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Message: 10828 Date: Fri, 16 Apr 2004 22:00:03 Subject: Re: On From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > > <<0, 0, 12, 0, 19, 28]], on the other hand, seems to be functionally > > the same as Jon Catler's '12-tone plus' tuning -- except that the > > offset is cleverly an eighthtone instead of a sixthtone, making for > > better 7:5s . . . right? > > One way to look at it is that it is 5-limit 12-equal with 7s tacked > on, which may as well be pure 7s (the TOP tuning.) TOP 12-et of course > has flat octaves, but this is hardly a requirement. "12 tone plus" > seems like a good description of this--is this Catler's idea? Apparently. See the microtones.com website. > Is > "catler" a good name for this temperament? He might object, since his extended 13-limit JI systems are clearly dearer to his heart.
Message: 10829 Date: Fri, 16 Apr 2004 22:43:43 Subject: Re: On From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > > <<0, 0, 12, 0, 19, 28]], on the other hand, seems to be functionally > > the same as Jon Catler's '12-tone plus' tuning -- except that the > > offset is cleverly an eighthtone instead of a sixthtone, making for > > better 7:5s . . . right? > > One way to look at it is that it is 5-limit 12-equal with 7s tacked > on, which may as well be pure 7s (the TOP tuning.) TOP 12-et of course > has flat octaves, but this is hardly a requirement. "12 tone plus" > seems like a good description of this--is this Catler's idea? Is > "catler" a good name for this temperament? I might use "catler" in my paper, but I'm having trouble with some of the other names. In 5-limit, <3 0 -7] is called "augmented" -- fine. In 7-limit, we have <3 0 6 -7 1 14] and <3 0 -6 -7 -18 -14] Why is one of these "augmented" and the other "tripletone"?
Message: 10830 Date: Sat, 17 Apr 2004 17:06:36 Subject: Re: On From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > > In 5-limit, <3 0 -7] is called "augmented" -- fine. > > > > In 7-limit, we have > > <3 0 6 -7 1 14] > > and > > <3 0 -6 -7 -18 -14] > > Why is one of these "augmented" and the other "tripletone"? > > By my proposal to use TOP to guide us, the second one should be called > "augmented" also, and the name "tripletone" retired. > > TOP generators: > > 128/125 [399.020, 93.145] > > <3 0 6 -7 1 14| [399.992 107.311] > > <3 0 -6 -7 -18 -14| [399.020 92.460] Close, but no cigar. In my paper, I will only use the same name if the tuning is exactly the same, because the names will be references to horagrams. > That leaves us without a name for the other system, which is also an > important one. I propose both names be some sort of variants of "augmented".
Message: 10833 Date: Mon, 19 Apr 2004 15:28:37 Subject: Re: On From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: > > > Close, but no cigar. In my paper, I will only use the same name if > > the tuning is exactly the same, because the names will be > references > > to horagrams. > > There's no reason not to use the same tuning for both, I consoled John Chalmers by telling him I'd use only one tuning -- the TOP tuning -- for each temperament. Are you suggesting a deviation from this strategy? Can you make it quantitative?
Message: 10834 Date: Mon, 19 Apr 2004 21:55:53 Subject: notation of monzos From: monz hello all, did we ever reach a consensus on the use or non-use of angle-brakets and/or pipe symbols in expressing monzos? i have not been following tuning-math much lately, and just want to be sure that the Encyclopaedia entry is up-to-date. -monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: Yahoo groups: /tuning-math/message/7527 * > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > > > I admit to being shamefully ignorant of these things. > > It sounds then like vals could use the angle-brackets. > > Physicists sometimes use bra and ket vectors; if you > did that, the monzo for 81/80 would be a ket, [-4 4 -1> > and the val for 5-limit 12-et would be a bra, <12 19 28]. > Putting them together would give the bra-ket, angle bracket, > or inner product: <12 19 28 | -4 4 -1> = 0. > > See: > > Contravariant Vector -- from MathWorld * > > Covariant Vector -- from MathWorld * > > Ket -- from MathWorld * > > Bra -- from MathWorld * > > One-Form -- from MathWorld * > > Angle Bracket -- from MathWorld *
Message: 10836 Date: Mon, 19 Apr 2004 22:02:39 Subject: Re: TM-reduced bases for ETs From: monz hello all, i've just updated the Encyclopaedia of Tuning entry for "TM-reduced lattice", to include examples of TM-reduced bases for ETs that were posted here by Gene at the beginning of November 2003. Tonalsoft Encyclopaedia of Tuning - TM-reduced lattice basis, (c) 2004 Tonalsoft Inc. * feedback appreciated. (paul, i know that i also need to quote your post about how certain ETs can have multiple equally-good/bad val mappings ... i will when i get more time.) -monz
Message: 10839 Date: Tue, 20 Apr 2004 19:52:24 Subject: Re: notation of monzos From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > > hello all, > > > > > > did we ever reach a consensus on the use or > > non-use of angle-brakets and/or pipe symbols > > in expressing monzos? > > Not really. Angle-brakets seem to be well accepted, but so far as "|" > versus "]" goes, some people use one, and some the other. The monzo would have a "[", not a "]". I'm using these, not pipe symbols or even vertical lines, in my paper.
Message: 10840 Date: Tue, 20 Apr 2004 22:33:12 Subject: Re: TM-reduced bases for ETs From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > > Tonalsoft Encyclopaedia of Tuning - TM-reduced lattice basis, (c) 2004 Tonalsoft Inc. * > Right off the bat: > > "A method for reducing the bases of a lattice." > > should read > > "A method for reducing the basis of a lattice." > > TM-reduction results in a *single* basis for the lattice, not > multiple _bases_. thanks. i'd really like to add a little more "regular English" to the opening part of that definition, describing exactly what TM-reduction does ... before heading into Gene's mathematical definition. can you or anyone else help? i'm thinking something like this: "A method for reducing the basis of a lattice to its most compact representation, with all unison-vectors as small as possible in prime-space." -monz
Message: 10841 Date: Tue, 20 Apr 2004 22:37:38 Subject: Re: notation of monzos From: monz hi paul and Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > > > hello all, > > > > > > > > > did we ever reach a consensus on the use or > > > non-use of angle-brakets and/or pipe symbols > > > in expressing monzos? > > > > Not really. Angle-brakets seem to be well accepted, > > but so far as "|" versus "]" goes, some people use one, > > and some the other. > > The monzo would have a "[", not a "]". I'm using these, > not pipe symbols or even vertical lines, in my paper. it was my understanding that the monzo by itself uses [...> and the val uses <...] , and that putting them together one would use the pipe symbol instead of the two square brackets thus: <...|...> . has this become established usage? also, what about the suggestion to use comma punctuation after the exponents of 3, 11, 19, 31, etc.? is that established at all? -monz
Message: 10842 Date: Tue, 20 Apr 2004 16:47:51 Subject: Re: TM-reduced bases for ETs From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > hello all, > > > i've just updated the Encyclopaedia of Tuning > entry for "TM-reduced lattice", to include examples > of TM-reduced bases for ETs that were posted here > by Gene at the beginning of November 2003. > > Tonalsoft Encyclopaedia of Tuning - TM-reduced lattice basis, (c) 2004 Tonalsoft Inc. * > > feedback appreciated. Right off the bat: "A method for reducing the bases of a lattice." should read "A method for reducing the basis of a lattice." TM-reduction results in a *single* basis for the lattice, not multiple _bases_. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service *
Message: 10843 Date: Wed, 21 Apr 2004 20:05:42 Subject: Re: 270 equal as the universal temperament From: Graham Breed Gene Ward Smith wrote: > If "four digit frequency ratios" (which from context I take to mean > superparticular ones) have no audible meaning, it seems like a nifty > idea to temper them out. In the 7-limit this gives ennealimmal, and in > the 11-limit hemiennealimmal. If we take Graham's view, which I think > has something to be said for it, we go up to the 13-limit but no > farther. In the 13-limit, there are twelve four digit superparticular > commas; the kernel of all of these taken together is 270-equal. This > sort of fact I've discussed before; it does seem there is some > justification for considering 270-et to be a sort of universal > replacement for just intonation. I did mean the complexity was too great (and obviously so). Probably harmonic commas can be heard for much smaller intervals -- particularly mistuned unisons. But commas aren't usually hidden within chords, are they? Such small commas should be melodically inaudible, especially if shared among a few chords. So an adaptive tuning scheme with these commas should be indistinguishable on a chord-by-chord basis from JI. In which case a 270 note system would be a replacement for JI if used with adaptive temperament. 13-prime limit is one place to stop. I think harmony works clearly up to the 9-limit. Then at the 11-limit you get exotic intervals like neutral thirds and seconds. The 13-limit is roughly more of the same, but you get 8:10:13 and 8:11:13 chords. These are theoretically important for a number of reasons: - They start with a power of two, and so may have a strong virtual pitch. - All intervals, including the implied octave, are larger than 7:8 (roughly a critical bandwidth). - Along with 4:5:6, they're the only such chords with the first number less than 16. Such chords may be useful in cadences, I still haven't decided. And slightly mistuned versions may work as well, if each interval is a better approximation to an 11-limit interval. But, anyway, they might be a reason for going to the 13-limit, at which point you may as well go to 15. It may even be possible to hear higher limits. With 16:19:24, you get that alternative tuning of a minor triad. Graham
Message: 10844 Date: Wed, 21 Apr 2004 20:45:16 Subject: Re: 270 equal as the universal temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: > Gene Ward Smith wrote: > > > If "four digit frequency ratios" (which from context I take to mean > > superparticular ones) have no audible meaning, it seems like a nifty > > idea to temper them out. In the 7-limit this gives ennealimmal, and in > > the 11-limit hemiennealimmal. If we take Graham's view, which I think > > has something to be said for it, we go up to the 13-limit but no > > farther. In the 13-limit, there are twelve four digit superparticular > > commas; the kernel of all of these taken together is 270-equal. This > > sort of fact I've discussed before; it does seem there is some > > justification for considering 270-et to be a sort of universal > > replacement for just intonation. > > I did mean the complexity was too great (and obviously so). Probably > harmonic commas can be heard for much smaller intervals -- particularly > mistuned unisons. But commas aren't usually hidden within chords, are > they? 64:63 is famous for being hidden within chords . . .
Message: 10845 Date: Wed, 21 Apr 2004 20:59:00 Subject: Re: 270 equal as the universal temperament From: Graham Breed Paul Erlich wrote: > 64:63 is famous for being hidden within chords . . . Yes, but it doesn't run to four digits, or not in each number. The neutral third comma makes it to three -- 243:242. Graham
Message: 10847 Date: Wed, 21 Apr 2004 20:58:28 Subject: Re: notation of monzos From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: > --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: > > > >However I don't use the word "monzo" > > > > Why not? > > > > -Carl > > I want the minimum of jargon, and the maximum sense > of "I could have thought of this myself and I probably > did at one point" in the reader. i guess i'm just being selfish, but i am disappointed, and wish you'd use "monzo" in your paper ... mainly because it's my feeling that its use in a paper by *you* would give the term a cachet that it doesn't currently have. :) as Gene pointed out when he first named the "vector of prime-factor exponents" after me, it's useful to have one word to replace a whole phrase, if you're going to be referring to it often. -monz
Message: 10849 Date: Wed, 21 Apr 2004 21:00:51 Subject: Re: 270 equal as the universal temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: > Gene Ward Smith wrote: > > > If "four digit frequency ratios" (which from context I take to mean > > superparticular ones) have no audible meaning, it seems like a nifty > > idea to temper them out. In the 7-limit this gives ennealimmal, and in > > the 11-limit hemiennealimmal. If we take Graham's view, which I think > > has something to be said for it, we go up to the 13-limit but no > > farther. In the 13-limit, there are twelve four digit superparticular > > commas; the kernel of all of these taken together is 270-equal. This > > sort of fact I've discussed before; it does seem there is some > > justification for considering 270-et to be a sort of universal > > replacement for just intonation. > > I did mean the complexity was too great (and obviously so). Probably > harmonic commas can be heard for much smaller intervals -- particularly > mistuned unisons. But commas aren't usually hidden within chords, are > they? Such small commas should be melodically inaudible, especially if > shared among a few chords. So an adaptive tuning scheme with these > commas should be indistinguishable on a chord-by-chord basis from JI. > In which case a 270 note system would be a replacement for JI if used > with adaptive temperament. Being just like JI (and ~2 cent errors don't bother me) doesn't strike me as a sufficient qualification for being a/the "universal temperament". In addition, multiple vals/breeds should be supported, including some useful non-micro-temperaments. My favorites among those include the ones that support omnitetrachordal scales and have relatively low complexity, such as meantone, double-meantone (injera), and pajara. You'll find my suggested "universal tuning" here: Tonalsoft Encyclopaedia of Tuning - equal-temperament, (c) 2004 Tonalsoft Inc. *
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