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Message: 8250 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:50:44 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> You started with 6 and ended up with 22. Where is your 22 note >> scale/chord? >> Ok, now we're on the right track, but I'm still not grokking > you. I started with six rationals and ended up with 6 integers. > What's the problem?Are your integers consecutive?

Message: 8251 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:14:20 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> time. It's a lot more complicated to find the best mapping for an >> inconsistent ET. >> right, but gene just did that (two days ago?), and hopefully will do > more.I was quite aware of that.>> It's just that Gene and George found it interesting to look at how >> complete chords can be mapped to a single octave of the ET of the > same>> cardinality as the chord. It turns out that the 11-limit otonality >> can't be. There is no mapping and no voicing of the chord that will > do >> this. >> let me just repeat dave and say that this has *nothing* to do with > the definition of vals -- it's a separate question that you can > safely ignore if you want to understand vals.I was quite aware of that. I was merely trying to answer Carl's questions. I understand vals now (as of this week), and would have understood them several _years_ ago, if the term prime-mapping had been used. Or since I have undertood prime-mappings for years (pre-Gene) you could say I understood vals the whole time (as applied to tuning). I just didn't have a clue that's what he was talking about. The new term and the pure-math definitions obscured this simple fact. For years!

Message: 8252 - Contents - Hide Contents Date: Fri, 14 Nov 2003 07:38:03 Subject: Re: Vals? From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> monz, this is a very specific, and perhaps even unusual, > application of the val or mapping concept. it may warrant > mention in an Encyclopedia but probably not in a dictionary.well, OK ... i don't have anything about it in the Dictionary yet, so i guess i'll leave it out, or just include it in the upcoming Encyclopedia.> p.s. would it be OK for me to attempt a modification of your page > > Definitions of tuning terms: EDO prime error, ... * [with cont.] (Wayb.) > > ? i realized that you *do* have the signed errors of the > primes in the text, despite your use of absolute values in > the graph. so if i just added the signed errors for the > odds that you omitted, i could then quickly locate any > inconsistency, since inconsistency occurs if and only if > the signed relative error of one odd differs by over 50% > from the signed relative error of another odd. for example, > in 43-equal, the error on 7 is, as you show, +28%; the error > on 9 is double that on 3, so about -30%; the difference > between these two signed percentages (and thus the implied > error on 9:7) is 58%; so 43-equal is inconsistent in the > 9-limit. what do you think? i think the page would be > sorely misleading, and much less useful, without this > information.paul, you already know that i think that the information given on my "EDO prime error" page is useful as it is. but if you envision a better version of the page, sure, send me the modification. you know i trust your judgment on tuning matters! :) -monz

Message: 8253 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:55:34 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> So are monzos are now kets written [ ... > ? >>>> and vals are bras written < ... ] ? >>>>>> I think that's a good suggestion. It is a standard (especially in >>> physics), clever notation due to Dirac. We just don't worry about >>> complex numbers and certainly not about quantum mechanics. >>>> But Dave is right that you weren't suggesting using the names >> "bra" and "ket", right? >> No--that would really lead to confusion anyway.but those are the names of the mathematical objects.

Message: 8254 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:14:24 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> the symbol normally indicates the cross-product, which is extremely >> useful in tuning: for example, if i take the monzo for the diaschisma >> >> [-4 4 -1> >> >> and cross it with the (transpose of the?) monzo for the syntonic comma >> >> <-11 4 2] >> Huh; I thought all monzos were supposed to be written | ... > from > here on out.if so, then the second is the transpose of a monzo.>> i get the val for the et where they both vanish: >> >> [12 19 28] >> >> not sure how gene would do this notationally, probably i did >> something terrible, but without it i could not have made those >> charts . . . >> And I thought all vals were supposed to be written < ... |. > > Did I miss something?that's why i said i probably did something terrible notationally!

Message: 8255 - Contents - Hide Contents Date: Fri, 14 Nov 2003 07:42:56 Subject: Re: Definition of microtemperament From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> You mean 11-odd limit? Well, meantone contained > excellent approximations to ratios of 7, but practically > no one considered them consonant historically.that's not true, and you know it: meantone gave good approximations to a 4:5:7 triad in its "augmented-6th" chord, which was used a *lot* in the "common-practice" era. true, no-one at the time analyzed these chords as consonant 4:5:7 chords, but in meantone, that's what they were, and they were perfectly acceptable in both theory and practice. i wrote something to the main tuning list fairly recently (perhaps this past spring?) that went into pretty good detail about augmented-6th chords and meantone versions of them. with your prodigious memory, you'll probably recall them ... -monz

Message: 8256 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:57:23 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> I'm still wondering about 6. In 22, the 11-prime-limit val > consistently maps the 9/8, and the resulting hexad taken as a > scale is a Constant Structure.no, it's still not a constant structure, since one 3:2 subtends more notes than the other ('9:6').

Message: 8257 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:13:29 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Let's leave matricies alone. As for covariant vs contravariant, if >> you change the basis for monzos to something other than primes, you >> have to make a complimentary change in basis for the val basis. The >> standard basis is that monzos have a basis e_2, e_3 etc. >> corresponding to primes, and vals v_2, v_3 corresponding to (whether >> we want to call them that or not) padic valuations. >> What happens if we change the bases to odd numbers. There's no > longer a unique monzo for any given interval, which seems bad. > What other bases did you have in mind?overtones of inharmonic spectra is one kind graham has used.

Message: 8258 - Contents - Hide Contents Date: Fri, 14 Nov 2003 08:04:30 Subject: Integrating the Riemann-Siegel Zeta function and ets From: Gene Ward Smith The Riemann-Siegel Zeta function Z(t) is defined here: Riemann-Siegel Functions -- from MathWorld * [with cont.] For our purposes we want to change scales, setting t = 2 pi x / ln(2), and use Z(x) instead. I integrated |Z(x)| between successive zeros, for zeros up to 100.409754. Below I list every inteval between zeros where the integral is greater than one. The point of this business is to give what you might call a generic goodness measure for ets; meaning one not attached to any particular prime limit. The result seems better than what we get for maximal values of |Z(x)|, and much better than what we can glean from gaps between the zeros. If we look at successively larger values, we get 2, 5, 7, 12, 19, 31, 41, 53, 72 ..., and this makes a lot of sense to me. The so-called "Omega theorems", about the rate of growth of the high values of |Z(x)|, do not seem strong enough to show this is an infinte list, though it starts out looking as if it is planning on being one. I think I'll write to some people more expert than I am and inquire. If we take the values over one by decades, we get: 1-10: 2, 3, 5, 7, 10 11-20: 12, 15, 17, 19 21-30: 22, 24, 26, 27, 29 31-40: 31, 34, 36 41-50: 41, 43, 46, 50 51-50: 53, 58, 60 61-70: 63, 65, 68 71-80: 72, 77, 80 81-90: 84, 87, 89 91-100: 94, 96, 99 It seems the density may be falling off slowly. [1.559311781, 2.319105165] 1.103823 [2.759142784, 3.356405400] 1.044063 [4.779747405, 5.295822634] 1.131648 [6.710827976, 7.183072612] 1.162332 [9.797225769, 10.20350285] 1.082282 [11.82260542, 12.24853409] 1.269599 [14.86604170, 15.23665791] 1.104057 [16.88134757, 17.22203271] 1.032175 [18.74431544, 19.13037920] 1.313799 [21.84461333, 22.20308465] 1.258178 [23.84734791, 24.16705528] 1.092055 [25.78054223, 26.09283267] 1.031155 [26.92536457, 27.26360905] 1.185939 [28.77144315, 29.07689211] 1.000619 [30.80395665, 31.16093004] 1.403777 [33.89177893, 34.21059373] 1.241437 [35.83815669, 36.12289081] 1.028887 [40.82320329, 41.15537120] 1.423937 [42.89664942, 43.18457394] 1.035628 [45.83210532, 46.15561125] 1.356067 [49.79781990, 50.08281814] 1.111229 [52.83584779, 53.15446302] 1.486620 [57.92538202, 58.23716835] 1.358357 [59.77541720, 60.04861404] 1.131000 [62.88678487, 63.14811332] 1.049023 [64.88227375, 65.16035560] 1.269821 [67.90486013, 68.18884771] 1.254592 [71.78033774, 72.10918271] 1.625363 [76.85025545, 77.12671468] 1.311364 [79.93215353, 80.20726288] 1.247325 [83.87267811, 84.13938972] 1.241945 [86.87178850, 87.15758094] 1.439474 [88.90088275, 89.15353029] 1.124501 [93.84133446, 94.11907762] 1.394050 [95.82981785, 96.06991440] 1.045052 [98.91449741, 99.20014010] 1.510412

Message: 8259 - Contents - Hide Contents Date: Fri, 14 Nov 2003 21:58:24 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hey paul, > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> >> wrote: >>>> p.s. would it be OK for me to attempt a modification of your page >>> >>> Definitions of tuning terms: EDO prime error, ... * [with cont.] (Wayb.) >>> >>> ? i realized that you *do* have the signed errors of the >>> primes in the text, despite your use of absolute values in >>> the graph. so if i just added the signed errors for the >>> odds that you omitted, i could then quickly locate any >>> inconsistency, since inconsistency occurs if and only if >>> the signed relative error of one odd differs by over 50% >>> from the signed relative error of another odd. for example, >>> in 43-equal, the error on 7 is, as you show, +28%; the error >>> on 9 is double that on 3, so about -30%; the difference >>> between these two signed percentages (and thus the implied >>> error on 9:7) is 58%; so 43-equal is inconsistent in the >>> 9-limit. what do you think? i think the page would be >>> sorely misleading, and much less useful, without this >>> information. >> >> >>>> paul, you already know that i think that the information >> given on my "EDO prime error" page is useful as it is. >> >> but if you envision a better version of the page, sure, >> send me the modification. you know i trust your judgment >> on tuning matters! :) > > >> take a look at this: > > Definitions of tuning terms: EDO 11-odd-limit ... * [with cont.] (Wayb.) > > > > > -monznice! replied on main tuning list.

Message: 8260 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:17:11 Subject: Re: Vals? From: Paul Erlich similarly, if i take the (transpose of the?) val for 12-equal: |12 19 28> and take the cross product with the val for 22-equal: <22 35 51| i get the monzo for the diaschisma, the interval that vanishes in both tunings: [-11 4 2] again, not sure what's going on notationally, but the numbers work . . . --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> >>> wrote: >>>>>>> If we are told that the mapping is for a tET then _which_ tET > it is>>>> for can be read straight out of the mapping, as the coefficient > for>>>> the prime 2 (the first coefficient). And the generator is > simply one>>>> step of that tET. >>>>>> just wondering why you keep saying "tET" -- 'If we are told that > the>>> mapping is for a tone equal temperament then . . .' ?? >>>> I agree it's awkward. Carl objected so vehemently to EDO and I > wanted>> to reserve ET for the most general term (including EDOs ED3s cETs). >> Perhaps this would be a misuse of ET. Do we have some other term for >> the most general category of 1D temperaments, i.e. any single >> generator temperament whether or not it is an integer fraction of > any>> ratio? I guess "1D-temperament" will do. >>>>> actually, > and < fit together and create a X (as in times) ! >>>> Oops. Well we could interpret that as the matrix-product as opposed > to>> the scalar-product (dot-product), but I don't know of any meaning > for>> that in tuning. >> the symbol normally indicates the cross-product, which is extremely > useful in tuning: for example, if i take the monzo for the diaschisma > > [-4 4 -1> > > and cross it with the (transpose of the?) monzo for the syntonic comma > > <-11 4 2] > > i get the val for the et where they both vanish: > > [12 19 28] > > not sure how gene would do this notationally, probably i did > something terrible, but without it i could not have made those > charts . . .

Message: 8261 - Contents - Hide Contents Date: Fri, 14 Nov 2003 09:55:09 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> A prime-mapping (or val with log-prime basis) simply maps each prime > number (or strictly-speaking the logarithm of each prime number) to an > integer multiple of some interval (log of frequency ratio) that we > call a generator.This is absolutely not what I mean by a val, which maps to integers.>> Note that I have no idea what the bra ket notation stuff is about. >> It's just a way of distinguishing prime-mappings (vals) from > prime-exponent-vectors (monzos) without having to say it in words > every time.It also shows the covariant vs contravariant aspect.

Message: 8262 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:03:18 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> That's Gene's proposal (that we should write them that way, not that >> we should call them bras and kets) and it seems like a reasonable > one >> to me. >> it'll help me, since i'm used to them.Sounds like we are achieving consensus on something!>> Except I wonder how we should write a complete mapping matrix for a > more-than-1D temperament. >> a matrix is a matrix, not a bra or a ket. i never understood > covariant vs. contravariant, though . . .Let's leave matricies alone. As for covariant vs contravariant, if you change the basis for monzos to something other than primes, you have to make a complimentary change in basis for the val basis. The standard basis is that monzos have a basis e_2, e_3 etc. corresponding to primes, and vals v_2, v_3 corresponding to (whether we want to call them that or not) padic valuations.

Message: 8263 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:18:11 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:>> let me just repeat dave and say that this has *nothing* to do with >> the definition of vals -- it's a separate question that you can >> safely ignore if you want to understand vals. >> I was quite aware of that.you should be, because as i said, i was just repeating you!

Message: 8264 - Contents - Hide Contents Date: Fri, 14 Nov 2003 02:00:11 Subject: Re: Vals? From: Carl Lumma>Yes! Congratulations!Yeah well, the choice of 6 here still hasn't been accounted for.>So it's proper, but not a constant structure. I was under the >misapprehension that proper always implied constant structure, i.e >that propriety was a stronger condition. Hmm.Nope. The set of all non-CS scales is equivalennt to the set of all non-strictly-proper scales. I remember surprise when John Chalmers first pointed this out to me, by way of this example: [private communication]>However, the Enharmonic of Archytas is. Translate the scale 28/27 x >36/35 x 5/4 x 9/8 x 28/27 x 36/35 x 5/4 into cents and generate the >D-matrix. > >63 49 386 204 63 49 386 >112 435 590 267 112 435 449 >498 639 653 316 498 498 498 >702 702 702 702 561 547 884 >765 751 1088 765 610 933 1088 >814 1137 1151 814 996 1137 1151 >1200 ....... -Carl

Message: 8265 - Contents - Hide Contents Date: Fri, 14 Nov 2003 15:22:45 Subject: Re: Vals? From: Carl Lumma>>>>>> > started with six rationals and ended up with 6 >>>>>> integers. What's the problem? >>>>>>>>>> Are your integers consecutive? >>>>>>>> No, and that's part of the def. of standard val, but what >>>> motivates it? >>what does that have to do with the definition of standard val?Sorry, it doesn't. I forgot the definition doesn't mention consecutive. It's just this particular case. Wait... is this true: 'For a scale with card k, if there is no standard val with n=k that consistently maps the scale, the scale is not a Constant Structure.' -Carl

Message: 8266 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:10:25 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> but those are the names of the mathematical objects.Not really. Monzos are a particular type of ket vector, with a particular interpretation, and vals a particular type (in both cases, composed of integers) of bra vector, with a particular interpretation. The "mathematical object" involves more than merely being a vector. Any integer bra vector can be interpreted as a val, and any integer ket vector as a monzo (and hence, a positive rational number) but we need the interpretation.

Message: 8267 - Contents - Hide Contents Date: Fri, 14 Nov 2003 10:00:58 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> It's proper and for that matter seems to fit to 6-tET reasonably > well.It's not a Constant Structure.> So are monzos are now kets written [ ... > ? > and vals are bras written < ... ] ?I think that's a good suggestion. It is a standard (especially in physics), clever notation due to Dirac. We just don't worry about complex numbers and certainly not about quantum mechanics.

Message: 8268 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:12:31 Subject: Re: "does not work in the 11-limit" (was:: Vals?) From: Dave Keenan George, I was already convinced that Constant Structure is a valuable melodic property of a scale. But what's wrong with using complete 11-limit hexads as vertical harmony within a larger CS scale? Why should we care that the hexads _themselves_ are not CS?

Message: 8269 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:37:36 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Except I wonder how we should write a complete mapping matrix for a > more-than-1D temperament. >> a matrix is a matrix, not a bra or a ket.True, but the dot product of two vectors generalises naturally to the product of a matrix with a column vector. When we go beyond 1D temperaments we have prime-mappings which are matrices (one row per generator) and we multiply that by the transpose of a ratio's prime-exponent-vector (monzo) to get a vector giving the count of each generator.> i never understood > covariant vs. contravariant, though . . .Me neither. Paul, we've gotta keep on Gene to tell us what things mean AS APPLIED TO TUNING. And for him to keep trying different explanations until we get it. I propose a new term for that, since we'll probably be writing it so often. BWDIMAATT. Pronounced "BWOOD-ee-mart" As in "Gene, BWDIMAATT" Meaning "Gene, But What Does It Mean As Applied To Tuning". ;-) And when one of us gets it we've gotta try to explain it to the rest, - if it actually has any relevance at all.

Message: 8270 - Contents - Hide Contents Date: Fri, 14 Nov 2003 02:08:11 Subject: Re: Vals? From: Carl Lumma>> >o are monzos are now kets written [ ... > ? >> and vals are bras written < ... ] ? >>I think that's a good suggestion. It is a standard (especially in >physics), clever notation due to Dirac. We just don't worry about >complex numbers and certainly not about quantum mechanics.But Dave is right that you weren't suggesting using the names "bra" and "ket", right? Carl

Message: 8271 - Contents - Hide Contents Date: Fri, 14 Nov 2003 22:13:22 Subject: Re: "does not work in the 11-limit" (was:: Vals?) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:> Is it important that a musical scale be > a constant structure, and if so, why?but we're talking about a chord, not a whole scale . . .> (By "scale", I am referring to a set of tones that may be used to > write a simple melody.exactly . . . the two champions would have to be the diatonic pentatonic and heptatonic scales . . .> If I'm using a pentatonic scale made from a 9-limit otonal chord: > 8 : 9 : 10 : 12 : 14 : 16 > then I have two intervals each of 2:3 (both pentatonic "4ths") and > 3:4 (both pentatonic "3rds").personally, i'm not fond of this as a scale or melodic entity at all - - when i improvise over a dominant ninth chord, simply using its notes is about the worst way to come up with a melody . . .> So far, so good. > > But if I try to use hexatonic scale made from an 11-limit otonal > chord: > 8 : 9 : 10 : 11 : 12 : 14 : 16 > then one of my 2:3s is a hexatonic "5th" and the other is a > hexatonic "4th", and likewise one of my 3:4s is a hexatonic "4th" and > the other is a hexatonic "3rd". Most attempts to transfer a melodic > figure beginning on a certain scale degree to another scale degree > (such as is required in the musical device called a "sequence") will > tend to produce undesirable consequences (such as listener > disorientation) due to the fact that the 2:3 and/or 3:4 must switch > degree-roles in the process.yet even an algorithmic composition program, such as those written by prent rodgers, can produce lovely music by simply using a single such hexad at a time, for both harmony and melody (or for polyphony). i'm not disputing your constant structure argument too vehemently, especially when it concerns such an important interval as 3:2, but note especially that the diatonic scale in 12-equal is not CS, and yet doesn't cause any more listener disorientation than the diatonic scale in, say, 19-equal or 17-equal, where it it CS.> Now we could go on to ask why this scale-member identity or > functionality is so important, and this is the point at which I > really had to dig deep for an answer. I believe that, at least with > the examples given above, it has something to do with the role that > the simplest ratios of 3 play in establishing the roots of chords. > If a chord contains a *single* 2:3 or 3:4 (whether just or tempered), > then I can almost guarantee that the tone represented by the 3 will > *never* be heard as the root of the chord.another commonality with my 22 paper.> (There are instances, > e.g., 8:10:15, that a tone not in the 2:3 interval will be perceived > as the root, but that's not critical to the point that I'm making.) > It is this property of the simple ratios of 3 that makes it possible > to *invert* many conventional triads and seventh chords *without* > changing our *perception* of which note of the chord functions as the > *root*. > > So it would not have been possible for the methods of conventional (5- > limit) harmony to have reached such sophistication if the major and > minor scales were not constant structures, because our whole method > of building chords (by 3rds) has depended on the fact that the simple > ratios of 3 would always be heptatonic 4ths and 5ths and that the > simple ratios of 5 would always be 3rds and 6ths.ah, but you're depending on the heptatonic scale here! if we used some sort of heptatonic or other scalar framework to understand 11- limit harmony, the same property might hold, despite the fact that the hexad itself is not CS. so the latter fact seems irrelevant.> One can similarly demonstrate that pentatonic melodies are perceived > as coherent, because pentatonic scales that contain multiple 3:4s and > 2:3s are also constant structures. > > But take away the property of constant structure while retaining > multiple 2:3s, and you invite confusion and disorientation. > > If you want 11-limit otonal harmony in a conherent scale, then I > think it will have to be at least heptatonic and that you're going to > have to fill that extra position with something or other, such as: > 8:9:10:11:12:27/2:14:16. > Hmmm, that's really not a bad choice, if you'll notice that 22:27:32 > is an isoharmonic triad. I remember that this scale works very > nicely in 31-ET, since the 27/2:16:20 ends up as an ordinary minor > triad. > > Likewise, you can have constant-structure 17 and 19-limit otonal > scales: > 16:17:18:20:21:22:24:26:28:30:32 (with chords built in > decatonic "4ths") > and > 16:17:18:19:20:21:22:24:25:26:28:30:32ok, but not much harmonic movement possible here. p.s. i enjoy using 8:9:10:11:12 as a consonant chord, and not only isn't it CS, it's not even proper! voice-leading can be tricky, though, when the consonant chord isn't spread roughly evenly over the octave . . .

Message: 8272 - Contents - Hide Contents Date: Fri, 14 Nov 2003 23:51:34 Subject: Re: Vals? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>>> Except I wonder how we should write a complete mapping matrix for a >> more-than-1D temperament. >>>> a matrix is a matrix, not a bra or a ket. >> True, but the dot product of two vectors generalises naturally to the > product of a matrix with a column vector. > When we go beyond 1D > temperaments we have prime-mappings which are matrices (one row per > generator) and we multiply that by the transpose of a ratio's > prime-exponent-vector (monzo) to get a vector giving the count of each > generator.can you show an example? obviously i'm plenty confused as to how to correctly notate these things . . .>> i never understood >> covariant vs. contravariant, though . . . >> Me neither. Paul, we've gotta keep on Gene to tell us what things mean > AS APPLIED TO TUNING. And for him to keep trying different > explanations until we get it.right, but i still want to understand it, since it was in my relativity textbooks . . .

Message: 8273 - Contents - Hide Contents Date: Sat, 15 Nov 2003 05:34:08 Subject: Re: Vals? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> So [-4 4 -1> (x) [-11 4 2> = <12 19 28] >> Or if you like, |-4 4 -1> ^ |-11 4 2> = <12 19 28|. Welcome to the > wonderful world of wedge products.Yes indeed. Hoorah! Although I think I'd rather use the square brackets and only use the pipe if you put them together so < ... ].[ ... > is equivalent to < ... | ... > The square brackets alook more enclosing, and can't be mistaken for a one.

Message: 8274 - Contents - Hide Contents Date: Sat, 15 Nov 2003 19:14:39 Subject: Re: Vals? From: Graham Breed Dave Keenan wrote:> So the wedge product is a generalisation of the 3D Cartesian product > or cross product. Awesome! There are really some light-bulbs coming on > in my head today. :-) Thanks Gene.No, it's the other way around. The 3-D dot and cross products are special cases of Grassman algebra, which came first. Graham

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8450
8500
8550
8600
8650
8700
8750
8800
8850
8900
8950

**8250 -**
8275 -