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Message: 10825 - Contents - Hide Contents Date: Fri, 16 Apr 2004 21:09:20 Subject: Re: On From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> So what's the wedgie, TOP primes, period, and generator for the > tuning where 531441/524288 and 5120/5103 vanish?wedgie <<0 12 12 19 19 -6|| mapping [<12 19 28 34|, <0 0 -1 -1|] TM basis [50/49 3645/3584] TOP tuning [1198.015473 1896.857833 2778.846497 3377.854234] TOP generators [99.83462277 16.52294019] Not the best of the "family", I guess.

Message: 10826 - Contents - Hide Contents Date: Fri, 16 Apr 2004 00:22:08 Subject: Six 13-limit nonoctave temperaments From: Gene Ward Smith Nonoctave temperaments of dimension n are essentially identical to octave temperaments of dimension n+1; in particular a linear nonoctave temperament is essentially the same beast as a planar temperament such that the kernel has only commas which are ratios of odd integers. I took four commas 245/243, 275/273, 847/845, 1575/1573 from Lehmer's N/(N-2) list, and added a comma from his N/(N-4) list, namely 4459/4455, and then took this in sets of three, finding the corresponding planar temperaments (equivalent to nonoctave linears.) Here is what I got, listing (planar) wedgie, mapping, and TOP tuning for six temperaments. [21, 9, 19, 13, -24, -18, -30, 14, 2, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[1, 0, 0, 0, 0, 0], [0, 1, 9, 5, 9, 7], [0, 0, 21, 9, 19, 13]] [1200.0, 1902.277228, 2785.467655, 3367.802970, 4150.708360, 4441.876018] [6, -3, -2, 13, -15, -16, 5, 3, 30, 33, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[1, 0, 0, 0, 0, 0], [0, 1, 5, 0, 1, 10], [0, 0, 6, -3, -2, 13]] [1200.0, 1903.771872, 2783.652046, 3367.603656, 4148.840976, 4444.769540] [0, 0, 13, 13, 0, 19, 19, 23, 23, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[1, 0, 0, 0, 0, 0], [0, 13, 19, 23, 0, 2], [0, 0, 0, 0, 1, 1]] [1200.0, 1904.187463, 2783.043215, 3368.947050, 4146.445241, 4439.397158] [9, 15, 10, 13, 6, -5, -2, -15, -12, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[1, 0, 0, 0, 0, 0], [0, 1, 4, 6, 5, 6], [0, 0, 9, 15, 10, 13]] [1200.0, 1902.661549, 2784.022381, 3371.596268, 4150.392396, 4444.179340] [12, -6, 9, 0, -30, -13, -28, 29, 14, -21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [[1, 0, 0, 0, 0, 0], [0, 3, 1, 7, 4, 7], [0, 0, 4, -2, 3, 0]] [1200.0, 1903.372996, 2784.236389, 3366.314293, 4150.164703, 4441.203655] ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> 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 * [with cont.] (Wayb.)

Message: 10827 - Contents - Hide Contents Date: Fri, 16 Apr 2004 21:15:49 Subject: Re: On From: Gene Ward Smith --- 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?

Message: 10828 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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: 10831 - Contents - Hide Contents Date: Sat, 17 Apr 2004 17:27:12 Subject: Re: On From: Gene Ward Smith --- 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, which argues for the same name for both. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> 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 * [with cont.] (Wayb.)

Message: 10832 - Contents - Hide Contents Date: Sat, 17 Apr 2004 05:44:00 Subject: Re: On From: Gene Ward Smith --- 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] That leaves us without a name for the other system, which is also an important one. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> 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 * [with cont.] (Wayb.)

Message: 10833 - Contents - Hide Contents 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 - Contents - Hide Contents 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 * [with cont.]> --- 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 * [with cont.] > > Covariant Vector -- from MathWorld * [with cont.] > > Ket -- from MathWorld * [with cont.] > > Bra -- from MathWorld * [with cont.] > > One-Form -- from MathWorld * [with cont.] > > Angle Bracket -- from MathWorld * [with cont.]

Message: 10835 - Contents - Hide Contents Date: Mon, 19 Apr 2004 15:47:19 Subject: Re: On From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> 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?I was talking about in practice, not for a theory paper. If we wanted to quantize it, perhaps a place to start would be to look at the ratio between the difference between the two top tunings over the error, for consonances which are not just. When the tuning difference is a lot less than the error, in practical terms it doesn't mean much. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> 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 * [with cont.] (Wayb.)

Message: 10836 - Contents - Hide Contents 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... * [with cont.] (Wayb.) 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: 10837 - Contents - Hide Contents Date: Mon, 19 Apr 2004 23:14:48 Subject: Re: notation of monzos From: Gene Ward Smith --- 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.

Message: 10838 - Contents - Hide Contents Date: Mon, 19 Apr 2004 23:16:06 Subject: Re: TM-reduced bases for ETs From: Gene Ward Smith --- 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.The title should be TM-reduced lattice basis, since the lattice is not being reduced.

Message: 10839 - Contents - Hide Contents 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 - Contents - Hide Contents 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:> 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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... * [with cont.] (Wayb.) > > 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/ * [with cont.] <*> 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 * [with cont.] (Wayb.)

Message: 10843 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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: 10846 - Contents - Hide Contents Date: Wed, 21 Apr 2004 00:09:40 Subject: 270 equal as the universal temperament From: Gene Ward Smith Over on metatuning Graham had this to say: "It doesn't matter if anybody can tell the difference, because there's no evidence that four digit frequency ratios have any audible meaning. The 13-limit is borderline..." 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. Incidentally, we have to stop here; if we try to add 729/728, the next smallest comma, we find this is a step of 270-et.

Message: 10847 - Contents - Hide Contents 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: 10848 - Contents - Hide Contents Date: Wed, 21 Apr 2004 00:13:08 Subject: Re: notation of monzos From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> 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?It seems to have helped a great deal on this list in clarifying matters. I think Paul should use it, myself.

Message: 10849 - Contents - Hide Contents 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-temp... * [with cont.] (Wayb.)

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