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Message: 8025 Date: Sat, 08 Nov 2003 20:50:27 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> What's the definition of "standard val"?
The vector consisting of round(n log2(p)) for primes p in ascending order up to the chosen prime limit, considered as defining a val.
Message: 8026 Date: Sat, 08 Nov 2003 12:57:41 Subject: Re: Eponyms From: Carl Lumma
>> What's the definition of "standard val"?
> >The vector consisting of round(n log2(p)) for primes p in ascending >order up to the chosen prime limit, considered as defining a val.
What's n? -C.
Message: 8027 Date: Sat, 08 Nov 2003 21:17:58 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> What's the definition of "standard val"?
> > > >The vector consisting of round(n log2(p)) for primes p in
ascending
> >order up to the chosen prime limit, considered as defining a val.
> > What's n?
The number you are finding a standard val for. We could adjust this number to the Zeta tuning or the Gram tuning, and have a Zeta val or a Gram val, by the way; in that case n would no longer be an integer.
Message: 8028 Date: Sat, 08 Nov 2003 22:52:44 Subject: Re: Comma size categories extended From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> I really haven't had very much time lately to participate in this > sort of discussion, but when it seemed that Dave had dropped out I > saw that there were things that hadn't been resolved, I felt that I > had to jump in and say something. If you want a couple of specific > examples to pursue this further, I can mention a couple of instances > from which we were trying to set the kleisma-comma boundary: > > 152:153 (~11.4c) is definitely a 17:19-kleisma, but 1114112:1121931 > (~12.1c) will either be either a (subordinate) 17:19-kleisma or the > 17:19-comma, depending on where the boundary is set. > > 135:136 (~12.8c) will either be the 5:17-kleisma or the 5:17-comma, > but 327680:334611 (~36.2c) also claims the name 5:17-comma with our > present comma/S-diesis boundary. > > These are issues that we still need to work out.
George, I hope you've seen my latest proposals by now in: Yahoo groups: /tuning-math/message/7458 * [with cont.] It resolves all the above issues. Previously we allowed our desire for the boundaries to fall between sagittal symbols, to distort our choices. The optimum size-category boundaries, for maximising the number of unique simple names, are always at the irrational square roots of 3-commas. And I now have a more complete proposal for naming commas when they do not have a unique name by simply using these size categories preceded by the ratio with powers of 2 and 3 removed. We simply prepend the following words in order of increasing absolute 3-exponent. complex supercomplex hypercomplex ultracomplex 5-complex 6-complex 7-complex ... My 10 year old son, Hunter, confirms that super, hyper, ultra is the correct progression, as this is used for the strengths of magic potions in Pokemon. :-) I don't expect these will ever really be needed. But here's a ridiculous example to make my intentions clear. Prime exponent vector Cents Name ----------------------------------------------- [-19 12] 23.46 Pythagorean comma [65 -41] 19.84 complex Pythagorean comma [-103 65] 27.08 supercomplex Pythagorean comma [149 -94] 16.23 hypercomplex Pythagorean comma [-187 118] 30.69 ultracomplex Pythagorean comma [233 -147] 12.61 5-complex Pythagorean comma [-271 171] 34.31 6-complex Pythagorean comma [298 -188] 32.46 7-complex Pythagorean comma [-336 212] 14.46 8-complex Pythagorean comma [382 -241] 28.84 9-complex Pythagorean comma [-420 265] 18.08 10-complex Pythagorean comma [466 -294] 25.23 11-complex Pythagorean comma
Message: 8029 Date: Sat, 08 Nov 2003 15:33:32 Subject: Re: Eponyms From: Carl Lumma
>> >> What's the definition of "standard val"?
>> > >> >The vector consisting of round(n log2(p)) for primes p in >> >ascending order up to the chosen prime limit, considered >> >as defining a val.
>> >> What's n?
> >The number you are finding a standard val for.
Then what's p!? Besides using two vals to find a lt, can you give an example of what a single standard val would be good for? -Carl
Message: 8030 Date: Sat, 08 Nov 2003 00:55:40 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> Mapping a harmonic series consistently into a 12-tone octave (without > skipping over any primes to reach other primes, and without skipping > over any odd harmonics to reach other odd harmonics) yields a 19- > limit set that's a favorite of mine: > > 16:17:18:19:20:21:22:24:25:26:28:30:32
As I remarked, the 3, 5, 7, 9 and 13 limits map respectivly to the 2, 3, 4, 5, and 7 standard vals. However, I had forgotten until I checked it again that the 11-limit doesn't map to anything; in fact the 13-limit seems to be the last odd limit whose complete otonal chord defines a val. However, we can proceed as George does above; he gives an incomplete 25-odd-limit chord, which defines the following 19-limit val: g12 = [12, 19, 28, 34, 42, 45, 49, 51]. While g12 isn't what I've called the "standard" val h12, it in fact is preferable, since it has a lower consistent badness score.
Message: 8031 Date: Sat, 08 Nov 2003 01:15:45 Subject: Re: Eponyms From: Carl Lumma
>As I remarked, the 3, 5, 7, 9 and 13 limits map respectivly to the >2, 3, 4, 5, and 7 standard vals. However, I had forgotten until I >checked it again that the 11-limit doesn't map to anything; in fact >the 13-limit seems to be the last odd limit whose complete otonal >chord defines a val.
You lost me. And you're site's apparently down. -Carl
Message: 8032 Date: Sat, 08 Nov 2003 06:39:53 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
> >As I remarked, the 3, 5, 7, 9 and 13 limits map respectivly to the > >2, 3, 4, 5, and 7 standard vals. However, I had forgotten until I > >checked it again that the 11-limit doesn't map to anything; in fact > >the 13-limit seems to be the last odd limit whose complete otonal > >chord defines a val.
> > You lost me.
Consider the otonal chord of the n-odd-limit. This has (n+1)/2 octave reduced elements, 1 < q[i] <= 2, where the q[i], i from 1 to (n+1)/2, are arranged in increasing size. The n-odd-limit has pi(n) primes; we may solve the (n+1)/2 linear equations for the val which sends q[1] to 1, q[2] to 2, up to q[(n+1)/2]=2 to (n+1)/2. These linear equations have a unique solution in the 3, 5, 7, 9, and 13 odd limits. For 3 we get [2, 3], for 5 [3, 5, 7] and so forth--the standard vals in the respective prime limits 3, 5, 7, 7, 11 for 2, 3, 4, 5, and 7.
Message: 8033 Date: Sat, 08 Nov 2003 09:00:25 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> For 3 we get [2, 3], for 5 [3, 5, 7] and so forth--the standard
vals
> in the respective prime limits 3, 5, 7, 7, 11 for 2, 3, 4, 5, and 7.
Should be prime limits 3, 5, 7, 7 and 13. To give a simple example, in the 5-limit, (5+1)/2 = 3, and we may start from the 3-chord [5/4, 3/2, 2]. If we solve for a val [a, b, c] such that 5/4, or [-2, 0, 1] is mapped to 1, 3/2 is mapped to 2, and 2 is mapped to 3 we get the equations a5 - 2 a2 = 1, a3 - a2 = 2, and a2 = 3, the solution of which is a2 = 3, a3 = 5, and a5 = 7, so the val in question is uniquely determined to be [3, 5, 7], the standard 3-val for the 5-limit.
Message: 8034 Date: Sat, 08 Nov 2003 09:56:51 Subject: _Alternative Rock Chord_ and 43 (was: Eponyms) From: monz hi George and paul, --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:
> > so the primes are arranged as > > 2 3 , 5 7 11 , 13 17 19 , 23 29 31 , 37 41 43 , 47 53 59 , > > 61 67 71 > > > > looks like the next comma after 31 makes sense too -- > > isn't 43 the highest limit used by george secor at least > > in some context?
> > If you're referring to something that I posted some time > ago, I think that was 41. (But I did happen to come across > a use for 43 when I was subsequently rummaging through some > of my old papers.) Trouble with 43 is that 32:43 readily > invites confusion with 3:4.
hmmm... come to think of it, i wrote a piece a few years back called _Alternative Rock Chord_, which in fact was inspired by a discussion i was having with paul on the tuning list. http://sonic-arts.org/monzo/altrock/altrock.mid - Type Ok * [with cont.] (Wayb.) the piece is minimalist, featuring two repeating phrases which have only one chord each. the first chord is tuned to 12edo, then the rest are JI. all voices but one stay on the same pitch for the rest of the pattern, but one voice moves 4/3 - 21/16, 43/32, 11/8, 43/32 - 21/16 - 4/3 then back to 12edo again, then the whole pattern repeats with new parts added each time. the frequencies of the center section of the moving part have the proportions 42:43:44:43:42, heard against the 4:5:6:7 drone of the other voices. it was my specific intention in this piece to tune up a variety of "4ths" and play them together with the 4:5:6:7 chord. traditional music-theory prohibits sounding the "4th" along with the "major-3rd", but that's exactly what a lot of alternative-rockers did in the 1990s. so the idea arose first in my imagination by wondering what the differently-tuned "4th"s would sound like and thinking about the numbers. since the piece is minimalist, it was easy to use copy-and-paste to create many similar sections, and then just add new instrumental parts and pitch-bend data. when i finished putting all the pitch-bend data into the MIDI file, there was still a big chunk of music at the end that was still in 12edo. strangely to my ears, and finally heard it, when the piece came back to 12edo at the end, it sounded like a resolution of some sort. so i decided to begin the piece in 12edo, and to frame each repetition by constantly returning back to 12edo. just some observations about this piece that i thought interesting. the point of posting it here is that i used 43 as part of my experiment in tuning the notes of that moving part, and after hearing 43, deliberately decided to keep it and not retune it to something simpler (i.e., lower prime factor). - monz
Message: 8035 Date: Sun, 09 Nov 2003 12:46:52 Subject: Re: Enharmonic diesis? From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> Monz, > > In > Definitions of tuning terms: 19edo and 1/3-com... * [with cont.] (Wayb.) > you refer to 625:648 (62.57 c) as the enharmonic diesis. > > But the enharmonic dieses you described in > Tutorial on ancient Greek Tetrachord-theory * [with cont.] (Wayb.) > would seem to be in the approximate range of 45 to 56 cents > (a fourth minus either a Pythagorean or just major third > and then divided by two). > > And elsewhere, including Margo Schulter here > Yahoo groups: /MakeMicroMusic/message/166 * [with cont.] > I find 125:128 (41.06 c) being called the enharmonic > diesis, or simply the enharmonic. > > In Scala, or here > Stichting Huygens-Fokker: List of intervals * [with cont.] (Wayb.) > we find > 31:32 (54.96 c) called the Greek enharmonic, and > 512:525 (43.41 c) called the Avicenna enharmonic diesis. > > Apart from your 19edo page, they all seem to point away > from considering anything as large as 625:648 (62.57 c) to > be an enharmonic diesis. > > Just thought you'd like to know.
thanks for that. i have to point out that the reason i called that 19edo (or 1/3-comma meantone ... they're essentially the same size) interval the "enharmonic diesis" is simply because that's the way that particular interval functions in those tunings, i.e., the difference between Ab and G#. you're correct that theory treatises generally refer to the "enharmonic diesis" as being somewhere in the neighborhood of 41 cents, which is roughly the size of the JI version 125:128 = [3, 5]-monzo [0, -3]. but in 19edo, treating it as 1/3-comma meantone and mapping the pitch-names according to generator "5ths", the difference between G#:Ab is that between the 12th and 13th degrees of 19edo, thus one degree of 19edo which equals ~63.15789474 cents. so in this particular case, the epithet "enharmonic" refers not to the size-range of the interval, but to its harmonic or scalar function. -monz
Message: 8037 Date: Sun, 09 Nov 2003 22:47:39 Subject: Re: Enharmonic diesis? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > Apart from your 19edo page, they all seem to point away > > from considering anything as large as 625:648 (62.57 c) to > > be an enharmonic diesis. > > > > Just thought you'd like to know.
> > > > thanks for that. > > i have to point out that the reason i called that 19edo > (or 1/3-comma meantone ... they're essentially the same size) > interval the "enharmonic diesis" is simply because that's > the way that particular interval functions in those tunings, > i.e., the difference between Ab and G#. > > you're correct that theory treatises generally refer to > the "enharmonic diesis" as being somewhere in the neighborhood > of 41 cents, which is roughly the size of the JI version > 125:128 = [3, 5]-monzo [0, -3]. > > but in 19edo, treating it as 1/3-comma meantone and mapping > the pitch-names according to generator "5ths", the difference > between G#:Ab is that between the 12th and 13th degrees of > 19edo, thus one degree of 19edo which equals ~63.15789474 > cents. > > so in this particular case, the epithet "enharmonic" refers > not to the size-range of the interval, but to its harmonic > or scalar function.
Silly me. I didn't read far enough. I was just searching on "enharmonic diesis" and saw the value of 62.565148 cents and the terms "great" and "diminished second" associated with it. But you are quite correct, in precise 1/3-comma-meantone (but not 19-EDO) the tempered size of 125;128 is precisely the untempered 625:648. So who was the turkey who called 125:128 the "great" diesis? The sooner we lose that name the better. I notice Scala doesn't use it. And is it correct to call it a diminished second? Isn't G# to Ab a _doubly_ diminished second? But considered as a doubly diminished second (i.e. a Pythagorean comma), its tempered size in 1/3-comma meantone is _minus_ 62.565148 cents.
Message: 8038 Date: Sun, 09 Nov 2003 23:08:55 Subject: Re: Enharmonic diesis? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> 125:128 = [3, 5]-monzo [0, -3].
I've been meaning to say: I don't think it is a good idea to use this terminology "[3, 5]-monzo". I always read the "[3, 5]" _as_ a monzo and then do a double-take when I find the _real_ monzo such as the "[0, -3]" above. After all, [3, 5]-monzo translates as 3^3*5^5-monzo or 84375-monzo. I think it would be better to write e.g. 125:128 = 3,5-monzo [0, -3]. or just 125:128 = [0, -3] After all, if the reader doesn't already know what the square brackets mean, they probably won't be much the wiser for being told it is a "3,5-monzo". "3,5-exponent vector" would be more useful. But I understand you're saying "3,5" here because there has recently been some vehement objection by one party, to the practice of leaving out the 2-exponent, and the recently-proposed selective use of commas(punctuation sense) can't be considered standard yet.
Message: 8039 Date: Sun, 09 Nov 2003 21:43:59 Subject: Re: Linear temperament names? From: Carl Lumma
>Carl Lumma wrote:
>> Nooooooooooooooooooooooo!
> >Thanks Carl. Rest assured that I've taken this carefully reasoned >defense into account in the above. ;-) > >-- Dave Keenan
That was to Paul, but I gotta say you're approaching "intervention" territory with this stuff. -Carl
Message: 8040 Date: Sun, 09 Nov 2003 02:11:51 Subject: Enharmonic diesis? From: Dave Keenan Monz, In Definitions of tuning terms: 19edo and 1/3-com... * [with cont.] (Wayb.) you refer to 625:648 (62.57 c) as the enharmonic diesis. But the enharmonic dieses you described in Tutorial on ancient Greek Tetrachord-theory * [with cont.] (Wayb.) would seem to be in the approximate range of 45 to 56 cents (a fourth minus either a Pythagorean or just major third and then divided by two). And elsewhere, including Margo Schulter here Yahoo groups: /MakeMicroMusic/message/166 * [with cont.] I find 125:128 (41.06 c) being called the enharmonic diesis, or simply the enharmonic. In Scala, or here Stichting Huygens-Fokker: List of intervals * [with cont.] (Wayb.) we find 31:32 (54.96 c) called the Greek enharmonic, and 512:525 (43.41 c) called the Avicenna enharmonic diesis. Apart from your 19edo page, they all seem to point away from considering anything as large as 625:648 (62.57 c) to be an enharmonic diesis. Just thought you'd like to know.
Message: 8041 Date: Mon, 10 Nov 2003 16:35:32 Subject: Re: Enharmonic diesis? From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >
> > G#:A is a minor-2nd, so G#:Ab is simply a diminished-2nd. > > > > examples of a doubly-diminished-2nd would be G#:Abb or Gx:Ab .
> > Yes, of course. Thanks for taking the time to correct me.
nothing to it. i teach my students this stuff every day. (... the minor-2nd and diminished-2nd, that is. only the more advanced students get into double-sharps and double-flats ... it usually takes a year or two of lessons before we get that far into theory.) -monz
Message: 8042 Date: Mon, 10 Nov 2003 16:40:31 Subject: Re: Enharmonic diesis? From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > > > i'm also trying to establish here a new standard usage > > of brackets. since using commas to separate groups of > > exponents in a monzo eliminates the need for contrasting > > brackets to indicate the presence or absence of prime-factor 2, > > i propose that we use angle-brackets for the prime-factors > > themselves, and square-brackets for the actual monzo of > > the exponents.
> > Well I suppose you could, but I don't think this is necessary, > and we could save the angle-brackets for something more > important. There is obviously no need for selective use of > commas between the primes themselves. They provide no > additional information there.
huh? they would simply separate the groups of primes to show how the exponents are grouped. i know that that's "a given", so i guess maybe you're right.
> I would simply call these 2,3,5-monzos.
simply using a comma between *every* prime, and no spaces. i suppose i like that. (i don't sound too convinced, tho.)
> Of course the important thing to know is whether they > are 2,...-monzos (complete monzos, octave specific > monzos) or 3,..-monzos (2-free monzos, octave equivalent > monzos).
of course ... which i why i continue to like labeling the monzo with its constituent prime-factors, altho the comma convention makes it unnecessary.
> Maybe a good use for angle-brackets would be for wedgies > since, as I understand it, these are in a _very_ different > domain from that of monzos, and the angle-brackets are > suggestive of wedges themselves.
great minds thinking alike! ;-) i had actually already thought of that too ... but since my understanding of wedgies is lagging far behind that of many of you others, i'll refrain from commenting further. -monz
Message: 8043 Date: Mon, 10 Nov 2003 20:59:41 Subject: Re: Comma size categories extended From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> I'm sorry, but I can't stop playing with the idea of systematic
comma
> names. ...
I see that you've finalized the boundaries (below), so least I'm getting something out of this.
> Here's a proposal to extend the systematic comma size categories out > to 115 cents. ... > > ... > Size category name Boundary Alternative name > ---------------------------------------------------------- > 0 c > schismina > [-84 53]/2 ~= 1.81 c > schisma > [317 -200]/2 ~= 4.50 c > kleisma > [-19 12]/2 ~= 11.73 c > comma > [-57 36]/2 ~= 35.19 c > minor-diesis or small-diesis > [8 -5]/2 ~= 45.11 c > diesis or medium-diesis > [-11 7]/2 ~= 56.84 c > major-diesis or large-diesis > [-30 19]/2 ~= 68.57 c > chromatic-semitone or small-semitone > [35 -22]/2 ~= 78.49 c > limma or medium-semitone > [-3 2]/2 ~= 101.96 c > diatonic-semitone or large-semitone > [62 -39]/2 ~= 111.88 c > apotome > [-106 67]/2 ~= 115.49 c
I was under the impression that "chromatic semitone" and "apotome" were alternate names for the same thing (at least in JI), likewise "diatonic semitone" and "limma". I think that "limma" and "apotome" are okay, but I would prefer to see different names for the other 2 categories.
> ... > George, > Please remind me why we didn't use the terms minor-diesis, diesis, > major-diesis?
We spent a lot of time looking for meaningful terms that would result
in abbreviations that would require different letters and would also
be easy to remember. Small, medium, and large ("S", "M", and "L")
was by far the best thing we could come up with (all in upper case).
("Schisma" is also abbreviated with "s", but that one, along with
kleisma, is lower case.)
--George
Message: 8044 Date: Mon, 10 Nov 2003 21:42:39 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> > wrote: >
> > Mapping a harmonic series consistently into a 12-tone octave
(without
> > skipping over any primes to reach other primes, and without
skipping
> > over any odd harmonics to reach other odd harmonics)
I meant to say, "and without skipping over any *non-prime* odd harmonics to reach other odd harmonics)"
> > yields a 19- > > limit set that's a favorite of mine: > > > > 16:17:18:19:20:21:22:24:25:26:28:30:32
> > As I remarked, the 3, 5, 7, 9 and 13 limits map respectivly to the > 2, 3, 4, 5, and 7 standard vals. However, I had forgotten until I > checked it again that the 11-limit doesn't map to anything;
This is why I personally desire to use JI (or a microtemperament) that is at least 13-limit.
> in fact > the 13-limit seems to be the last odd limit whose complete otonal > chord defines a val. However, we can proceed as George does above;
he
> gives an incomplete 25-odd-limit chord, which defines the following > 19-limit val: g12 = [12, 19, 28, 34, 42, 45, 49, 51]. While g12
isn't
> what I've called the "standard" val h12, it in fact is preferable, > since it has a lower consistent badness score.
It's also possible to make a 17-limit decatonic set (or incomplete 21- odd-limit chord) by the same method: 16:17:18:20:21:22:24:26:28:30:32 Taking every 3rd tone of the above scale results in a chord that returns to the starting tone at the triple-octave. These decatonic "4ths" are the same general size as heptatonic "3rds", and there are 4 triads in the scale that have perfect fifths (hence four useful modes for this scale). 4:5:6:15/2:9:11:14:17:21:26:32 --George
Message: 8045 Date: Mon, 10 Nov 2003 04:27:05 Subject: Re: Linear temperament names? From: Dave Keenan Paul Erlich wrote:
> Despite his opinion above, Gene first gave the data for all of these, > as well as much more complex ones, and i just named the ones simpler > than the atom of Kirnberger so that one would have something other > than numbers, numbers, numbers, to refer to on Monz's ET page.
To remind other readers: We're talking about the diagram and table in Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) Dave Keenan:
> > Whether we have sharp cutoffs or gradual > > rolloffs on error and complexity, there has to be _some_ such for > > naming purposes. Doesn't there?
Paul Erlich:
> The atom of kirnberger is of historical interest as a result of the > means of setting one of its associated temperaments, namely 12-equal. > It's a bit outside the usual direct application of temperament, which > is why i put 12 in paretheses in the table.
I think these parentheses are a bit subtle - easily missed. An asterisk and corresponding footnote might be better. However ... The atom of Kirnberger may be of historical interest, but why would anyone be interested in using the 5-limit linear temperament in which it vanishes, as opposed to using an ET that happens to be a member of it? If all 12 chains were extended equally you'd need 108 notes before a single minor third appeared! Even if this linear temperament (and not just its comma) is of interest because the comma was used in the tuning of 12-equal, I don't see why that should imply that all less-complex 5-limit temperaments are also of interest. I'm suspecting that blind faith in log-flat badness measures may have something to do with this. I think we should take some account of the fact that we're supposed to be considering musical purposes for actual human beings here, not pure mathematics. Please explain to me why anyone would want to make musical use of any 5-limit linear temperament where you need more than 10 generators to approximate some 5-limit ratio, when we have schismic that can get us errors of less than a quarter of a cent, with only 9 generators? I'll grant that Aristoxenean is a special case and might be included despite needing 12 generators, because of its relationship to 12-ET. Don't bother with a "you just never know" type of answer, because if you're going to clutter an already cluttered (but very clever) diagram with lines corresponding to them, and give them meaningless or cryptic names and fill up a huge table (43 temperaments!), then you'd better do a lot better than that. Particularly given the fact that we can now generate temperaments on demand, for specific purposes, using Graham's Breed's online temperament finder. Linear Temperament Finding Home * [with cont.] (Wayb.) The same goes for 5-limit linear temperaments having any error greater than the one that uses the same neutral third generator as both major and minor third. Why would anyone care about such so-called 5-limit temperaments? ------- Now we're talking about the one called "minortone" in the abovementioned table. Dave Keenan:
> > Well I dunno about anyone else, but saying "two chains of 183 c > > generators" is something I can associate with a helluvalot better,
Paul Erlich:
> But it doesn't uniquely signify a temperament. I believe Graham has > discussed 13-limit generators which are fifths within a fraction of a > cent in optimal size. In order to actually temper, a temperament must > have a mapping associated with it.
Sure. But giving a period and generator (and odd-limit) gets you to a small list of possible mappings which can easily be ranked by error or complexity. The best one (within reasonable limits (or rolloffs) of error and complexity) can be given the simplest name, e.g. "minortone", and the less accurate ones can be called "inaccurate minortone", "super-inaccurate minortone" and the more complex ones "complex minortone", "supercomplex minortone", etc. That way it would be rare to ever need to refer to those with the longer names. Dave Keenan:
> > IMHO the 5-limit temperament you describe above is so complex it > > doesn't need a name at all.
Paul Erlich:
> It describes 46, 125, 171, 217 equal temperaments, barely used so far > but why not?
So it includes some interesting ETs, but these ET's are also included in many other temperaments. It's what's special about the actual temperament (the line between the ETs) that we should be asking here. And the answer is "practically nothing".
> Look, if it'll make you happy i'll replace all the names > on Monz's page with SINGLE LETTERS except for meantone and schismic > (or whatever you say) . . . that way the *signification purpose* is > not lost . . .
I earlier wrote: "I certainly don't want you to use letters instead of names in those wonderful diagrams. I'm just saying I think it has gone far enough, and besides there is at least _some_ kind of logic to most of those names." But I hadn't looked very closely at that stage, and I think I just wanted out, with no hard feelings. Now I want to address those temperaments where the names you've given have no discernable logic (and a few that do). These are (25): father, beep, misty, escapade, amity, parakleismic, semisuper, vulture, enneadecal, semithirds, vavoom, tricot, counterschismic, ennealimmal (5-limit), minortone, kwazy, astro, whoosh, monzismic, egads, senior, gross, pirate, raider, atomic. Note that this includes everything that comes after schismic in the table. In my opinion, the diagram (and table) would be greatly improved if these did not appear at all, because they are either too inaccurate or too complex to be of any likely musical use. By removing these you would reduce clutter on the diagram and focus attention on the temperaments that deserve it. If anyone really wants a 5-limit temperament that has an error less than a thousanth of a cent, and they don't mind if they need more notes than a grand piano to get a single triad, then you can direct them to Graham's temperament finder. But if you really think these 25 should stay, then I'll take you up on your offer and ask that you refer to them with single letters, except where they are 5-limit subsets of higher-limit temperaments that _do_ deserve a name, in which case their names should be followed by "(5-limit)". Carl Lumma wrote:
> Nooooooooooooooooooooooo!
Thanks Carl. Rest assured that I've taken this carefully reasoned defense into account in the above. ;-) -- Dave Keenan
Message: 8046 Date: Mon, 10 Nov 2003 21:59:52 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> Tossing out powers such as 6561/6400 is what I'd recommend also,
i thought that went without saying.
> though Dave might not go for it.
why not? dave has expressed interest in tuning systems where a single just lattice does not suffice as a derivation for all the pitches, but tempering out 6561/6400 simply leads to torsion and not to such a tuning system.
Message: 8047 Date: Mon, 10 Nov 2003 22:00:19 Subject: Re: Comma size categories extended From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...> wrote:
> > I really haven't had very much time lately to participate in this > > sort of discussion, but when it seemed that Dave had dropped out
I
> > saw that there were things that hadn't been resolved, ... > > ... > > These are issues that we still need to work out.
> > George, > > I hope you've seen my latest proposals by now in: > Yahoo groups: /tuning-math/message/7458 * [with cont.] > > It resolves all the above issues. Previously we allowed our desire
for
> the boundaries to fall between sagittal symbols, to distort our > choices. The optimum size-category boundaries, for maximising the > number of unique simple names, are always at the irrational square > roots of 3-commas.
Good, I'm glad to see that you have been able to finalize this.
> And I now have a more complete proposal for naming commas when they
do
> not have a unique name by simply using these size categories
preceded
> by the ratio with powers of 2 and 3 removed. We simply prepend the > following words in order of increasing absolute 3-exponent. > > complex > supercomplex > hypercomplex > ultracomplex > 5-complex > 6-complex > 7-complex > ... > > My 10 year old son, Hunter, confirms that super, hyper, ultra is the > correct progression, as this is used for the strengths of magic > potions in Pokemon. :-) > > I don't expect these will ever really be needed. But here's a > ridiculous example to make my intentions clear. > > Prime > exponent > vector Cents Name > ----------------------------------------------- > [-19 12] 23.46 Pythagorean comma > [65 -41] 19.84 complex Pythagorean comma > [-103 65] 27.08 supercomplex Pythagorean comma > [149 -94] 16.23 hypercomplex Pythagorean comma > [-187 118] 30.69 ultracomplex Pythagorean comma > [233 -147] 12.61 5-complex Pythagorean comma > [-271 171] 34.31 6-complex Pythagorean comma > [298 -188] 32.46 7-complex Pythagorean comma > [-336 212] 14.46 8-complex Pythagorean comma > [382 -241] 28.84 9-complex Pythagorean comma > [-420 265] 18.08 10-complex Pythagorean comma > [466 -294] 25.23 11-complex Pythagorean comma^
These could be even be abbreviated 3C, 3C-x, 3C-2x, 3C-3x, etc. (or something similar), which goes to show that you can have comma names with abbreviations that are short enough to label positions on a lattice! How 'bout that! --George
Message: 8048 Date: Mon, 10 Nov 2003 06:35:21 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Carl Lumma wrote:
> >> Nooooooooooooooooooooooo!
> > > >Thanks Carl. Rest assured that I've taken this carefully reasoned > >defense into account in the above. ;-) > > > >-- Dave Keenan
> > That was to Paul, but I gotta say you're approaching "intervention" > territory with this stuff.
I don't understand what you mean by ""intervention" territory". Please explain.
Message: 8049 Date: Mon, 10 Nov 2003 22:03:23 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...>
> wrote:
> > So I think that there are many that would agree with 11 as a
> boundary. > > I've recently done some pieces with full 13-limit chords as
harmony;
> the results may convince people that stopping earlier is a good
plan.
> So far as complete otonal and utonal chords go, however, 5-limit
has
> a natural 3-et triadic nature, 7-limit 4-et tetradic, 9-limit 5-et > quintadic, and 13-limit 7-et septadic. For 11-limit complete
harmony,
> we are stuck with a 6 val, which is a little ungainly.
i think george secor was trying to make a similar point in explaining why the 11-odd-limit was not of much use for him for compositional purposes. on the other hand, if you listen to prent rodgers' music, it becomes really hard to deny as a harmonic possibility.
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