------------------
> skhisma schisma
> kleisma kleisma
> comma comma
> small diesis comma
> great diesis diesis
> small semitone ediesis (and other larger anomalies)
Will it cause confusion to call any small interval which vanishes in a given temperament a "comma"?
Some word is needed, and I'm not keen on "unison vector".
Message: 6301 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 15:22:33
Subject: Comment on notation
From: Gene Ward Smith
If we take the Hemiennealimmal temperament, which one gets by ignoring the "schismninas" sagittal
proposes to ignore, and then zeros out one of the commas associated to
11-limit symbols as well, one gets the following list of vals:
729/704 [198, 312, 457, 554, 684]
33/32 [18, 28, 41, 50, 62]
36/35 [18, 30, 44, 52, 63]
6561/6400 [90, 142, 208, 252, 311]
45/44 [108, 170, 249, 302, 373]
45927/45056 [306, 484, 709, 858, 1058]
55/54 [36, 58, 85, 102, 125]
81/80 [90, 142, 208, 252, 311]
5120/5103 [198, 314, 460, 556, 685]
All of these except for the last, for 5120/5103, are non-standard.
Have you two notated the 198-et, by the way?
If we take the ratio of two continguous intervals on the above list,
and toss 5120/5103 which we have already considered, we get the
following vals, all of which are standard, and in which the notation
simplies itself:
413343/409600 [72, 114, 167, 202, 249]
2200/2187 [126, 200, 293, 354, 436]
243/242 [72, 114, 167, 202, 249]
385/384 [72, 114, 167, 202, 249]
8019/8000 [72, 114, 167, 202, 249]
1240029/1239040 [342, 542, 794, 960, 1183]
Message: 6302 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 18:43:51
Subject: "Sagittal" temperaments
From: Gene Ward Smith
Here is a list of 11-limit temperaments which arise from triplets of
commas, each of which is either one of the 11-limit sagittal notation
commas, or a quotient of such commas. They all, therefore, should
simplfy the sagittal notation in their case.
Septimal
[0, 0, 7, 0, 0, 11, 0, 16, 0, -24]
rms 22.636347 comp 26.058106 bad 5184.550161
[1, -3, -2, -1, -7, -6, -5, 4, 8, 4]
rms 54.982509 comp 15.875495 bad 5513.617843
Porcupine
[3, 5, -6, 4, 1, -18, -4, -28, -8, 32]
rms 11.793935 comp 41.067070 bad 5765.207400
[2, 1, 3, 5, -3, -1, 1, 4, 8, 4]
rms 55.689539 comp 16.340735 bad 5859.936955
[2, 1, 3, -2, -3, -1, -10, 4, -8, -16]
rms 52.234534 comp 17.254513 bad 6018.139493
Arnold
[1, 4, -2, -1, 4, -6, -5, -16, -16, 4]
rms 39.863722 comp 21.483576 bad 6618.437353
[2, 1, -4, 5, -3, -12, 1, -12, 8, 28]
rms 27.107413 comp 27.157168 bad 6651.131418
[1, 4, -2, 6, 4, -6, 6, -16, 0, 24]
rms 28.872265 comp 26.432118 bad 6771.748430
[2, 1, -4, -2, -3, -12, -10, -12, -8, 8]
rms 42.344437 comp 21.310039 bad 6935.909992
Semififth
[2, 8, -11, 5, 8, -23, 1, -48, -16, 52]
rms 6.681355 comp 64.720989 bad 6970.647533
[3, 5, 1, 4, 1, -7, -4, -12, -8, 8]
rms 37.194175 comp 23.910689 bad 7381.224379
Hexidecimal
[1, -3, 5, -1, -7, 5, -5, 20, 8, -20]
rms 30.938504 comp 27.201522 bad 7611.810986
[6, 17, -26, 15, 13, -58, 3, -108, -24, 132]
rms 1.860428 comp 152.101752 bad 8063.122648
Flattone
[1, 4, -9, 6, 4, -17, 6, -32, 0, 48]
rms 13.051258 comp 48.332860 bad 8369.956700
[1, 4, 5, 6, 4, 5, 6, 0, 0, 0]
rms 55.787056 comp 20.476771 bad 8550.061093
[4, 9, -15, 10, 5, -35, 2, -60, -8, 80]
rms 5.007160 comp 88.019435 bad 8720.788247
[2, 8, -4, 5, 8, -12, 1, -32, -16, 28]
rms 17.726670 comp 43.670435 bad 9600.039564
Amity
[5, 13, -17, 9, 9, -41, -3, -76, -24, 84]
rms 4.265603 comp 104.931875 bad 9957.709517
Message: 6303 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 20:40:10
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
> ... [re boundaries]
> To summarise:
> 0
> schismina
> 0.98
> schisma
> 4.50
> kleisma
> 13.58
> comma
> 37.65
> carcinoma
> 45.11
> diesis
> 56.84
> ediasis
> 68.57
> ...
> On second thoughts, 13.47 cents might be a better choice for the
kleisma-comma boundary.
I don't know where you're getting the numbers 13.58 and 13.47 cents.
I indicated earlier a rationale for a lower limit for a comma:
<< The point here is that I thought that the comma (120:121,
~14.367c) between the next smaller pair of superparticular ratios
(10:11 and 11:12) should be smaller than the lower size limit for a
comma. >>
So I suggest that the upper limit for a kleisma should be 120:121
(~14.367c), and that a comma would be anything infinitesimally larger
than that, unless there is something between 13.47 and 14.37 cents
that we need to have in the comma category.
--George
Message: 6304 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 20:48:15
Subject: Re: Schisminas and rational notation
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> If the four "schisminas" 2401/2400, 3025/3024, 4375/4374 and
9801/9800
> vanish, we are in the Hemiennealimmal temperament or something
compatible with it, such as the 612 et. I proposed some time back a
notation for 11-limit JI based on nine nominals for which these are
ignored, and Graham has a similar 10-nominal system based on 494. Is
there some kind of convergence going on here?
Only that we're all wisely economizing our use of symbols by relating
them to very small commas -- schisminas, if you will -- that happen
to vanish in both of these excellent divisions. Since sagittal is
based on 7 nominals, I'd say that that's as far as the resemblance
goes.
--George
Message: 6305 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 22:40:38
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>>>>> Is it any more proportional to log(d) than log(n) in this
>>>>> case? Since n~=d?
>>>>
>>>> No, and the spreadsheet sorted by d is also sorted by n.
>>>
>>> So it could just as well be (n-d)/(d*log(n))?
>>
>> a very different sorting. that would be heuristic error, not
>> heuristic complexity. of course it's a very different sorting,
>> since knowing log(n) or log(d) tells you nothing about (n-d).
>
> ? I was asking if the *error* heuristic could become
> (n-d)/(d*log(n)) if we substituted log(n) for log(d).
oh . . . yeah sure. these are almost identical. i use odd limit for
both occurences of n or d in the denominator.
Message: 6306 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 22:44:24
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>>>>>> And in that case it would have been easier to go straight
>>>>>> to log(n*d).
>>>>>
>>>>> Straight to where (do you see log(n*d))?
>>>>
>>>> log(n*d) = log(n) + log(d)
>>>
>>> Of course... so we're coming from log(n*d), not going to it.
>>
>> what do you mean, we're coming from log(n*d)??
>
> In your original message, you *start from* "is proportional to
> log(n) + log(d)" and *arrive at* "Hence the amount of tempering
> implied by the unison vector is approx. proportional to
> (n-d)/(d*log(d))".
>
> -Carl
right . . . well if you temper the octave too, then you'd use the
tenney metric which is indeed log(n) + log(d) . . . but typically
we've not been tempering the octave, so it's the van prooijen metric,
or log of the "odd limit" of the ratio, which is really relevant
here . . .
Message: 6307 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 22:51:32
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
>
>> Monz George and Dave
>> --------------------------------
>> skhisma schisma
>> kleisma kleisma
>> comma comma
>> small diesis comma
>> great diesis diesis
>> small semitone ediesis (and other larger anomalies)
>
> Will it cause confusion to call any small interval which vanishes
>in a given temperament a "comma"? Some word is needed, and I'm not
>keen on "unison vector".
what's wrong with "commatic unison vector"?
Message: 6308 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 22:52:27
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote [#5712]:
> ... But I really don't like using ( for 5:7-kleisma up. ...
> As you say, we're scraping the bottom of the barrel. It can't be an
> uppercase character. In approximate keyboard order: It can't be
> `,~!|@#%^&()+-{}{}\/'.";?<>. It can't be qwtyuosdfhjxcvbnm. Already
used or
> rejected for any use.
>
> That only leaves $*_=:eripagklz.
>
> A lowercase character shouldn't be used unless it has a descender,
or
> no-ascender and is open at the bottom. That eliminates eaklz
leaving ripg.
> p and g are too big to represent something that small. Cant use $
because
> it is wavy not concave. I want to reserve colon for placing between
notes
> to form chords. _ is obviously down, not up. = suggests no
direction and is
> utterly unlike an arrow.
>
> That leaves *ri.
>
> I note that k isn't a bad looking down symbol and might be paired
with p
> for some use, for lack of anything else to pair it with and because
p's
> obvious partner b is already taken. I also note that e and a or g
and a
> might make a pair, and possibly $ and z. But none of these suit a
small
> right concave flag.
>
> r is more like |) or )|). i looks like an inverted ! which should
at least
> make it an up symbol, but I'm inclined to go with * because of its
> smallness and upwardness and because it seems better to use special
> characters rather than letters when possible.
>
> Do we want to consider something other than c for its partner? k
bears a
> vague resemblance to *, but it seems a bit too big. What do you
think?
I think that if we can't think of anything else, then it will have to
be either * and k or * and c. I agree that the k character is rather
large, so I would tend to prefer * and c, even if they aren't very
good as opposites.
I like the pair a and e, because the letters are small. True, they
don't look much like the 5:7 kleima symbols, but at this point
nothing will. I don't know which one should be up and which down --
a (for ascending) and e for (d_e_scending) might therefore be as good
a choice as any.
But of the two above, I think I would go for * and c, which, like the
17 comma, have a special character and letter as a pair.
--George
Message: 6309 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 23:52:20
Subject: Re: A common notation for JI and ETs
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> what's wrong with "commatic unison vector"?
I'd prefer not to call positive rational numbers "vectors" in general.
Message: 6310 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 03:02:49
Subject: Re: heuristic and straightness
From: Carl Lumma
>>>> >s it any more proportional to log(d) than log(n) in this
>>>> case? Since n~=d?
>>>
>>> No, and the spreadsheet sorted by d is also sorted by n.
>>
>> So it could just as well be (n-d)/(d*log(n))?
>
>a very different sorting. that would be heuristic error, not
>heuristic complexity. of course it's a very different sorting,
>since knowing log(n) or log(d) tells you nothing about (n-d).
? I was asking if the *error* heuristic could become
(n-d)/(d*log(n)) if we substituted log(n) for log(d).
>> The Mercator series?? And all the stuff on this page applies
>> only to ln, not log in general (which is what I assume Paul
>> meant), right?
>
>i meant ln. i always use matlab, in which "log" means ln.
Oh. How odd.
-Carl
Message: 6311 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 03:07:34
Subject: Re: heuristic and straightness
From: Carl Lumma
>>>>> >nd in that case it would have been easier to go straight
>>>>> to log(n*d).
>>>>
>>>> Straight to where (do you see log(n*d))?
>>>
>>> log(n*d) = log(n) + log(d)
>>
>> Of course... so we're coming from log(n*d), not going to it.
>
>what do you mean, we're coming from log(n*d)??
In your original message, you *start from* "is proportional to
log(n) + log(d)" and *arrive at* "Hence the amount of tempering
implied by the unison vector is approx. proportional to
(n-d)/(d*log(d))".
-Carl
Message: 6312 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 03:10:55
Subject: Re: heuristic and straightness
From: Carl Lumma
>> > do see that... you were already using the term for the
>> complexity heuristic at that time, right?
>
> no, gene introduced the word "heuristic".
K.
>>> w=log(n/d)
>>
>> Got that.
>>
>>> w~=n/d-1
>>
>> How do you get this from that?
>
> standard taylor series approximation for log . . . if x is
> close to 1, then log(x) is close to x-1 (since the derivative
> of log(x) near x=1 is 1/1 = 1).
Cool.
>>> w~=(n-d)/d
>>
>> Ditto.
>
> arithmetic. n/d - 1 = n/d - d/d = (n-d)/d.
Yeah, I thought the 1 was in the denominator.
-C.
Message: 6313 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 03:29:37
Subject: Re: heuristic and straightness
From: Carl Lumma
>> >k, but I still don't get how the "Mercator series" shown in (8)
>> dictates the rules for this approximation.
>
>Oh, I thought you had followed that.
>
>It's usually called the Taylor series. I don't know what
>Mercator's got to do with it.
On the mathworld page, it says "the Mercator series gives a
Taylor series for the natural logarithm", and in fact makes
it look like the Taylor series is the Mercator series.
>But anyway it's
>
> ln(1+x) = x - x**2/2 + x**3/3 + ...
>
> where x is small. Because x is small, x**2 must be even
> smaller, so you can use the first order approximation
>
> ln(1+x) =~ x
>
> In this case, 1+x is n/d, so x = n/d - 1
>
> ln(n/d) =~ n/d - 1
Thanks!
-Carl
Message: 6314 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 04:02:23
Subject: Re: heuristic and straightness
From: Carl Lumma
>> >t's usually called the Taylor series. I don't know what
>> Mercator's got to do with it.
>
>On the mathworld page, it says "the Mercator series gives a
>Taylor series for the natural logarithm", and in fact makes
>it look like the Taylor series is the Mercator series.
...by the way the page is formatted.
Turns out that the Mercator series is what we're talking
about; a special case of the Taylor series.
Taylor Series -- from MathWorld * [with cont.]
"A Taylor series is a series expansion of a function about
a point. ... "
-Carl
Message: 6315 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 18:01:03
Subject: Re: A common notation for JI and ETs
From: David C Keenan
Paul Erlich wrote:
Thanks. Those were very useful. In case anyone is wondering, the usage I
gave, that is referred to on Monz's page above, was that of Scala's
intnam.par which is that the difference between 3 major thirds and an
octave is the minor diesis (41.06 c untempered) and the difference between
4 minor thirds and an octave is the major diesis (62.57 c untempered).
Since for our purposes, we _are_ basing it strictly on "JI cents
considerations" (i.e. the untempered size), I think we can take it that the
"minimal diesis" at 27.66 c and the "small diesis" at 29.61 cents can
afford to be considered as some kind of comma, since there's a big gap
between them and all the other 5-prime-limit dieses. Given that the prime
exponent vectors for these are [5 -9 4] and [-10 -1 5] respectively, there
is unlikely to be much call for using them to notate rational pitches, and
certainly not for notating temperaments.
The next largest is the minor diesis of 41.06 c. By the way, calling this a
great or major diesis just seems silly to me since there are two larger
than it, and they are not particularly obscure or complex. I think
Mandelbaum and/or Würschmidt and/or Helmholtz and/or Ellis screwed up.
Monz's chart showing his size categories for 5-prime-limit "anomalies" is
very interesting and has the following agreement with ours so far.
The skhisma-kleisma boundary is between 2 and 6 cents.
The kleisma-comma boundary is between 13 and 18 cents.
There is a boundary between 53 and 59 cents.
Differences are:
We have found no need of a boundary between 23 and 28 cents.
We don't have a boundary between 34 and 39 cents, although we should
probably move our 40 cent boundary down to here.
We could use a boundary between 43 and 47 cents .
We have one between 65 and 69 cents that Monz doesn't.
The places where we want different or additional boundaries, _do_ appear as
gaps on Monz's chart, even when he has not seen fit to make them nominal
boundaries.
Here's how the names correspond so far:
Monz George and Dave
--------------------------------
skhisma schisma
kleisma kleisma
comma comma
small diesis comma
great diesis diesis
small semitone ediesis (and other larger anomalies)
>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
>> There are two slightly more popular pairs that would benefit from a
>higher
>> kleisma-comma boundary.
>>
>> N kleisma comma
>> 245 14.19 37.65
>> 7:13 14.61 38.07
>>
>> I note that neither of us is willing to bring the comma diesis
>boundary
>> down below 38.07 or 37.65 cents.
>
>But I'm willing to consider it if there's a good reason for it.
>> ...
Monz's chart has helped me to see that lowering the comma-diesis boundary
to just below 37.65 cents is a better solution than raising the
kleisma-comma boundary above 14.61.
Then we have
N comma diesis
245 14.19 37.65
7:13 14.61 38.07
>> I'll use the term ediasis (pron. ed-I-as-is, not ee-DI-as-is) for a
>diasis
>> larger than a half apotome, until someone tells me they like
>something else
>> better.
>
>Okay. (Somebody, *please* suggest something better; ediasis sounds
>too much like a disease.)
You saw my other suggestions didn't you? "biesis" and "diesoma". I think
they sound even more like diseases. There's not a lot you can do about that
when "diesis" itself sounds like a disease?
The prefix "edi" or "oedi" meaning swollen has the same etymology as
"Edipus" of the ancient Greek story. Edipus is literally "swollen feet"
(from walking so much).
>> Some things in the kleisma size range have been called semicommas.
>
>I just think that "kleisma" sounds better.
Me too. Good ole Shohe' Tanaka.
>>> ...
>>> The point here is that I thought that the comma (120:121, ~14.367c)
>>> between the next smaller pair of superparticular ratios (10:11 and
>>> 11:12) should be smaller than the lower size limit for a comma. If
>>> they were used as the two ("whole") tones in a tetrachord, their
>sum
>>> would be 5:6, which would leave 9:10 as the remaining interval
>>> (or "semitone") of the tetrachord. But to have a "semitone" in a
>>> tetrachord that is larger than either of the "whole" tones is
>absurd,
>>> hence a practical basis for a boundary.
>>
>> I find this argument interesting but not convincing. Why must the
>whole
>> tones be superparticular?
>
>They don't have to, but since there are actual examples of ancient
>Greek tetrachords with diatonic steps of 7:8 with 8:9 (Archytas), of
>8:9 with 9:10 (Didymus), and of 9:10 with 10:11 (Ptolemy's hemiolon),
>these being all of the possible cases, I found that it was possible
>to draw a conclusion from them.
>
>Interestingly, the next larger pair, 6:7 with 7:8 -- difference of
>48:49, ~35.697c -- adds up to an exact 3:4, leaving a semitone that
>vanishes.
A tetrachord with only 3 notes isn't exactly kosher is it? And we don't
have any other reason to go quite that low.
> Their difference is still a bit smaller than we were
>considering for the comma-diesis boundary, which makes me wonder if
>40 cents is still too large.
Yes. I'm down to 37.65 c now. Just under the 245-diesis.
>> Why must they even be simple ratios?
>
>Because a comma, by definition, is the difference between two
>rational intervals similar in size.
However they don't have to be _simple_. But never mind.
>> Anyway, some very small intervals have been called commas for a
>long time.
>> e.g. We have Mercator's comma at about 3.6 cents and Wuerschmidt's
>comma at
>> about 11.4 cents. These are from Scala's intnam.par.
>
>These are just examples that the term can have a broad or generic
>usage in addition to the more specific definition that we're seeking.
I'll buy that.
>>> You want the boundary to be somewhere between what we have been
>>> calling the 17 comma (~8.7c) and 17' comma (~14.730c). To
>>> accommodate both of these requirements, we could put the lower
>>> boundary for a comma at infinitesimally above 120:121, >14.37c.
>>> Would this be too large an upper limit for a "kleisma?" If so,
>why?
>>
>> No I can't really argue that, although it is getting close to
>double the
>> size of _the_ kleisma. I now want to put the boundary even a little
>higher
>> than you suggest, at just above 28431:28672 (or 14.614 c) so we
>have a 7:13
>> kleisma and a 7:13 comma (38.07 c) as mentioned above.
>>
>> This does mean we have the 17-comma and the 7:13-kleisma being
>notated with
>> the same symbol ~|(
>
>But it doesn't make much sense, though.
You're right.
>> but I can probably live with that. Or would you rather
>> have two 7:13 commas?
>
>This might be a good reason to make the comma-diesis boundary
>somewhere around 37 to 38 cents.
Yes.
> This would then put 1deg31 in the
>diesis range (at the lower end) -- 1deg31 is also functions as a 7-
>comma, but I think its dual use demonstrates that it's appropriate to
>have the boundary somewhere around this size.
But the size in cents of 1deg31 is irrelevant because it's tempered. I
expect it is called a diesis because it corresponds to some 5-prime-limit
comma whose untempered size would have made it a diesis even with our
earlier 40 cent cutoff. In the case of sagittal it remains a diesis because
it's a tempered 11-diesis.
>If 1664:1701 (~38.073c) is the 7:13 diesis, then 1377:1408 (~38.543c)
>and 44:45 (~38.906c) would become the 11:17 and 5:11 dieses. Do you
>know of any potential problems with these designations?
No problem.
>If nobody else says anything, then that's what's going to happen. I
>can't see making another boundary between semicomma and kleisma
>without a good reason.
I agree there is no need for a separate semicomma category.
>> diesis 1 diesis 2
>> N cents cents
>> ------------------------------
>> 5:13 43.83 46.39
>> 37 42.79 47.43
>> 11:19 40.33 49.89
>> 25:77 44.66 45.56
>>
>> It doesn't seem like a good idea to have a category that only
>covers the
>> range 40.00 to 45.11 cents, but that's what we need for the above.
>
>I'll have to look these over to see if there's really any musical
>need some of these. For example, the 46.39-cent 5:13 diesis would
>notate 10:13 as an interval of a third, but I can't imagine that
>anyone would want a third this large in a diatonic or heptatonic
>scale very often.
It _does_ seem like a good idea to me now to have a category of small
dieses between 37.65 c and 45.11 c. For now I'll call these "carcinomas".
That should make folks think _real_hard_ to come up with a better term. :-)
With this category, we don't have anything with two useful "anomalies" in
the same category until we get down to 32:49, which is waaaay down the
popularity list at number 145 with a 0.02% ocurrence. It has two kleismas,
from C, Cb - 10.81 c and B + 12.65 c.
Although, when I say that, I am not counting 11:35 which is considerably
more popular and has been mentioned before and has, from C, Cb - 11.64 c
and B + 11.82 c. But we can get away with this because these are
essentially a single kleisma.
To summarise:
0
schismina
0.98
schisma
4.50
kleisma
13.58
comma
37.65
carcinoma
45.11
diesis
56.84
ediasis
68.57
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 6316 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 18:08:32
Subject: Re: A common notation for JI and ETs
From: David C Keenan
On second thoughts, 13.47 cents might be a better choice for the
kleisma-comma boundary.
Message: 6317 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 18:08:32
Subject: Re: A common notation for JI and ETs
From: David C Keenan
On second thoughts, 13.47 cents might be a better choice for the=20
kleisma-comma boundary.
Message: 6318 - Contents - Hide Contents
Date: Thu, 06 Feb 2003 12:20:30
Subject: Re: A common notation for JI and ETs
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:
> Commatic UV seems okay to me. The terminology comes from
> Fokker, and refers to a quantity with magnitude and direction.
> What would you suggest Gene?
We
aren't in a vector space, so mathematically it's awfully dubious to be
talking about vectors. I think it is confusing and excessively
verbose. "Comma" is short and sweet.
Message: 6319 - Contents - Hide Contents
Date: Thu, 06 Feb 2003 12:25:03
Subject: Re: A common notation for JI and ETs
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
Ratios are one of the insufficiently
> general representations -- they don't work for inharmonic timbres.
Isn't this a complete red herring? We are now talking about lists or sets of partial tones, not vectors.
Message: 6320 - Contents - Hide Contents
Date: Thu, 06 Feb 2003 17:04:18
Subject: Re: A common notation for JI and ETs
From: Graham Breed
Me:
> Ratios are one of the insufficiently
>> general representations -- they don't work for inharmonic timbres.
Gene:
> Isn't this a complete red herring? We are now talking about lists or sets of partial tones, not vectors.
Intervals are defined as vectors in terms of a minimal subset of the
partials relative to the fundamental (which, for inharmonic timbres,
will probably be the whole set).
The only part of the definition here:
Vector -- from MathWorld * [with cont.]
they don't comply with is that you can't construct unit vectors. But
that can be fixed if we have to as there are cases where fractions do
creep in. They can be added, subtracted and multiplied by scalars. A
list, it appears, is merely an ordered set, and so doesn't support these
operations.
If all vectors are also lists, so they must be lists. But they look
like vectors as well.
As for sets, from
Set -- from MathWorld * [with cont.]
"A set is a finite or infinite collection of objects in which order has
no significance, and multiplicity is generally also ignored (unlike a
list or multiset)"
Well, we certainly can't ignore multiplicity. [1, 2, 0] is very
different to [1, 1, 0]. Even if we don't, are partials allowed to be
present a negative number of times? Even if that is a set, we're
writing it as a list or vector (except when we write it as a ratio, but
that won't work in general).
I'm sorry the CGI doesn't support this yet. For now, here's an old list
of tubulong temperaments:
2 4 6 10 14 16 18 20 22 26 30 34 36 38 44 46 5... * [with cont.] (Wayb.)
The unison vectors are not frequency ratios, so there's no other way to
represent them. The partials are:
2.82843 5.42326 8.77058 12.86626 17.70875 23.29741
The first unison vector is [-3, 2, 0, 0] That's -3*log2(2) +
2*log2(2.82843) = 2.9 microoctaves. I know it isn't quite a unison, but
if it isn't a vector, what is it?
Graham
Message: 6321 - Contents - Hide Contents
Date: Thu, 06 Feb 2003 17:12:06
Subject: Re: A common notation for JI and ETs
From: Graham Breed
I wrote:
> 2.82843 5.42326 8.77058 12.86626 17.70875 23.29741
>
> The first unison vector is [-3, 2, 0, 0] That's -3*log2(2) +
> 2*log2(2.82843) = 2.9 microoctaves. I know it isn't quite a unison, but
> if it isn't a vector, what is it?
Bwahahaha!!!!
This particular unison vector, as an emergent property of the system,
really is a unison.
Graham
Message: 6322 - Contents - Hide Contents
Date: Thu, 06 Feb 2003 19:17:49
Subject: Success at last
From: Gene Ward Smith
I finally got Clifford, the Maple Clifford algebra package to work in
my updated Maple. This means Graham and I should be on the same page
so far as wedge products go.
Message: 6323 - Contents - Hide Contents
Date: Thu, 06 Feb 2003 19:47:06
Subject: Re: Comment on notation
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
>> Have you two notated the 198-et, by the way?
>
> No, but I just tried, and it is difficult. The biggest problem is in
> finding a valid symbol for 10deg198. Best I can come up with is
>
> 198: )|( |~ ~|( /| |\ /|~ (|( ~|\ /|\ .(|\ (|)
>
> You will see that I have resorted to using a 5-schisma flag to
notate
> 10deg198 as the 7-ediesis 27:28. It could equally be '/|) the 7-
diesis
> 57344:59049. This is rather ugly either way.
Ugly is a good way of putting it, since it is evident that (/| and
|\) aren't a cure-all for our half-apotome problems. The cleanest
way to do it here (as well as in a lot of those divisions that
require something very close to 1/2-apotome) would be '|)). The
thing that has been keeping us from using it is that we don't have a
rational complement for |)). But should that stop us from
having '|)), which is its own rational complement?
I hesitate to suggest this, but with the pinch we're in, we could
possibly allow ''|)) as the rational complement of |)) -- if we could
this once allow a double-5-schisma, just as we allowed a double-5
comma. I've noticed that the 19-schisma is only rarely twice the
number of degrees in an ET as the 5-schisma, or I might have
suggested )|)), even if it's a three-flagger.
--George
Message: 6324 - Contents - Hide Contents
Date: Thu, 06 Feb 2003 20:01:48
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
> <d.keenan@u...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
>>
>>> I think that if we can't think of anything else, then it will
have to
>>> be either * and k or * and c. I agree that the k character is
rather
>>> large, so I would tend to prefer * and c, even if they aren't
very
>>> good as opposites.
>>>
>>> I like the pair a and e, because the letters are small. True,
they
>>> don't look much like the 5:7 kleima symbols, but at this point
>>> nothing will. I don't know which one should be up and which
down --
>>> a (for ascending) and e for (d_e_scending) might therefore be
as good
>>> a choice as any.
>>>
>>> But of the two above, I think I would go for * and c, which,
like the
>>> 17 comma, have a special character and letter as a pair.
>>
>> Sold!
>
> I'm going to push on here and suggest we use $ and z for the 23-
comma
> symbols |~ and !~
Yes, I'll buy that!
> That way we have a single ascii character for every flag, and could
> could then have at most two-character abbreviations for all the
> single-shaft symbols.
This also gives us single-character abbreviations for at least one
comma for each of the primes through 29.
--George
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