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Message: 3350 - Contents - Hide Contents Date: Fri, 18 Jan 2002 22:37:35 Subject: Re: Carl's Mathworld-Complete Correspondence Theorem From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> MCCT- For all pairs of terms [t_i, t_(i+1)], from finite > alphabet [gamma], it is possible to select a pair of > terms [m_i, m_(i+1)] from the Mathworld website which > are identical.My brother Robin was complaining to me about this, and brought up the word "pencil". It turned out he was *not* referring to http://mathworld.wolfram.com/Pencil.html * [with cont.] but had picked a word at random. He was pretty triumphant when he found out it had also been made off with by mathematicians, though I think the mathematical use is older than those yellow things.

Message: 3351 - Contents - Hide Contents Date: Fri, 18 Jan 2002 22:40:59 Subject: Re: Carl's Mathworld-Complete Correspondence Theorem From: clumma>My brother Robin was complaining to me about this, and brought >up the word "pencil". It turned out he was *not* referring to > > http://mathworld.wolfram.com/Pencil.html * [with cont.] > >but had picked a word at random. He was pretty triumphant when he >found out it had also been made off with by mathematicians, though >I think the mathematical use is older than those yellow things.It says 1960...? -Carl

Message: 3352 - Contents - Hide Contents Date: Fri, 18 Jan 2002 22:46:57 Subject: Re: Carl's Mathworld-Complete Correspondence Theorem From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> My brother Robin was complaining to me about this, and brought >> up the word "pencil". It turned out he was *not* referring to >> >> http://mathworld.wolfram.com/Pencil.html * [with cont.] >> >> but had picked a word at random. He was pretty triumphant when he >> found out it had also been made off with by mathematicians, though >> I think the mathematical use is older than those yellow things. >> It says 1960...? > > -CarlCremona was 1960. Desargues lived 1591-1661.

Message: 3353 - Contents - Hide Contents Date: Sat, 19 Jan 2002 09:37:24 Subject: The consonant heptads of Tweedledee and Tweedeldum From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> 16. Tweedledee > > [3, 5, 9, 4, 1, 6, -4, 7, -8, -20] > > ets 15 > > [55/54, 56/55, 100/99, 121/120, 126/125, 3025/3024] > > [[0, -3, -5, -9, -4], [1, 2, 3, 4, 4]] > > [.1329702752, 1] > > a = 1.9946/15 = 159.5643303 cents > > badness 255.7850727 > rms 22.12985764 > g 4.342481185The heptad in 15-et is 2222223 in its various flavors, and this is pretty much a 15-et system, though [7,7,7,7,7,7,11] in the 53-et and [9,9,9,9,9,9,14] in the 68-et are interesting alternatives.> 17. Tweedledum > > [3, 5, -6, 4, 1, -18, -4, -28, -8, 32] > > [55/54, 64/63, 100/99, 121/120, 176/175, 385/384] > > ets 7, 15, 22 > > [[0, -3, -5, 6, -4], [1, 2, 3, 2, 4]] > > [.1357721305, 1] > > a = 2.9870/22 = 162.9265567 cents > > badness 262.2914819 > rms 11.79393546 > g 6.430951940The 22-et version of this is 3333334, but 8/59 is closer to the rms optimal generator, and has the same 7s and 11s as the 118-et; this version of it goes [8,8,8,8,8,8,11].

Message: 3354 - Contents - Hide Contents Date: Sat, 19 Jan 2002 09:54:15 Subject: Tweedle From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: This one is related to Dum and Dee, so I guess it's just Tweedle.

Message: 3355 - Contents - Hide Contents Date: Sat, 19 Jan 2002 07:52:11 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: monz Hi Gene,> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, January 18, 2002 8:22 PM > Subject: [tuning-math] A top 20 11-limit superparticularly generatedlinear temperament list> > > I look all the 11-limit superparticulars >= 49/48 and > found all the 11-limit linear temperaments they generated; > there turned out to be 319 of them. The following is > the top 20 in terms of low (logarithmically flat) badness, > plus a special guest star "Monzo" which is what 45/44, > 64/63 and 81/80 will give you. If Joe objects I will > quit calling it that.Cool! I never object to having my name on a tuning thing (something like the old show-biz dictum "even bad publicity is good publicity"). But ... it would be really nice if you could explain, as only tw examples, exactly what all this means. Since I've already played around with these particular unison-vectors, explaining what you did here would help me a lot to understand the rest of your work.> > 19. Monzoid > > [1, 4, -2, -1, 4, -6, -5, -16, -16, 4] > > [55/54, 64/63, 81/80, 385/384] > > ets 5, 7 > > [[0, -1, -4, 2, 1], [1, 2, 4, 2, 3]] > > [.4181947520, 1] > > a = 5.0183/12 = 501.8337024 cents > > badness 269.9708171 > rms 39.86372247 > g 3.150963571 > > ... > > Number 46 Monzo > > > [64/63, 81/80, 100/99, 176/175] > > ets 7, 12 > > [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]] > > [.4190088422, 1] > > a = 5.0281/12 = 502.8106107 cents > > badness 312.5112733 > rms 28.87226550 > g 4.174754057 -monz _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3356 - Contents - Hide Contents Date: Sat, 19 Jan 2002 08:01:26 Subject: Re: ERROR IN CARTER'S SCHOENBERG (Re: badly tuned remote overtones) From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, January 18, 2002 1:04 PM > Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned remote overtones) > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> I think you misunderstand me, Paul. I just mean that there's >> probably a good chance that at least some of the time, Schoenberg >> thought of the "Circle of 5ths" in a meantone rather than a >> Pythagorean sense. >> I doubt it. For Schoenberg, the circle of 5ths closes after 12 > fifths -- which is closer to being true in Pythagorean than in most > meantones.I understand that, Paul ... but if one is trying to ascertain the potential rational basis behind Schoenberg's work, how does one decide which unison-vectors are valid and which are not? Schoenberg was very clear about what he felt were the "overtone" implications of the diatonic scale (and later, the chromatic as well), but as I showed in my posts, the only "obvious" 5-limit unison-vector is the syntonic comma, and it seemed to me that there always needed to be *two* 5-limit unison-vectors in order to have a matrix of the proper size (so that it's square). (I realize that by transposition it need not be a 5-limit UV, but I'm not real clear on what else *could* be used, except for the 56:55 example Gene used.)>> This reference to you is only meant to credit you for opening >> my eyes to the strong meantone basis behind a good portion of >> the "common-practice" European musical tradition. >> OK -- but you're confusing two completely unrelated facts -- that > 128:125 is just in 1/4-comma meantone, and that 128:125 is one of the > simplest unison vectors for defining a 12-tone periodicity block.OK, I'm willing to take note of your point, but ... *why* are these two facts "completely unrelated"? Isn't it possible that there *is* some relation between them that no-one has noticed before? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3357 - Contents - Hide Contents Date: Sat, 19 Jan 2002 08:20:00 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, January 19, 2002 7:52 AM > Subject: Re: [tuning-math] A top 20 11-limit superparticularly generatedlinear temperament list> > > Hi Gene, >>> From: genewardsmith <genewardsmith@xxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Friday, January 18, 2002 8:22 PM >> Subject: [tuning-math] A top 20 11-limit superparticularly generated> linear temperament list >> >> > But ... it would be really nice if you could explain, as > only tw examples, exactly what all this means. Since I've > already played around with these particular unison-vectors, > explaining what you did here would help me a lot to > understand the rest of your work. >>>> Number 46 Monzo >> >> >> [64/63, 81/80, 100/99, 176/175] >> >> ets 7, 12 >> >> [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]] >> >> [.4190088422, 1] >> >> a = 5.0281/12 = 502.8106107 cents >> >> badness 312.5112733 >> rms 28.87226550 >> g 4.174754057 > > > > > -monzOK, I gave this a whirl thru my spreadsheet and this is what I got: kernel 2 3 5 7 11 unison vectors ~cents [ 1 0 0 0 0 ] = 2:1 0 [ 4 0 -2 -1 1 ] = 176:175 9.864608166 [ 2 -2 2 0 -1 ] = 100:99 17.39948363 [ 6 -2 0 -1 0 ] = 64:63 27.2640918 [-4 4 -1 0 0 ] = 81:80 21.5062896 adjoint [ 0 0 0 -0 0 ] [ 0 1 1 -1 0 ] [ 0 4 4 -4 0 ] [ 0 -2 -2 2 0 ] [ 0 6 6 -6 0 ] determinant = | 0 | mapping of Ets (top row above) to Uvs [ 1 1/3 2/3 -2/3 0 ] [ 4 2&2/3 2&2/3 -2&2/3 0 ] [ 2 1&1/3 1&1/3 -1&1/3 0 ] [ 6 4 4 -4 0 ] [-4 -2&2/3 -2&2/3 2&2/3 0 ] I don't really understand what this is saying either. (Many of the "0"s were actually given by Excel as tiny numbers such as "2.22045 * 10^-16", which is what it actually gave as the determinant.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3358 - Contents - Hide Contents Date: Sat, 19 Jan 2002 10:05:03 Subject: Re: Schoenberg's 1927/34 "Problems of Harmony" theory From: monz Graham, Gene, Paul, Can you please verify that what I said here is correct, or fix and explain if it's not?> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, January 18, 2002 2:52 AM > Subject: [tuning-math] Schoenberg's 1927/34 "Problems of Harmony" theory > > > ... >>> [Schoenberg] 1934 _Problems of Harmony_ 13-limit system > > ... > > matrix >> [ 1 0 0 0 0 ] = 2:1 > [ -2 0 0 -1 1 ] = 104:99 > [ 2 0 -1 0 1 ] = 117:112 > [ -2 0 -1 0 0 ] = 64:63 > [ 4 -1 0 0 0 ] = 81:80 > [ 2 1 0 -1 0 ] = 45:44 > > > adjoint: > > [ 12 0 0 0 0 0 ] > [ 19 -1 1 -1 1 1 ] > [ 28 -4 4 -4 -8 4 ] > [ 34 2 -2 -10 -2 -2 ] > [ 42 -6 6 -6 -6 -6 ] > [ 44 4 8 -8 -4 -4 ] > > > OK, I see that the first column-vector gives a typical > 12-EDO mapping. Interestingly, now the 11th harmonic > is mapped to 42 degrees of 12-EDO -- if "C" is n^0, this > is "F#", the opposite of how Schoenberg mapped it in 1911 > (as "F"), and indeed this is exactly how Schoenberg now > notates 11 in "Problems of Harmony". And the "new" 13th > harmonic is mapped to the 44th degree ("Ab"), which again > is how Schoenberg notates it. > > > As for the other column-vectors: > > I can see that all of them map 3 = 1 generator, the "5th", > typical of both meantone and Pythagorean. > > Columns 2, 3, 4, and 6 map 5 = 4 generators, also typical > of meantone, and the 5th column maps 5 = -8 generators, > typical of a Pythagorean-based schismic temperament. > > Columns 2, 3, 5, and 6 map 7 to -2 generators, the "minor 7th", > the closest approximation in Pythagorean. Column 4 maps > 7 = 10 generators, the "augmented 6th", which is a typical > meantone mapping. > > Columns 2, 3, and 4 map 11 = 6 generators, the "tritone" or > "augmented 4th", a meantone-like approximation. Columns > 5 and 6 map 11 = -6 generators, the Pythagorean "diminished 5th", > again only an approximation. > > Columns 2, 5, and 6 map 13 = -4 generators, the "minor 6th", > a meantone-like approximation, and columns 3 and 4 map > 13 = 8 generators, the "augmented 5th", a Pythagorean > approximation. ----------I appreciated Graham's explanation of this ...>> adjoint >> >> [ 12 0 0 0 0 ] >> [ 19 -1 0 0 3 ] >> [ 28 -4 0 0 0 ] >> [ 34 2 0 -12 -6 ] >> [ 41 1 12 0 -3 ] >> >> determinant = | 12 | >> See the third column has all zeros except for a 12 right at the bottom. > That means, trivially, it has a greatest common divisor of 12. (We don't > count zeros in the gcd.) As a rule of thumb, the GCD of a column tells you > how many *equal* steps the equivalence interval is being divided into. In > this case, we're dividing the octave into 12 equal steps. > > Dividing the octave this way is the same as defining a new period to be a > fraction of the original equivalence interval. Divide the whole column > through by the GCD, and you get the mapping within the period. In this > case, that gives [0 0 0 0 1]. The first zero tells you that the octave > takes the same value as it does in 12-equal. Not much of a surprise. The > next four zeros tell you that 3:1, 5:1 and 7:1 are also taken from 12-equal. > And the 1 in the last column tells you that 11:1 is one generator step away > from it's value in 12-equal. For 11:1 to be just, you'd have a 51 cent > generator. > > This is a fairly trivial example, and not much use as a temperament. But > you could realise it by having two keyboards tuned a quartertone apart. To > play an 11-limit otonality, you'd have C-E-G-Bb-D on one keyboard, and F on > the other.... and would appreciate some insight into how it would apply to my description of the 1927/34 13-limit system shown at the top of this post. Also, Graham's explanation here is of the matrix which I believe is a good candidate for modeling Schoenberg's 1911 theory. Can someone please expand on what Graham says and show what it might have to do with Schoenberg's adoption of 12-EDO as his preferred tuning? I've shown that Schoenberg rejected microtonality on practical grounds (lack of instruments, impact on his own financial situation, etc.), but his acceptance of 12-EDO eventually became so strong that, in addition to the known numerological motives behind many of his choices, my hunch is that there's probably a strong aesthetic motive as well, and an explanation like this would help to formulate that. Just trying to make sure that I understand this stuff. Thanks. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3359 - Contents - Hide Contents Date: Sat, 19 Jan 2002 11:40:56 Subject: deeper analysis of Schoenberg unison-vectors From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, January 19, 2002 8:01 AM > Subject: Re: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned remote overtones) > > > ... but if one is trying to ascertain > the potential rational basis behind Schoenberg's work, how does > one decide which unison-vectors are valid and which are not? > > Schoenberg was very clear about what he felt were the "overtone" > implications of the diatonic scale (and later, the chromatic > as well), but as I showed in my posts, the only "obvious" > 5-limit unison-vector is the syntonic comma, and it seemed > to me that there always needed to be *two* 5-limit unison-vectors > in order to have a matrix of the proper size (so that it's square). > > (I realize that by transposition it need not be a 5-limit UV, > but I'm not real clear on what else *could* be used, except for > the 56:55 example Gene used.)To make clear what I'm trying to say: Let's begin with the unison-vectors clearly implied by Schoenberg's 1911 diagram.> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 16, 2002 3:43 AM > Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG > > ... the total list of unison-vectors implied by Schoenberg's > 1911 diagram [on p 23 of the original 1911 edition of > _Harmonielehre_, p 24 in the Carter 1978 English edition] is: > > Bb 11*4=44 : Bb 7*6=42 = 22:21 > F 16*4=64 : F 7*9=63 = 64:63 > F 11*6=66 : F 16*4=64 = 33:32 > F 11*6=66 : F 7*9=63 = 22:21 > A 9*9=81 :(A 20*4=80) = 81:80 > C 11*9=99 :(C 24*4=96) = 33:32 > > But because 22:21, 33:32, and 64:63 form a dependent triplet > (any one of them can be found by multiplying the other two), > this does not suffice to create a periodicity-block, which > needs another independent unison-vector.But now let's try to find the other unison-vector we need from Schoenberg's musical examples. If our 1/1 is called "C", in his overtone diagram, Schoenberg calls "Eb" the "6th overtone [= 7th harmonic] of F", so that its ratio is 7/6. But then Schoenberg leaves the discussion of implied 7- and higher-limit harmonies to the later chapters, and devotes several chapters to explaining the diatonic major scale and its harmonies, using C as a reference pitch and C-majoras the reference scale and key. The diagram immediately before the one referred to above is one in which he derives the diatonic major scale from the first 6 harmonics of F, C, and G : 5:3-----5:4----15:8 A E B / \ / \ / \ / \ / \ / \ 4:3-----1:1-----3:2-----9:8 F C G D This is standard stuff, going back to Zarlino (1558). And as everyone here knows, a description of standard diatonic chord progressions is going to bump into the syntonic comma wherever a II-V progression is found, which would imply a new D on our lattice at 10/9. According to the expanded diagram of Schoenberg's explanation of the overtone theory on p 23 [p 24 in Carter] (going up to the 12th harmonics), the one I refer to at the beginning, the 81:80 syntonic comma is already a part of the system anyway. He shows A as the 5th and 10th harmonics of F and as the 9th harmonic of G, which are the ratios 81:20 and 81:40, which in turn are the syntonic comma plus 2 "8ves" and 1 "8ve" respectively. So we already have the unison-vector of 81:80 = [-4 4 1] included in our kernel. Schoenberg first introduces chromatic pitches in the chapter "Die Molltonart" [p 110-128 in the original edition, p 95-111 in the Carter edition]: F# and G# in the context of A-minor. To me, his explanation clearly implies a basis somewhere between meantone and 5-limit JI: A-minor is seen as the relative of C-major, so the note A is ~5/3. G# is always regarded as a "leading-tone" and is assumed to be a consonant ~5/4 above the "dominant" E, which is assumed to be ~3/2 above the tonic A. F# is always ~5/4 above D, the "subdominant", which is assumed to be ~4/3 above the tonic A; thus, the 10/9 version of D is the one in effect here. So our diatonic minor-scale paradigm lattice is: 25:18----( )---25:16 F# G# / \ / \ / \ / \ / \ / \ 10:9----5:3-----5:4----15:8 D A E B \ / \ / \ / \ / \ / \ / 4:3-----1:1-----3:2 F C G and again the syntonic comma is in effect because, according to Schoenberg's list of available minor-key chords on p 115 [p 99 in Carter], B can also be 50/27, F# can also be 45/32, and D can still also be 9/8. In a tiny handful of examples Schoenberg also introduces C# as a sharpened "3rd" (= ~5/4) in the II chord in the key of G-major. So altogether up to this point we have this lattice: 50:27---25:18---25:24---25:16 B F# C# G# \ / \ / \ / \ \ / \ / \ / \ 10:9----5:3-----5:4----15:8---45:32 D A E B F# \ / \ / \ / \ / \ / \ / \ / \ / 4:3-----1:1-----3:2-----9:8 F C G D These are the only pitches implied in any of Schoenberg's explanations until the chapter "Modulation" [p 169-198 in the original edition, p 150-174 in the Carter edition]. Thus, excluding the prefatory chapters on aesthetics, about 1/3 of _Harmonielehre_ devoted to discussion of this simple harmonic paradigm. On p 184 [p 161 in Carter], music example number 110, we see a D# in a musical example for the first time in _Harmonielehre_. The first chord is a C-major triad, or I in the key of C-major, which Schoenberg also designates simultaneously as VI in E-minor. The second chord is a V in E-minor, which is a B-major triad, and so its ~5/4 is D# 75/64 : 50:27---25:18---25:24---25:16---75:64 B F# C# G# D# \ / \ / \ / \ / \ \ / \ / \ / \ / \ 10:9----5:3-----5:4----15:8---45:32 D A E B F# \ / \ / \ / \ / \ / \ / \ / \ / 4:3-----1:1-----3:2-----9:8 F C G D So comparing this D# 75/64 with our Eb 7/6, now we finally have a canditate for another 7-limit unison-vector, namely 225:224 = [-5 2 2 -1] . So as of p 184 in _Harmonielehre_, we can construct as system valid for Schoenberg's theories, as follows: kernel 2 3 5 7 11 unison vectors ~cents [ 1 0 0 0 0 ] = 2:1 0 [-5 2 2 -1 0 ] = 225:224 7.711522991 [-4 4 -1 0 0 ] = 81:80 21.5062896 [ 6 -2 0 -1 0 ] = 64:63 27.2640918 [-5 1 0 0 1 ] = 33:32 53.27294323 adjoint [ 12 0 0 0 0 ] [ 19 1 2 -1 0 ] [ 28 4 -4 -4 0 ] [ 34 -2 -4 -10 0 ] [ 41 -1 -2 1 12 ] determinant = | 12 | mapping of ETs to UVs [ 12 -7 12 0 12 ] [ 0 1 0 1 -2 ] [ 0 0 0 0 1 ] [ 0 0 0 1 0 ] [ 0 0 1 0 0 ] This last matrix shows that 12-ET maps all of the unison-vectors except 225:224 to 0 or 12 (i.e., unison), correct? And that the last three do not temper out the 81:80, 64:63, and 33:32 respectively, correct? Further illumation would be appreciated. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3360 - Contents - Hide Contents Date: Sat, 19 Jan 2002 20:06:58 Subject: Re: deeper analysis of Schoenberg unison-vectors From: joemonz --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Message 2798 > From: "monz" <joemonz@y...> > Date: Sat Jan 19, 2002 2:40 pm > Subject: deeper analysis of Schoenberg unison-vectors > > ... > > So as of p 184 in _Harmonielehre_, we can construct as system > valid for Schoenberg's theories, as follows: > > kernel > > 2 3 5 7 11 unison vectors ~cents > > [ 1 0 0 0 0 ] = 2:1 0 > [-5 2 2 -1 0 ] = 225:224 7.711522991 > [-4 4 -1 0 0 ] = 81:80 21.5062896 > [ 6 -2 0 -1 0 ] = 64:63 27.2640918 > [-5 1 0 0 1 ] = 33:32 53.27294323 > > adjoint > > [ 12 0 0 0 0 ] > [ 19 1 2 -1 0 ] > [ 28 4 -4 -4 0 ] > [ 34 -2 -4 -10 0 ] > [ 41 -1 -2 1 12 ] > > determinant = | 12 | > > > mapping of ETs to UVs > > [ 12 -7 12 0 12 ] > [ 0 1 0 1 -2 ] > [ 0 0 0 0 1 ] > [ 0 0 0 1 0 ] > [ 0 0 1 0 0 ] > > > This last matrix shows that 12-ET maps all of the > unison-vectors except 225:224 to 0 or 12 (i.e., unison), > correct? > > And that the last three do not temper out the 81:80, 64:63, > and 33:32 respectively, correct? > > > Further illumation would be appreciated.Specificially: what is that second line saying? It looks like 225:224 and 64:63 are 1 step, and 33:32 is -2 steps. What tuning is that? Would these be an example of a mapping to two keyboards where 1 is 12-tET, and the second is mistuned by some amount that renders ~1/6-tone (i.e., an amount which makes 7-limit harmonies accurate) as 1 step, and ~1/4-tone (i.e., to make 11-limit harmonies accurate) as -2 steps ? deeply curious, -monz

Message: 3361 - Contents - Hide Contents Date: Sat, 19 Jan 2002 20:56:56 Subject: Re: deeper analysis of Schoenberg unison-vectors From: joemonz --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> Message 2798 > From: "monz" <joemonz@y...> > Date: Sat Jan 19, 2002 2:40 pm > Subject: deeper analysis of Schoenberg unison-vectors > > > > Let's begin with the unison-vectors clearly implied by > Schoenberg's 1911 diagram. > >>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Wednesday, January 16, 2002 3:43 AM >> Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG >> >> ... the total list of unison-vectors implied by Schoenberg's >> 1911 diagram [on p 23 of the original 1911 edition of >> _Harmonielehre_, p 24 in the Carter 1978 English edition] is: >> >> Bb 11*4=44 : Bb 7*6=42 = 22:21 >> F 16*4=64 : F 7*9=63 = 64:63 >> F 11*6=66 : F 16*4=64 = 33:32 >> F 11*6=66 : F 7*9=63 = 22:21 >> A 9*9=81 :(A 20*4=80) = 81:80 >> C 11*9=99 :(C 24*4=96) = 33:32 >> >> But because 22:21, 33:32, and 64:63 form a dependent triplet >> (any one of them can be found by multiplying the other two), >> this does not suffice to create a periodicity-block, which >> needs another independent unison-vector. > >> But now let's try to find the other unison-vector we need > from Schoenberg's musical examples. > > If our 1/1 is called "C", in his overtone diagram, Schoenberg calls > "Eb" the "6th overtone [= 7th harmonic] of F", so that its > ratio is 7/6. > > But then Schoenberg leaves the discussion of implied 7- and > higher-limit harmonies to the later chapters, and devotes > several chapters to explaining the diatonic major scale and > its harmonies, using C as a reference pitch and C-majoras the > reference scale and key. The diagram immediately before the > one referred to above is one in which he derives the diatonic > major scale from the first 6 harmonics of F, C, and G : > > > 5:3-----5:4----15:8 > A E B > / \ / \ / \ > / \ / \ / \ > 4:3-----1:1-----3:2-----9:8 > F C G D > > > This is standard stuff, going back to Zarlino (1558). > > And as everyone here knows, a description of standard > diatonic chord progressions is going to bump into the > syntonic comma wherever a II-V progression is found, > which would imply a new D on our lattice at 10/9.I thought it worth pointing out that from the very beginning of his descriptions of the diatonic scale, the 81:80 must be tempered out, so that the proper lattice for at least the first 184 pages of _Harmonielehre_ would be a cylindrical meantone-based one.> So comparing this D# 75/64 with our Eb 7/6, now we finally > have a canditate for another 7-limit unison-vector, namely > 225:224 = [-5 2 2 -1] .Also worth pointing out: 224/224 is neither a divisor nor product of any of the other potential unison-vectors <22:21, 33:32, 63:64, 81:80>, thus it satisfies the condition we need for the unison-vector we're seeking, namely, that it be independent of all the others.> mapping of ETs to UVs > > [ 12 -7 12 0 12 ] > [ 0 1 0 1 -2 ] > [ 0 0 0 0 1 ] > [ 0 0 0 1 0 ] > [ 0 0 1 0 0 ]What does the -7 mean in the first row? It's telling us something significant about how 12-tET handles 225:224 in this kernel, but what? -monz

Message: 3362 - Contents - Hide Contents Date: Sat, 19 Jan 2002 20:56:22 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> But ... it would be really nice if you could explain, as > only tw examples, exactly what all this means. Since I've > already played around with these particular unison-vectors, > explaining what you did here would help me a lot to > understand the rest of your work. > > >> >> 19. MonzoidThis was number 19 on the list, in terms of a badness measure.>> [1, 4, -2, -1, 4, -6, -5, -16, -16, 4]This is the "wedgie". When standardized, there is a unique wedgie corresponding to each (non-torsion, non-equal) temperment. This is a linear temperament wedgie; in the 11-limit the planar temperament wedgies also have ten dimensions; however this wedgie is computed from three unisons or two ets, whereas a planar would be computed from two unisons or three ets.>> [55/54, 64/63, 81/80, 385/384]These are all the 11-limit superparticulars equal to or less than 49/48 which are commas of the temperament--meaning they are tempered out. Since there are four of them, there is a linear dependency, but we can generate Monzoid from three independent ones.>> ets 5, 7These are "standard" ets, which round off to the nearest integer when mapping primes; 12 is not on the list, but h5+h7 would be if I listed anything "nonstandard".>> [[0, -1, -4, 2, 1], [1, 2, 4, 2, 3]]This is the period matrix, in a way easier to print than as a 4x2 matrix. The first list is the first column, giving maps to primes of the generator, the second column is the octaves.>> [.4181947520, 1] >> a = 5.0183/12 = 501.8337024 centsThese are the two generators, the second being merely an octave, and the first being a slightly sharp fourth.>> badness 269.9708171 >> rms 39.86372247 >> g 3.150963571"Badness" is the flat badness measure, "rms" is an average value for how much, in cents, the 11-limit consonances are off (40 cents!), and g is the average number of generator steps to get to a consonance (a mere 3.)>> Number 46 Monzo >> >> >> [64/63, 81/80, 100/99, 176/175]45/44 does not appear only because 45/44 > 49/48, which I used as a cut-off.>> ets 7, 12This time, the "standard" h12 12-et map makes its appearance.>> [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]] >> >> [.4190088422, 1] >> >> a = 5.0281/12 = 502.8106107 centsA small difference in the size of the optimal generator, because Monzo maps 11 differently than Monzoid.>> badness 312.5112733 >> rms 28.87226550 >> g 4.174754057The different 11-map makes the 11-limit more accurate, but it takes more steps on average because 11 maps to -6 and not 1.

Message: 3363 - Contents - Hide Contents Date: Sat, 19 Jan 2002 00:09:37 Subject: Re: Hi Dave K. From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> Thanks, but I don't think RMS will work. That implies a Euclidean > metric, but a "taxicab" metric seems to be what we want here.Yes of course. Sorry. Just replace every ocurrence of "rms" with "taxicab" in what I wrote.

Message: 3364 - Contents - Hide Contents Date: Sat, 19 Jan 2002 21:34:47 Subject: Re: deeper analysis of Schoenberg unison-vectors From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Bb 11*4=44 : Bb 7*6=42 = 22:21 >> F 16*4=64 : F 7*9=63 = 64:63 >> F 11*6=66 : F 16*4=64 = 33:32 >> F 11*6=66 : F 7*9=63 = 22:21 >> A 9*9=81 :(A 20*4=80) = 81:80 >> C 11*9=99 :(C 24*4=96) = 33:32 >> >> But because 22:21, 33:32, and 64:63 form a dependent triplet >> (any one of them can be found by multiplying the other two), >> this does not suffice to create a periodicity-block, which >> needs another independent unison-vector.What it creates, in fact, is Monzoid, as shown by the fact that the above list is compatible with both h5 and h7. Taking triples, we find that the Monzoid wedgie results from (81/80,63/63,33/32), (81/80,64/63,22/21), and (64/63,33/32,22/21).> mapping of ETs to UVs > > [ 12 -7 12 0 12 ] > [ 0 1 0 1 -2 ] > [ 0 0 0 0 1 ] > [ 0 0 0 1 0 ] > [ 0 0 1 0 0 ]What is this?

Message: 3365 - Contents - Hide Contents Date: Sat, 19 Jan 2002 04:22:05 Subject: A top 20 11-limit superparticularly generated linear temperament list From: genewardsmith I look all the 11-limit superparticulars >= 49/48 and found all the 11-limit linear temperaments they generated; there turned out to be 319 of them. The following is the top 20 in terms of low (logarithmically flat) badness, plus a special guest star "Monzo" which is what 45/44, 64/63 and 81/80 will give you. If Joe objects I will quit calling it that. "Arabic", by the way, came in #32 but clearly would be much higher if we forgot about 7. Things to note are temperaments which don't seem to have much to do with "good" ets and temperaments which are close relatives of other temperaments. The first on the list, Hemiennealimmal, could certainly claim to be able to produce authentic Partch tunings of the 11-limit, and it would be interesting to check in how many keys 72 notes tempered in this way could play the Partch 43-tone scale to extreme accuracy. It is also interesting to note how many of the top temperaments are compatible with 72. 1. Hemiennealimmal [36, 54, 36, 18, 2, -44, -96, -68, -145, -74] [2401/2400, 3025/3024, 4375/4374, 9801/9800] ets 72, 198, 270, 342, 612 [[0, 2, 3, 2, 1], [18, 12, 17, 34, 54]] [.4591217954, 1/18] a = 33.0568/72 = 280.9825/612 = 550.9491544 cents badness 78.02778100 rms .1987978829 g 36. 2. Miracle [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] [225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024] ets 10, 31, 41, 72 [[0, 6, -7, -2, 15], [1, 1, 3, 3, 2]] [.9722688696e-1, 1] a = 7.0003/72 = 116.6722643 cents badness 125.5016755 rms 1.901465778 g 12.35198075 3. Octoid [24, 32, 40, 24, -5, -4, -45, 3, -55, -71] [540/539, 3025/3024, 4375/4374, 9801/9800] ets 72, 80, 152, 224, 296 [[0, -3, -4, -5, -3], [8, 16, 23, 28, 31]] [.1383934690, 1/8] a = 9.9643/72 = 31.0001/224 = 166.0721626 badness 147.3854996 rms .7687062948 g 23.42160176 4. Undecimal augmented fifth [12, 22, -4, -6, 7, -40, -51, -71, -90, -3] [385/384, 441/440, 3025/3024, 4375/4374, 9801/9800] ets 26, 46, 72, 118, 190 [[0, 6, 11, -2, -3], [2, -1, -3, 7, 9]] a = 25.0090/72 = 416.8172169 cents [.3473476807, 1/2] badness 169.9769111 rms 1.249416902 g 19.06380265 5. [12, 34, 20, 30, 26, -2, 6, -49, -48, 15] [243/242, 441/440, 540/539, 2401/2400, 9801/9800] ets 58, 72, 130 [[0, -6, -17, -10, -15], [2, 4, 7, 7, 9]] [.6933142420e-1, 1/2] a = 4.9919/72 = 83.1977090 cents badness 179.9856041 rms 1.462301383 g 17.95231779 6. [12, -2, 20, -6, -31, -2, -51, 52, -7, -86] [225/224, 385/384, 540/539, 9801/9800] ets 22, 50, 72, 94 [[0, 6, -1, 10, -3], [2, 1, 5, 2, 8]] [.1806533524, 1/2] a = 13.0070/72 = 216.7840228 cents badness 195.0280356 rms 1.584514315 g 17.95231779 7. Orwell [7, -3, 8, 2, -21, -7, -21, 27, 15, -22] [99/98, 121/120, 176/175, 225/224, 385/384, 540/539] ets 9, 22, 31, 53 [[0, 7, -3, 8, 2], [1, 0, 3, 1, 3]] [.2262038561, 1] a = 11.9888/53 = 271.4446272 cents badness 210.4954018 rms 5.548614670 g 8.860022575 8. Semihemimeantone [4, 16, 9, 10, 16, 3, 2, -24, -32, -3] [81/80, 99/98, 121/120, 243/242, 441/440, 540/539, 2401/2400] ets 31 [[0, -4, -16, -9, -10], [1, 3, 8, 6, 7]] [.3548751316, 1] a = 11.0011/31 = 425.8501579 cents badness 218.9540099 rms 6.965622568 g 7.914724072 9. [9, 5, -3, 7, -13, -30, -20, -21, -1, 30] ets 15, 31, 46 [121/120, 126/125, 176/175, 385/384, 441/440, 3025/3024] [[0, 9, 5, -3, 7], [1, 1, 2, 3, 3]] [.6494333856e-1, 1] a = 2.0132/31 = 77.9320062 cents badness 223.3668950 rms 4.418576095 g 10.52547929 10. [1, -1, 3, 4, -4, 2, 3, 10, 13, 1] [55/54, 56/55, 99/98, 3025/3024] ets 5 [[0, -1, 1, -3, -4], [1, 2, 2, 4, 5]] [.3798204598, 1] 11.0148/29 = 455.7845520 cents badness 235.8100854 rms 44.34125247 g 2.725540575 11. [6, 10, 10, 8, 2, -1, -8, -5, -16, -12] [50/49, 55/54, 99/98, 100/99, 121/120, 540/539, 9801/9800] ets 22 [[0, -3, -5, -5, -4], [2, 4, 6, 7, 8]] [.1375489239, 1/2] a = 3.0261/22 = 165.0587086 cents badness 238.7261371 rms 11.89273384 g 6.047431569 12. Nonkleismic [10, 9, 7, 25, -9, -17, 5, -9, 27, 46] [126/125, 176/175, 243/242, 441/440, 540/539, 2401/2400] ets 31, 58, 89 [[0, 10, 9, 7, 25], [1, -1, 0, 1, -3]] [.2584558979, 1] a = 23.0026/89 = 310.1470775 cents badness 240.3019988 rms 3.316530191 g 13.06303399 13. Magic [5, 1, 12, -8, -10, 5, -30, 25, -22, -64] [100/99, 225/224, 385/384, 540/539] ets 19, 22, 41 [[0, 5, 1, 12, -8], [1, 0, 2, -1, 6]] [.3172615104, 1] 13.0077/41 = 380.7138126 cents badness 242.7224832 rms 4.730404304 g 10.62006188 14. Septimal [0, 0, 7, 0, 0, 11, 0, 16, 0, -24] [55/54, 81/80, 100/99, 121/120, 243/242] ets 7 [[0, 0, 0, -1, 0], [7, 11, 16, 21, 24]] [.2141802354, 1/7] a = 257.0162824 cents badness 245.8506632 rms 22.63634705 g 4.183300133 15. Meanertone [1, 4, 3, -1, 4, 2, -5, -4, -16, -13] [55/54, 56/55, 81/80, 3025/3024] ets 5 [[0, -1, -4, -3, 1], [1, 2, 4, 4, 3]] [.4194849382, 1] a = 13.0040/31 = 503.3819256 cents badness 252.8666930 rms 47.54854253 g 2.725540575 16. Tweedledee [3, 5, 9, 4, 1, 6, -4, 7, -8, -20] ets 15 [55/54, 56/55, 100/99, 121/120, 126/125, 3025/3024] [[0, -3, -5, -9, -4], [1, 2, 3, 4, 4]] [.1329702752, 1] a = 1.9946/15 = 159.5643303 cents badness 255.7850727 rms 22.12985764 g 4.342481185 17. Tweedledum [3, 5, -6, 4, 1, -18, -4, -28, -8, 32] [55/54, 64/63, 100/99, 121/120, 176/175, 385/384] ets 7, 15, 22 [[0, -3, -5, 6, -4], [1, 2, 3, 2, 4]] [.1357721305, 1] a = 2.9870/22 = 162.9265567 cents badness 262.2914819 rms 11.79393546 g 6.430951940 18. Pentoid [2, 3, 1, -2, 0, -4, -10, -6, -15, -9] [49/48, 56/55, 99/98, 385/384] ets 4, 5, 9 [[0, -2, -3, -1, 2], [1, 2, 3, 3, 3]] [.2183480607, 1] a = 5.0220/23 = 262.0176727 cents badness 267.0829245 rms 40.16092708 g 3.116774888 19. Monzoid [1, 4, -2, -1, 4, -6, -5, -16, -16, 4] [55/54, 64/63, 81/80, 385/384] ets 5, 7 [[0, -1, -4, 2, 1], [1, 2, 4, 2, 3]] [.4181947520, 1] a = 5.0183/12 = 501.8337024 cents badness 269.9708171 rms 39.86372247 g 3.150963571 20. Catakleismic [6, 5, 22, -21, -6, 18, -54, 37, -66, -135] [225/224, 385/384, 540/539, 4375/4374] ets 19, 72 [[0, 6, 5, 22, -21], [1, 0, 1, -3, 9]] [.2639230436, 1] a = 19.0025/72 = 316.7076522 cents badness 271.0589693 rms 1.697136764 g 20.98979344 Number 46 Monzo [64/63, 81/80, 100/99, 176/175] ets 7, 12 [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]] [.4190088422, 1] a = 5.0281/12 = 502.8106107 cents badness 312.5112733 rms 28.87226550 g 4.174754057

Message: 3366 - Contents - Hide Contents Date: Sat, 19 Jan 2002 21:40:07 Subject: Re: deeper analysis of Schoenberg unison-vectors From: genewardsmith --- In tuning-math@y..., "joemonz" <joemonz@y...> wrote:> Also worth pointing out: 224/224 is neither a divisor > nor product of any of the other potential unison-vectors > <22:21, 33:32, 63:64, 81:80>, thus it satisfies the condition > we need for the unison-vector we're seeking, namely, that > it be independent of all the others.For independence, you need that any product to rational powers won't give you 225/224, or equivalently, that any product to integral powers will not give you any power of 225/224. You can check this with your linear algebra package, by taking the rank (or in this case, also the determinant) of the matrix of row vectors.

Message: 3367 - Contents - Hide Contents Date: Sat, 19 Jan 2002 23:35:22 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: Sorry about cutting your statement/question up as follows but I want to address the accuracy and complexity points separately.> The first on the list,Hemiennealimmal, could certainly claim to be able to produce authentic Partch tunings of the 11-limit, ... to extreme accuracy.>Yes. But even Partch didn't require such accuracy. I understand that he couldn't tell the difference between his scale and either 41-tET or 72-tET versions of it. So the extreme accuracy doesn't mitigate the badness of the extreme complexity, and Miracle leaves Hemiennealimmal for dead with any reasonable badness measure that relates to human beings.> ... and it would be interesting to checkin how many keys 72 notes tempered in this way could play the Partch 43-tone scale ...>I think the answer is zero. But the question seems fairly irrelevant of any temperament, since no-one I know wants to have as many as 72 notes per octave on a keyboard or fretboard and no composer I know wants to have to deal with that many notes (choosing always some more manageable subset). Also I don't find it likely that anyone would want to play Partch's scale in more than one "key" per piece. A more relevant question is how many notes of a given temperament does it take to include _one_ version of Partch's scale (without conflating any notes)? Since Partch's scale contains the 11-limit diamond, if I'm reading your cryptic lists of numbers correctly, the answer for hemiennealimmal cannot be less than (2*3+1)*18 = 126. The answer for Miracle is (2*22+1)*1 = 45.> 1. Hemiennealimmal > > [36, 54, 36, 18, 2, -44, -96, -68, -145, -74] > > [2401/2400, 3025/3024, 4375/4374, 9801/9800] > > ets 72, 198, 270, 342, 612 > > [[0, 2, 3, 2, 1], [18, 12, 17, 34, 54]] > > [.4591217954, 1/18] > > a = 33.0568/72 = 280.9825/612 = 550.9491544 cents > > badness 78.02778100 > rms .1987978829 > g 36. > > > 2. Miracle > > [6, -7, -2, 15, -25, -20, 3, 15, 59, 49] > > [225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024] > > ets 10, 31, 41, 72 > > [[0, 6, -7, -2, 15], [1, 1, 3, 3, 2]] > > [.9722688696e-1, 1] > > a = 7.0003/72 = 116.6722643 cents > > badness 125.5016755 > rms 1.901465778 > g 12.35198075

Message: 3368 - Contents - Hide Contents Date: Sun, 20 Jan 2002 01:45:43 Subject: Re: A top 20 11-limit superparticularly generated linear temperament list From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> Yes. But even Partch didn't require such accuracy.If I remember correctly, he did require such accuracy. He considered the fifth of 53-et to be good enough, and the third to be close, but no cigar. The fifth is flat by .068 cents, and the major third is flat 1.408 cents. The rms for Hemiennealimmal is 1/5 cent, so it is in this range and well towards the small side. Hence it seems reasonable to conclude that using this tuning would be authentically Partch. Of course, there's the question of how accurately his own instruments were tuned, which would give another handle on what authenticity would mean in this context. I understand that> he couldn't tell the difference between his scale and either 41-tET or > 72-tET versions of it.The question is one of his requirements, not his hearing.> I think the answer is zero.Here it is: 1--81/80--33/32--21/20--16/15--12/11--11/10--10/9--9/8--8/7--7/6 32/27--6/5--11/9--5/4--14/11--9/7--21/16--4/3--27/20--11/8--7/5 10/7--16/11--40/27--3/2--32/21--14/9--11/7--8/5--18/11--5/3--27/16 12/7--7/4--16/9--9/5--20/11--11/6--15/8--40/21--64/33--160/81--(2) This is mapped to [[0, 5, 3, 1, -5, 1, -2, -1, 4, -2, 0, -6, -1, -3, 3, 1, 2, 4, -2, 3, 1, -1, 1, -1, -3, 2, -4, -2, -1, -3, 3, 1, 6, 0, 2, -4, 1, 2, -1, 5, -1, -3, -5] , [0, -41, -24, -7, 43, -6, 19, 11, -30, 20, 4, 54, 13, 30, -19, -2, -10, -26, 24, -17, 0, 17, 1, 18, 35, -6, 44, 28, 20, 37, -12, 5, -36, 14, -2, 48, 7, -1, 24, -25, 25, 42, 59]] by Hemiennealimmal, so you are right, it won't fit--the first generator ranging from -6 to 6. But the question seems fairly irrelevant> of any temperament, since no-one I know wants to have as many as 72 > notes per octave on a keyboard or fretboard and no composer I know > wants to have to deal with that many notes (choosing always some more > manageable subset).This is assuming that you must be using a keyboard or fretboard. Partch, after all, *did* have 43 actual tones per octave in play, so I don't see how this theory holds up. Also I don't find it likely that anyone would want> to play Partch's scale in more than one "key" per piece.Even if they did not, tuning Partch's scale in this way would give you some equivalences for free (deriving from 2401/2400, 3025/3024, 4375/4374 and 9801/9800) which would make tempering Partch's 43 tones in this way a perfectly reasonable option.

Message: 3369 - Contents - Hide Contents Date: Sun, 20 Jan 2002 02:51:44 Subject: A comparison of Partch's scale in RI and Hemiennealimmal From: genewardsmith RI: edges: 18, 32, 64, 88 connectivity: 0, 0, 0, 2 Hemiennealimmal: edges: 64, 106, 159, 219 connectivity: 0, 0, 0, 4 The numbers are edges/connectivity in the 5, 7, 9 and 11-limits. I conclude that a great deal is gained by tempering in this way, and nothing significant is conceded in terms of quality of intonation. Of course, 72-et would do much better yet, but then some concessions will have been made.

Message: 3370 - Contents - Hide Contents Date: Sun, 20 Jan 2002 06:55:42 Subject: maple presentation? From: clumma Gene, have you seen this? Article Information * [with cont.] (Wayb.) -Carl

Message: 3371 - Contents - Hide Contents Date: Sun, 20 Jan 2002 00:08:37 Subject: Re: deeper analysis of Schoenberg unison-vectors From: monz> Message 2798 > From: "monz" <joemonz@y...> > Date: Sat Jan 19, 2002 2:40 pm > Subject: deeper analysis of Schoenberg unison-vectors Yahoo groups: /tuning-math/message/2798 * [with cont.] > > ... > > So as of p 184 in _Harmonielehre_, we can construct as system > valid for Schoenberg's theories, as follows: > > kernel > > 2 3 5 7 11 unison vectors ~cents > > [ 1 0 0 0 0 ] = 2:1 0 > [-5 2 2 -1 0 ] = 225:224 7.711522991 > [-4 4 -1 0 0 ] = 81:80 21.5062896 > [ 6 -2 0 -1 0 ] = 64:63 27.2640918 > [-5 1 0 0 1 ] = 33:32 53.27294323 > > adjoint > > [ 12 0 0 0 0 ] > [ 19 1 2 -1 0 ] > [ 28 4 -4 -4 0 ] > [ 34 -2 -4 -10 0 ] > [ 41 -1 -2 1 12 ] > > determinant = | 12 |Then I had something after this, about which Gene asked (and rightly so, as will be seen):> Message 2802 > From: "genewardsmith" <genewardsmith@j...> > Date: Sat Jan 19, 2002 4:34 pm > Subject: Re: deeper analysis of Schoenberg unison-vectors Yahoo groups: /tuning-math/message/2802 * [with cont.] > >>> mapping of ETs to UVs >> >> [ 12 -7 12 0 12 ] >> [ 0 1 0 1 -2 ] >> [ 0 0 0 0 1 ] >> [ 0 0 0 1 0 ] >> [ 0 0 1 0 0 ] >> What is this?Something I got from Graham. (I've been searching for an hour in both the tuning-math and tuning archives, and in my private emails, looking for it, and unfortunately can't find it!) He explained how the adjoint shows the mapping, and included this after it. Here's how it works: ... Oh no! My bad! That last one, the "mapping" matrix, was supposed to look like this: [ 1 0 0 0 0 ] [ 0 1 0 0 0 ] [ 0 0 1 0 0 ] [ 0 0 0 1 0 ] [ 0 0 0 0 1 ] I don't know what happened to give me that wrong matrix, and I'm glad Gene asked about it, because otherwise I wouldn't have realized that I made an error. Anyway, this is how it works: Look again at the first two matrices (the kernel and its adjoint). I divide the number from each successive row of the left column of the adjoint by the determinant, to get the proper numbers of the inverse, then multiply each of those quotients by each respective number in the top row of the kernel, add all of those products, and put the sum in the top row of the left column of the new "mapping" matrix. Then I go thru the left column of the inverse again, this time multiplying each row of that column by each number in the second row of the kernel, and put that sum down in the second row of the left column of the "mapping matrix". And so on for all the other rows of that column. Then repeat the same procedure for the second column-vector of the adjoint; the third column-vector of the adjoint; etc. So, using this example, the top row of the kernel is the vector for 2:1 = [ 1 0 0 0 0 ], so the left column of the inverse multiplied by this row gives (12/12)*1 and everything else times zero, so the sum is 1, which is set down as the first number of the left column. The next operation multiplies the left column of the inverse with the second row of the kernel: [ 12/12 ] * [-5 2 2 -1 0 ] [ 19/12 ] [ 28/12 ] [ 34/12 ] [ 41/12 ] = -5 + 19/6 + 28/6 - 17/6 + 0 = 0 So a zero is set down in the second row of the left column. And so on. According to what I remember Graham saying, correlating each row of the "mapping" matrix with the corresponding row of the kernel, each column of this "mapping" matrix shows which unison-vector is not tempered out by the temperament shown in the corresponding column of the adjoint. Thus, the "1" in the top row of the left column shows that the 12-tET does not temper out the 2:1 (?), the "1" in the second row of the second column shows that the [ 0 ] [ 1 ] [ 4 ] [ -2 ] [ -1 ] temperament does not temper out the 225:224, etc. Gene or Graham, can you explain what's going on here? And Paul: based on my summaries of _Harmonielehre_, do you agree with me that this PB accurately describes Schoenberg's theory up to at least p 184 of that book? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3372 - Contents - Hide Contents Date: Sun, 20 Jan 2002 00:27:34 Subject: more questions about adjoints and mappings From: monz Here's a simple example: Ellis's Duodene kernel 2 3 5 unison vectors ~cents [ 1 0 0 ] = 2:1 1200 [ -4 4 -1 ] = 81:80 21.5062896 [ 7 0 -3 ] = 128:125 41.05885841 adjoint [ -12 0 0 ] [ -19 -3 1 ] [ -28 0 4 ] determinant = | -12 | "mapping" of UVs [ 1 0 0 ] [ 0 1 0 ] [ 0 0 1 ] So here, I can see that the h12 mapping does not temper out the 2:1 ... and I still don't understand what that means. Is it simply because any "8ve"-based ET must include 2:1 by definition? I can also see that the third column of the adjoint specifies some kind of meantone, which tempers out the 2:1 and the 81:80, but not the diesis 128:125. Is there a way to tell what flavor of meantone it is? And the middle column of the adjoint specifies some temperament which does not temper out the 81:80. But can someone explain what kind of tuning this is? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3373 - Contents - Hide Contents Date: Sun, 20 Jan 2002 00:30:30 Subject: Re: deeper analysis of Schoenberg unison-vectors From: monz Hi Gene,> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, January 19, 2002 1:40 PM > Subject: [tuning-math] Re: deeper analysis of Schoenberg unison-vectors > > > --- In tuning-math@y..., "joemonz" <joemonz@y...> wrote: >>> Also worth pointing out: 224/224 is neither a divisor >> nor product of any of the other potential unison-vectors >> <22:21, 33:32, 63:64, 81:80>, thus it satisfies the condition >> we need for the unison-vector we're seeking, namely, that >> it be independent of all the others. >> For independence, you need that any product to rational > powers won't give you 225/224, or equivalently, that any > product to integral powers will not give you any power of > 225/224.Can you explain this in a little more detail, by using examples relevant to the Schoenberg PB I presented?> You can check this with your linear algebra package,Don't have one ... I do all this on an Excel spreadsheet.> by taking the rank (or in this case, also the determinant) > of the matrix of row vectors.By "matrix of row vectors" you mean the kernel, right? And what's the "rank"? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 3374 - Contents - Hide Contents Date: Sun, 20 Jan 2002 02:39:56 Subject: Re: deeper analysis of Schoenberg unison-vectors From: monz> >> Message 2798>> From: "monz" <joemonz@y...> >> Date: Sat Jan 19, 2002 2:40 pm >> Subject: deeper analysis of Schoenberg unison-vectors > Yahoo groups: /tuning-math/message/2798 * [with cont.] >> >> ... >> >> So as of p 184 in _Harmonielehre_, we can construct as system >> valid for Schoenberg's theories, as follows: >> >> kernel >> >> 2 3 5 7 11 unison vectors ~cents >> >> [ 1 0 0 0 0 ] = 2:1 0 >> [-5 2 2 -1 0 ] = 225:224 7.711522991 >> [-4 4 -1 0 0 ] = 81:80 21.5062896 >> [ 6 -2 0 -1 0 ] = 64:63 27.2640918 >> [-5 1 0 0 1 ] = 33:32 53.27294323 >> >> adjoint >> >> [ 12 0 0 0 0 ] >> [ 19 1 2 -1 0 ] >> [ 28 4 -4 -4 0 ] >> [ 34 -2 -4 -10 0 ] >> [ 41 -1 -2 1 12 ] >> >> determinant = | 12 |> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, January 20, 2002 12:08 AM > Subject: Re: [tuning-math] deeper analysis of Schoenberg unison-vectors UV map > [ 1 0 0 0 0 ] > [ 0 1 0 0 0 ] > [ 0 0 1 0 0 ] > [ 0 0 0 1 0 ] > [ 0 0 0 0 1 ]So in other words, the way Gene would write it: h12(225/224) = h12(81/80) = h12(64/63) = h12(33/32) = 0 h12(2/1) = 1 But how do you label those other four columns? Well, for the time being, I'll call them h0, g0, f0, and e0, respectively from left to right, so that: h0(2/1) = h0(81/80) = h0(63/64) = h0(33/32) = 0 , h0(225/224) = 1 g0(2/1) = g0(225/224) = g0(63/64) = g0(33/32) = 0 , g0(81/80) = 1 f0(2/1) = f0(225/224) = f0(81/80) = f0(33/32) = 0 , f0(64/63) = 1 e0(2/1) = e0(225/224) = e0(81/80) = e0(64/63) = 0 , e0(33/32) = 1 So, the 2nd and 4th column-vectors in the adjoint (h0 and f0, respectively) define two versions of meantone: - one (h0) in which 7 maps to the "minor 7th" = -2 generators, and which tempers out all the UVs except 225/224; - one (f0) in which 7 maps to the "augmented 6th" = +10 generators, and which tempers out all the UVs except 64/63; and both of which map 11 to the "perfect 4th" = -1 generator. But what about the 3rd and 5th column-vectors in the adjoint (g0 and e0, respectively)? What tunings are they? I don't get it. And what relevance to these other mappings have to Schoenberg's theory? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

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