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Message: 7925 Date: Sat, 01 Nov 2003 11:13:04 Subject: I get the message! From: Dave Keenan I guess I'm a bit slow on the uptake, but I think I'm getting the message now, from both Gene and Carl. If I may be allowed a little exaggeration: Dave, we don't want to know about any systematic naming schemes for temperaments or commas. Stop spoiling our fun. We were having a fine time making up silly names for everything we could churn out, until you turned up. Well, I shall go and leave you in peace. Peer review? More like peer insult-exchange. Gene, I apologise again for the personal insult. You said you did not call me names, and that's technically true. However I pride myself on my teaching and writing ability (perhaps mistakenly) as well as my system design ability, and so I'm afraid I do take it as a personal insult when someone who hasn't even read my article makes assumptions about what my method of exposition will be and declares them "sloppy", using the word three times, no less. You also said I was "going about it in a very, very bad way", and implied that my readers would need a "secret decoder ring" to understand my article. All without having read any of it. Yahoo groups: /tuning-math/message/7236 * [with cont.] I hope you can now understand why I found all this far more hurtful even than being called an anal retentive. But even if I felt insulted, I should not have responded in kind. I'm sorry. By the way, you probably don't realise that before you arrived, practically _all_ the prime-exponent vectors we used were 2-free, and no one had a problem with that. Paul, 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. Regards, -- Dave Keenan

Message: 7926 Date: Sat, 01 Nov 2003 12:00:05 Subject: Re: Eponyms From: monz hi Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >

> > so if they're being described as monzos, just leave out > > the first exponent of the vector and the first prime-factor > > of the label. > > > > > > ... looks like Gene and i support each other on this method > > of description.

> > I think we need a way of distinguishing 2-free monzos > from complete information monzos. I suggest <4, -1> vs > [-4, 4, -1] to distinguish the two ways of representing > 81/80; the corresponding octave class could be (4, -1). > The rule would be [] represents an interval, <> represents > an interval in the standard octave 1 <= q < 2, and () > represents the octave class whose represetative is given > by the corresponding <>.

i like that idea a lot !!!!! :) :) awesome! -monz

Message: 7927 Date: Sun, 02 Nov 2003 07:28:02 Subject: huge list of 11-limit intervals From: monz spurred on by the work i did on Dave's list of "kommas", i've made a huge list of all the 11-limit intervals where the <3, 5, 7, 11>-monzo exponents are <-15...+15, -10...+10, -4...+4, -2...+2>, arranged in order by cents. it takes quite a while to download ... there are over 28,000 intervals in the list. i hope it will be very useful as a reference. Definitions of tuning terms: interval, (c) 199... * [with cont.] (Wayb.) -monz

Message: 7928 Date: Sun, 02 Nov 2003 17:33:47 Subject: Re: huge list of 11-limit intervals From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >

> > it takes quite a while to download ... there are over > > 28,000 intervals in the list. i hope it will be very > > useful as a reference.

> > I am, of course, expecting a name for each and every one of them.

of course! ... but this time, we use *my* naming convention! ;-) -monz

Message: 7929 Date: Sun, 02 Nov 2003 08:13:21 Subject: Re: Capstone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi paul, > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:

> > it's the same thing that's shown in the linear temperament > > table on your et dictionary page -- the mapping from > > generators to primes. so for example, the third entry in > > each of the vectors, 22 and -3 respectively, refer to the > > fact that the third prime, 7, is approximated in the > > temperament by stacking 22 of the first generator > > (otherwise known as the period) and -3 of the second > > generator (otherwise known as, simply, the generator). > >

> > > and is [1/9, 1/24] meant to represent 2^(1/9) and 2^(1/24)?

> > > > yes.

> > > > oh, OK. thanks. > > > > > -monz

sorry, the third prime is 5, of course . . .

Message: 7931 Date: Sun, 02 Nov 2003 10:13:34 Subject: Re: A remarkable property of 270 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> > in all cases where the five commas are linearly independent.

> Sometimes

> > we get what we might regard as 540, but this is just a doubling

of

> 270. > > in other words, torsion?

Fraid so, if you try to use it for a block.

Message: 7932 Date: Sun, 02 Nov 2003 10:21:38 Subject: Re: huge list of 11-limit intervals From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> it takes quite a while to download ... there are over > 28,000 intervals in the list. i hope it will be very > useful as a reference.

I am, of course, expecting a name for each and every one of them.

Message: 7933 Date: Sun, 02 Nov 2003 00:15:19 Subject: Re: Capstone temperament From: monz hi Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> Before we get to the range which defines the > supertemperament (where the nullspace of the matrix > of monzos has dimension one, defining a val, and thus > an equal temperament) we have a range where the > nullspace is dimension two, and defines a linear > temperament--which I've mentioned before, under > the name of the capstone temperament. > Here are mappings for capstones up to the 19-limit, and > generators in terms of the corresponding supertemperament. > (This last is a little crude, but does get us into the > ballpark.) > > 5 limit: 81/80 meantone > > 7 limit: [[9, 15, 22, 26], [0, -2, -3, -2]] ennealimmal > generators: [1/9, 1/24]

this looks important to me, and i want to understand it. can you help, by labeling these lists of numbers? in this example, i think i recognize [0, -2, -3, -2] as a monzo, but it doesn't make sense as a [2, 3, 5, 7]-monzo. and is [1/9, 1/24] meant to represent 2^(1/9) and 2^(1/24)? if not, then what? and what about [9, 15, 22, 26]? what's that? Gene, it would help me a lot if you would *always* put a brief legend before lists like this. pretty please? -monz

Message: 7934 Date: Sun, 02 Nov 2003 00:17:37 Subject: Re: I get the message! From: monz hi Dave, --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>

> > [Dave Keenan] > > However I pride myself on my teaching and writing ability

> (perhaps mistakenly) as well as my system design ability, > > I think you're a great teacher, writer, and system designer. > > -Carl

i'll second that, particularly the "system design" part. -monz

Message: 7935 Date: Sun, 02 Nov 2003 00:28:56 Subject: Re: Eponyms From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >

> > and as i've already been arguing with the ratios, forget it. > > > > there's almost always nothing valuable about retaining the > > data for prime-factor 2, unless it need be considered for > > (to cite two examples i can think of quickly): > > > > - actual orchestral scoring where the 8ve-register must > > be considered, or > > > > - analyzing ancient Greek and Roman theory, which was > > based on 4:3 "perfect-4ths" and always specified 8ves, > > and gave different names to notes an 8ve apart.

> > if you don't retain data for prime-factor 2, how is your > software able to handle torsion?

ah, thanks for mentioning a third important example! the software retains the data for 2, but the default in *displaying* the data to the user is to start the prime-series with 3. if the user wants to see the data for 2, that option is available. for that matter, if the user wants to see ratios, that's available too. the default notation right now is what we're calling "prime ratios". for example, the 15/8 ratio is displayed by default as 3^1 5^1 / 2^3 ... well, that's the best i can do in ASCII equivalent, but the software uses actual superscripts instead of carets, and puts the numerator and denominator on separate lines with the separator line running horizontally. i hope to be able to get regular monzo notation, i.e. where 15/8 would be [-3 1 1], as soon as possible ... but it will entail a lot of work with the graphics, so the first release will probably not have it. i'm disappointed about that, but my partner does the coding and he has a much larger role than i in determining the priorities on the coding work. -monz

Message: 7936 Date: Sun, 02 Nov 2003 04:59:31 Subject: Re: Capstone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi Gene, > > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"

<gwsmith@s...>

> wrote: >

> > Before we get to the range which defines the > > supertemperament (where the nullspace of the matrix > > of monzos has dimension one, defining a val, and thus > > an equal temperament) we have a range where the > > nullspace is dimension two, and defines a linear > > temperament--which I've mentioned before, under > > the name of the capstone temperament. > > Here are mappings for capstones up to the 19-limit, and > > generators in terms of the corresponding supertemperament. > > (This last is a little crude, but does get us into the > > ballpark.) > > > > 5 limit: 81/80 meantone > > > > 7 limit: [[9, 15, 22, 26], [0, -2, -3, -2]] ennealimmal > > generators: [1/9, 1/24]

> > > > this looks important to me, and i want to understand it. > can you help, by labeling these lists of numbers? > > in this example, i think i recognize [0, -2, -3, -2] as > a monzo, but it doesn't make sense as a [2, 3, 5, 7]-monzo.

it's the same thing that's shown in the linear temperament table on your et dictionary page -- the mapping from generators to primes. so for example, the third entry in each of the vectors, 22 and -3 respectively, refer to the fact that the third prime, 7, is approximated in the temperament by stacking 22 of the first generator (otherwise known as the period) and -3 of the second generator (otherwise known as, simply, the generator).

> and is [1/9, 1/24] meant to represent 2^(1/9) and 2^(1/24)?

yes.

Message: 7937 Date: Sun, 02 Nov 2003 07:24:36 Subject: Re: Capstone temperament From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> it's the same thing that's shown in the linear temperament > table on your et dictionary page -- the mapping from > generators to primes. so for example, the third entry in > each of the vectors, 22 and -3 respectively, refer to the > fact that the third prime, 7, is approximated in the > temperament by stacking 22 of the first generator > (otherwise known as the period) and -3 of the second > generator (otherwise known as, simply, the generator). >

> > and is [1/9, 1/24] meant to represent 2^(1/9) and 2^(1/24)?

> > yes.

oh, OK. thanks. -monz

Message: 7938 Date: Mon, 03 Nov 2003 17:01:07 Subject: Re: TM-reduced bases for ETs From: monz hi Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >

> > have you already posted the TM-reduced bases for a number > > of common ETs in various prime-limits? if so, can you > > point me to them or repost them?

> > You can do exactly what I would need to do, namely search for them, > but I'll try to chase this down.

thanks ... mainly because i'm not even sure *how* to search for them. i'm thinking that a nice list would be good to add to the "TM-reduced lattice basis" Dictionary page. -monz

Message: 7939 Date: Mon, 03 Nov 2003 17:46:19 Subject: Re: TM-reduced bases for ETs From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi Gene, > > > have you already posted the TM-reduced bases for a number > of common ETs in various prime-limits? if so, can you > point me to them or repost them? > > right now, i'm particularly interested in 7-limit 41-ET. > i've tried putting various unison-vectors into my > software, but while i always get 41-ET for the temperament, > i keep getting 82-tone periodicity-blocks for the JI. > i don't know if these are cases of torsion (which is > what i suspect), or if there's a bug somewhere. > > > but aside from that specific example, i'd like to > have a nice list of TM-reduced bases for, say, > 12, 13, 15, 16, 17, 19, 22, 24, 31, 34, 36, 41, 43, 46, > 48, 50, 53, and 72-ETs, in the 5-, 7-, and 11-prime-limits. > > (and please, label as much stuff as you can.) > > > thanks. > > > > -monz

hi monz, i found this: Yahoo groups: /tuning-math/message/3151 * [with cont.] but for some of the ETs you're asking about (namely the ones inconsistent in the relevant limit), gene is going to assume the standard val, even when better vals are available . . . to get an "untainted" answer, you should provide a list of vals, not merely a list of ETs.

Message: 7940 Date: Mon, 03 Nov 2003 08:52:17 Subject: TM-reduced bases for ETs From: monz hi Gene, have you already posted the TM-reduced bases for a number of common ETs in various prime-limits? if so, can you point me to them or repost them? right now, i'm particularly interested in 7-limit 41-ET. i've tried putting various unison-vectors into my software, but while i always get 41-ET for the temperament, i keep getting 82-tone periodicity-blocks for the JI. i don't know if these are cases of torsion (which is what i suspect), or if there's a bug somewhere. but aside from that specific example, i'd like to have a nice list of TM-reduced bases for, say, 12, 13, 15, 16, 17, 19, 22, 24, 31, 34, 36, 41, 43, 46, 48, 50, 53, and 72-ETs, in the 5-, 7-, and 11-prime-limits. (and please, label as much stuff as you can.) thanks. -monz

Message: 7941 Date: Mon, 03 Nov 2003 18:54:07 Subject: Re: TM-reduced bases for ETs From: monz thanks, paul! awesome! i do want these for my webpages ... but i have to be honest, the main reason i wanted them was so that i could have some fun popping the unison-vectors into our software and seeing the 3-D periodicity-blocks which result. so 7-limit is perfect! :) -monz --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> > hi Gene, > > > > > > have you already posted the TM-reduced bases for a number > > of common ETs in various prime-limits? if so, can you > > point me to them or repost them? > > > > right now, i'm particularly interested in 7-limit 41-ET. > > i've tried putting various unison-vectors into my > > software, but while i always get 41-ET for the temperament, > > i keep getting 82-tone periodicity-blocks for the JI. > > i don't know if these are cases of torsion (which is > > what i suspect), or if there's a bug somewhere. > > > > > > but aside from that specific example, i'd like to > > have a nice list of TM-reduced bases for, say, > > 12, 13, 15, 16, 17, 19, 22, 24, 31, 34, 36, 41, 43, 46, > > 48, 50, 53, and 72-ETs, in the 5-, 7-, and 11-prime-limits. > > > > (and please, label as much stuff as you can.) > > > > > > thanks. > > > > > > > > -monz

> > hi monz, i found this: > > Yahoo groups: /tuning-math/message/3151 * [with cont.] > > but for some of the ETs you're asking about (namely the > ones inconsistent in the relevant limit), gene is going > to assume the standard val, even when better vals are > available . . . to get an "untainted" answer, you should > provide a list of vals, not merely a list of ETs.

Message: 7942 Date: Mon, 03 Nov 2003 09:42:35 Subject: Re: TM-reduced bases for ETs From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> have you already posted the TM-reduced bases for a number > of common ETs in various prime-limits? if so, can you > point me to them or repost them?

You can do exactly what I would need to do, namely search for them, but I'll try to chase this down.

Message: 7943 Date: Mon, 03 Nov 2003 18:58:42 Subject: Re: TM-reduced bases for ETs From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> hi monz, i found this: > > Yahoo groups: /tuning-math/message/3151 * [with cont.]

Thanks, Paul; this is exactly what's needed.

> but for some of the ETs you're asking about (namely the ones > inconsistent in the relevant limit), gene is going to assume the > standard val, even when better vals are available . . . to get > an "untainted" answer, you should provide a list of vals, not

merely

> a list of ETs.

There isn't a problem with the ones I did in that posting--namely, 7-limit 9, 10, 12, 15, 19, 22, 27, 31, 41, 68, 72, 99, 130, 140. Monz wants 13, 16, 17, 24, 34, 36, 43, 46, 48, 50, 53 as well, and also the 5- and 11-limits. Some of these *will* lead to problems, so a val is going to be required in some cases.

Message: 7944 Date: Mon, 03 Nov 2003 19:05:47 Subject: Re: TM-reduced bases for ETs From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >

> > hi monz, i found this: > > > > Yahoo groups: /tuning-math/message/3151 * [with cont.]

> > Thanks, Paul; this is exactly what's needed.

you also had this followup: "I accidentally left off 171: [2401/2400, 4375/4374, 32805/32768] Wouldn't want to do that--look at those three high-powered commas!"

Message: 7945 Date: Mon, 03 Nov 2003 19:07:35 Subject: More TM base postings From: Gene Ward Smith Here are some other messages to check: Yahoo groups: /tuning-math/message/3189 * [with cont.] Yahoo groups: /tuning-math/message/5053 * [with cont.] Yahoo groups: /tuning-math/message/5761 * [with cont.]

Message: 7946 Date: Mon, 03 Nov 2003 19:21:09 Subject: Re: More TM base postings From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> Here are some other messages to check: > > Yahoo groups: /tuning-math/message/3189 * [with cont.] > > Yahoo groups: /tuning-math/message/5053 * [with cont.] > > Yahoo groups: /tuning-math/message/5761 * [with cont.]

the first one assumes the standard val (or something) for inconsistent ets; i'd at least show the results for other vals.

Message: 7947 Date: Mon, 03 Nov 2003 19:39:52 Subject: Re: More TM base postings From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> the first one assumes the standard val (or something) for > inconsistent ets; i'd at least show the results for other vals.

The 5-limit needs to be what it is in order to include 81/80, and for some of these, the 7-limit needs to be what it is to avoid torsion.

Message: 7948 Date: Mon, 03 Nov 2003 19:48:25 Subject: Re: More TM base postings From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>

wrote:

>

> > the first one assumes the standard val (or something) for > > inconsistent ets; i'd at least show the results for other vals.

> > The 5-limit needs to be what it is in order to include 81/80, and

for

> some of these, the 7-limit needs to be what it is to avoid torsion.

yes, that process should be made explicit though -- i'd rather explain that 24-equal sometimes doesn't have a reasonable basis due to torsion, and in those cases is best understood as an equal halving of each 12-equal step, instead of using hidden rules to provide a nice-looking answer.

Message: 7949 Date: Mon, 03 Nov 2003 20:20:40 Subject: Re: 7-limit MT reduced bases for ets From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "genewardsmith" <genewardsmith@j...> wrote:

> For any prime limit, we could consider the most characteristic

linear temperament of a particular et to be the one leaving off the last member of the MT reduced basis. Let's follow up on that: 9: 21/20 ^ 27/25 = [2, 3, 1, 0, -4, -6] mapping: [[1,2,3,3], [0,-2,-3,-1]] 10: 25/24 ^ 28/27 = [2, 1, 6, -3, 4, 11] mapping: [[1,1,2,1], [0, 2, 1, 6]] 12: 36/35 ^ 50/49 = [4, 4, 4, -3, -5, -2] mapping: [[4,6,9,11], [0,1,1,1]] Diminished 15: 28/27 ^ 49/48 = [0, 5, 0, 8, 0, -14] mapping: [[5,8,12,14], [0,0,-1,0]] Blackwood 19: 49/48 ^ 81/80 = [2, 8, 1, 8, -4, -20] mapping: [[1,2,4,3], [0,-2,-8,-1]] Hemifourth 22: 64/63 ^ 50/49 = [2, -4, -4, -11, -12, 2] mapping: [[2,3,5,6], [0,1,-2,-2]] Pajara 27: 126/125 ^ 64/63 = [3, 0, -6, -7, -18, -14] mapping: [[3,5,7,8], [0,-1,0,2]] Tripletone 31: 126/125 ^ 81/80 = [1, 4, 10, 4, 13, 12] mapping: [[1,2,4,7], [0,-1,-4,-10]] Meantone 41: 225/224 ^ 245/243 = [5, 1, 12, -10, 5, 25] mapping: [[1,0,2,-1], [0,5,1,12]] Magic 68: 245/243 ^ 2048/2025 = [4, -8, 14, -22, 11, 55] mapping: [[2,3,5,5], [0,2,-4,7]] Shrutar 72: 225/224 ^ 1029/1024 = [6, -7, -2, -25, -20, 15] mapping: [[1,1,3,3], [0,6,-7,-2]] Miracle 99 and 130: 2401/2400 ^ 3136/3125 = [16, 2, 5, -34, -37, 6] mapping: [[1,-1,2,2], [0,16,2,5]] Hemiwuerschmidt 140: 5120/5103 ^ 2401/2400 = [2, 25, 13, 35, 15, -40] mapping: [[1,1,-5,-1], [0,2,25,13]] Hemififth 171: 4375/4374 ^ 2401/2400 = [18, 27, 18, 1, -22, -34] mapping: [[9,15,22,26], [0,-2,-3,-2]] Ennealimmal

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