Tuning-Math Archive Section 9: 8725

Message: 8725 - Contents - Hide Contents Date: Mon, 08 Dec 2003 20:56:48 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> It's *way* preferable. The latter is based on a false view of octave- > reducing the tenney lattice, at best. Do you think 5:3 and 15:8 > should count as equally 'distant' octave-equivalence classes from > 1:1? What I was asking about is supported by Partch, octave- > equivalent harmonic entropy, and pretty straighforward explanations I > posted for Maximiliano on the tuning list . .

Is the measure in question one which involves removing all factors of two, reducing to lowest form p/q, and taking max(p,q)?

Message: 8726 - Contents - Hide Contents Date: Mon, 08 Dec 2003 21:00:17 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

>> It's *way* preferable. The latter is based on a false view of > octave-

>> reducing the tenney lattice, at best. Do you think 5:3 and 15:8 >> should count as equally 'distant' octave-equivalence classes from >> 1:1? What I was asking about is supported by Partch, octave- >> equivalent harmonic entropy, and pretty straighforward explanations > I

>> posted for Maximiliano on the tuning list . . >

> Is the measure in question one which involves removing all factors of > two, reducing to lowest form p/q, and taking max(p,q)? yes.

Message: 8728 - Contents - Hide Contents Date: Mon, 08 Dec 2003 21:14:10 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

>> Is the measure in question one which involves removing all factors > of

>> two, reducing to lowest form p/q, and taking max(p,q)? > > yes.

Another possibility would be a variant on the Euclidean reduced scales I did once--minimal geometric complexity.

Message: 8729 - Contents - Hide Contents Date: Mon, 08 Dec 2003 21:35:03 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

>>> Is the measure in question one which involves removing all > factors >> of

>>> two, reducing to lowest form p/q, and taking max(p,q)? >> >> yes. >

> Another possibility would be a variant on the Euclidean reduced > scales I did once--minimal geometric complexity.

Do you want to try answering the question before changing it?

Message: 8730 - Contents - Hide Contents Date: Mon, 08 Dec 2003 23:24:44 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

>> Another possibility would be a variant on the Euclidean reduced >> scales I did once--minimal geometric complexity. >

> Do you want to try answering the question before changing it?

Har har har--little do you know, my friend! I first Tenney reduced in the range -35 to 36 steps, and shifted anything less than 1 up an octave. I then reduced this scale for minimum geometric complexity. Finally reduced it your way. The result in all three cases turns out to be *exactly the same*! Take that. :) I think under the circumstances I am justified in calling this the canonical 11-limit reduced 72-epimorphic JI scale, in whatever order you prefer those words in. ! red72_11.scl Canonical 11-limit reduced scale 72 ! 81/80 45/44 33/32 25/24 21/20 35/33 77/72 175/162 35/32 54/49 49/44 55/49 198/175 8/7 81/70 64/55 33/28 25/21 6/5 40/33 11/9 99/80 5/4 63/50 14/11 77/60 35/27 21/16 175/132 147/110 66/49 49/36 48/35 242/175 88/63 99/70 63/44 175/121 35/24 72/49 49/33 220/147 264/175 32/21 54/35 120/77 11/7 100/63 8/5 160/99 18/11 33/20 5/3 42/25 56/33 55/32 140/81 7/4 175/99 98/55 88/49 49/27 64/35 324/175 144/77 66/35 40/21 48/25 64/33 88/45 160/81 2/1

Message: 8731 - Contents - Hide Contents Date: Mon, 08 Dec 2003 23:38:41 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> I think under the circumstances I am justified in calling this the > canonical 11-limit reduced 72-epimorphic JI scale, in whatever order > you prefer those words in.

I see however it doesn't contain either a 9/8 or a 10/9, so I'd better check to see if I've totally goofed again.

Message: 8732 - Contents - Hide Contents Date: Tue, 9 Dec 2003 16:27:01 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul Here's an attempt, but it's possible that Gene finds a still better one. It's also strictly proper. 0: 1/1 0.000 unison, perfect prime 1: 126/125 13.795 small septimal comma 2: 45/44 38.906 1/5-tone 3: 33/32 53.273 undecimal comma, 33rd harmonic 4: 25/24 70.672 classic chromatic semitone, minor chroma 5: 21/20 84.467 minor semitone 6: 35/33 101.867 7: 77/72 116.234 8: 27/25 133.238 large limma, BP small semitone 9: 49/45 147.428 BP minor semitone 10: 11/10 165.004 4/5-tone, Ptolemy's second 11: 125/112 190.115 classic augmented semitone 12: 9/8 203.910 major whole tone 13: 25/22 221.309 14: 55/48 235.677 15: 81/70 252.680 Al-Hwarizmi's lute middle finger 16: 7/6 266.871 septimal minor third 17: 33/28 284.447 undecimal minor third 18: 25/21 301.847 BP second, quasi-tempered minor third 19: 6/5 315.641 minor third 20: 175/144 337.543 21: 27/22 354.547 neutral third, Zalzal wosta of al-Farabi 22: 99/80 368.914 23: 5/4 386.314 major third 24: 63/50 400.108 quasi-equal major third 25: 14/11 417.508 undecimal diminished fourth or major third 26: 77/60 431.875 27: 35/27 449.275 9/4-tone, septimal semi-diminished fourth 28: 98/75 463.069 29: 175/132 488.180 30: 75/56 505.757 31: 27/20 519.551 acute fourth 32: 15/11 536.951 undecimal augmented fourth 33: 11/8 551.318 undecimal semi-augmented fourth 34: 25/18 568.717 classic augmented fourth 35: 7/5 582.512 septimal or Huygens' tritone, BP fourth 36: 99/70 600.088 2nd quasi-equal tritone 37: 10/7 617.488 Euler's tritone 38: 81/56 638.994 39: 35/24 653.185 septimal semi-diminished fifth 40: 81/55 670.188 41: 49/33 684.379 42: 3/2 701.955 perfect fifth 43: 121/80 716.322 44: 55/36 733.722 45: 77/50 747.516 46: 14/9 764.916 septimal minor sixth 47: 243/154 789.631 48: 35/22 803.822 49: 45/28 821.398 50: 81/50 835.193 acute minor sixth 51: 18/11 852.592 undecimal neutral sixth 52: 33/20 866.959 53: 5/3 884.359 major sixth, BP sixth 54: 42/25 898.153 quasi-tempered major sixth 55: 56/33 915.553 56: 55/32 937.632 57: 210/121 954.459 58: 7/4 968.826 harmonic seventh 59: 99/56 986.402 60: 98/55 1000.020 61: 9/5 1017.596 just minor seventh, BP seventh 62: 49/27 1031.787 63: 11/6 1049.363 21/4-tone, undecimal neutral seventh 64: 231/125 1063.158 65: 15/8 1088.269 classic major seventh 66: 125/66 1105.668 67: 21/11 1119.463 68: 27/14 1137.039 septimal major seventh 69: 35/18 1151.230 septimal semi-diminished octave 70: 55/28 1168.806 71: 99/50 1182.601 72: 2/1 1200.000 octave Manuel

Message: 8733 - Contents - Hide Contents Date: Tue, 09 Dec 2003 16:32:20 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> > George Secor's paper includes a big 72-equal keyboard diagram. It's > marked with ratios, and I don't like them :)

So you don't like ratios, eh? So why did you ever join tuning- math? ;-) --George

Message: 8735 - Contents - Hide Contents Date: Tue, 09 Dec 2003 17:15:30 Subject: Re: TnI types From: monz hi Jon, --- In tuning-math@xxxxxxxxxxx.xxxx jon wild <wild@f...> wrote:

> And invoking set-classes doesn't mean you're suddenly > not allowed to additionally distinguish the L and R versions > of a chord (I call them "left-handed" and "right-handed", > because I find "prime" and "inverted" a bit obnoxious, > as if one's more important than the other).

bravo! that's one of best refinements of music-theory terminology that i've ever seen! -monz

Message: 8736 - Contents - Hide Contents Date: Tue, 09 Dec 2003 17:25:48 Subject: Re: Enumerating pitch class sets algebraically From: monz hi Dante and paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dante Rosati" <dante@i...> wrote: >>>>

>>>> As far as tonal theory being a science, you only >>>> have to look at or try to analyze some Brahms passages, >>>> or Wagner et al to see that it is far from >>>> being so (IMO). >>>

>>> It won't be any more scientific simply to look at sets >>> of equivalences class when analyzing Brahms, will it? >>> Or are you saying Brahms wrote unscientific music? >>

>> No, I'm saying Brahms wrote music that, at times, >> exhibits ambiguity when subjected to traditional harmonic >> analysis. And no, I'm not saying Fortean analysis will >> tell you anything here. My point was that the ambiguity >> demonstrates that harmonic analysis is more of an art >> than a science. >> >> Dante >

> Or it might just demonstrate that Bramhs's music exhibits > ambiguity -- maybe because he wanted it to! Anyway, I don't > think any of these modalities of musical analysis are > anywhere near a "science", but certainly ambiguity is > something that can be understood, described, and predicted > in a scientific way. For example, the pitch of an inharmonic > spectrum, as I've been discussing with Kurt on the tuning > list lately.

i think the main reason harmonic analysis would be characterised as an "art" is precisely *because* of the ambiguity available to a composer like Brahms, whether his intended tuning is 12edo or a meantone (the only two likely possibilities for Brahms IMO). my point: that *temperament* allows composers to play the kinds of games ("punning") that aren't possible in JI. and of course JI is the tuning which offers the straightforward "scientific" approach to harmonic analysis. in reality, JI harmonic analysis is merely a lot simpler than those cases in which temperament must considered. but harmonic analysis of tempered music can indeed be put on a similarly scientific basis, which i think is mostly what's going on on this list every day. ... and yes, for the JI enthusiasts out there: it is certainly possible to compose harmonically ambiguous music in JI as well. as with temperaments, the business of periodicity-blocks and unison-vectors tells the story. but in contrast to temperaments, JI music like this *exposes* the tiny discrepancies rather than eliminates them. -monz

Message: 8738 - Contents - Hide Contents Date: Tue, 09 Dec 2003 20:16:18 Subject: Re: Enumerating pitch class sets algebraicallyy From: monz hi Carl and Dante, --- In tuning-math@xxxxxxxxxxx.xxxx "Dante Rosati" <dante@i...> wrote:

> Hi Carl: >

>> () Was it [the term "pitch class"] started/coined by Babbitt? >

> Forte attributes the term and concept of pc sets to Babbitt. > I'm not sure how much else he came up with prior to Forte's > systematization in "The Structure of Atonal Music". There's > also Lewin's "Musical Intervals and Transformations"

the term "pitch-class" was coined by Babbitt in his 1960 paper "Twelve-Tone Invariants as Compositional Determinants" (The Musical Quarterly, vol 46, p 247). thanks for asking about it, Carl ... it prompted me to add a citation to Babbitt in my definition: Definitions of tuning terms: pitch-class, (c) ... * [with cont.] (Wayb.)

>> () What's the best piece for a beginner to start with, and what >> should he listen for? >

> Not sure what you mean here. It starts with Schoenberg > and Webern, developed by Boulez and Babbitt Carter etc, > and then carried on by multitudes of others in various > approaches. I love Boulez's radical "total serialization" > works from the 50s- Piano Sonata 2 and the two books of > Structures for two pianos, Marteau Sans Maitre, etc. He's > still going strong- recent works include Sur Incise & > Explosante fixe, though I'm not sure how serial or pc > set his process is these days. Babbitt and Carter I can > do without for the most part, although I'm open to learning > to hear what they're doing some day. I suppose Babbitt is > the ultimate example of this type of thinking applied to > composition.

Dante is right, Carl ... start at the beginning, with Schoenberg's _Suite for Piano, op. 25_ (1921-23), the first dodecaphonic serial piece. Arnold Schoenberg * [with cont.] (Wayb.) the introduction of serialism also ushered in Schoenberg's neoclassical style, which i never really cared for. the pieces he composed from then until his "temporary death" in 1946 are the ones of his which i like least. but then, *after* he was revived ... i'd say that by far the best introductions to Schoenberg's serial work are his two late compositions _String Trio_ (1946) and _A Survivor From Warsaw_ (1947). both of those should knock your socks off. (i've raved about both of them many times on these lists.) here's a good "list of works" with links to recordings, of Schoenberg's work: Arnold Schoenberg Recordings: List of Works * [with cont.] (Wayb.) then follow the development thru Berg and Webern. Berg's _Lyric Suite_ for string quartet (1925-26) was the first piece in which he used serialism (only in half of the six movements), and of course his opera _Lulu_ (1929-35, left unfinished at his death) and _Violin Concerto_ (1935) are masterpieces. (and of course there's also the non-serial opera _Wozzeck_, which you shouldn't miss if you're going to delve into this repertoire.) *see* the operas if possible. you can't get a proper understanding of Webern's life-work if you don't give enough attention to his vocal works, which make up more than half of his total output, and which unfortunately is exactly what has happened in the literature. but that said, the volume of literature with an analytical focus on Webern's _Symphonie, op. 21_ (1927-28) and _Piano Variations, op. 27_ (1935-36) does IMO do justice to the quality of those two pieces. in addition, his _String Quartet, op. 28_ (1936-38)is a marvel. Boulez's _Le Marteau Sans Maitre_ (19654-55) is a stunning virtuosic display of total serialism, and a piece which i absolutely love, and see live every time i find a performance of it. Babbitt's work is similarly virtuosic in compositional technique, and at the same time i find it very listenable. in particular, i like his _Three Compositions for Piano_ (1947) and _2nd String Quartet_ (1954). Babbitt is a huge fan of Broadway shows, and i think that somehow that populist lyricism breaks thru his convoluted theoretical apparatus. a lot more has happened since 1955, but those are pretty much the big "classics" of serialism. a good book offering an alternative theoretical summary to Forte's is George Perle's _Serial Composition and Atonality_, published originally in 1962. Amazon.com Error Page * [with cont.] (Wayb.) /0520074300/ref=pd_sxp_f/104-5664280-6894348?v=glance&s=books OR Amazon.com: Books: Serial Composition and Aton... * [with cont.] (Wayb.)

> I believe a piece of music must work as sound first, and > not rely on familiarity with techniques to appreciate it. > If a piece is only interesting if you know how it was put > together, then it doesn't cut it for me. Certain symmetries > and transformations are audible in, e.g, Webern's Symphonie, > STring Quartet and Piano Variations due to the simplicity of > presentation. Mostly you can't really hear this stuff, > especially in dense presentations, but thats neither here > nor there- either it sounds good to you or not- thats the > bottom line. It does take alot of listening to give it > a fair chance, though. > > Dante

i find that as the years pass and i listen to and study Webern and his music, it appeals to me more and more. this is something that also continues to be true of Mahler ... altho i certainly loved his music a lot right from the very beginning. but what i find interesting is that very parallel, because Webern admired Mahler with a fanaticism just like mine. i hope that means that people will continue to admire *my* compositions more and more as they study them! ;-) -monz

Message: 8741 - Contents - Hide Contents Date: Tue, 09 Dec 2003 15:39:15 Subject: Re: Enumerating pitch class sets algebraically From: Carl Lumma

>i'd say that by far the best introductions to >Schoenberg's serial work are his two late compositions >_String Trio_ (1946) and _A Survivor From Warsaw_ (1947). >both of those should knock your socks off. (i've raved >about both of them many times on these lists.) // >Berg's _Lyric Suite_ for string quartet (1925-26) was the >first piece in which he used serialism (only in half of the >six movements), and of course his opera _Lulu_ (1929-35, >left unfinished at his death) and _Violin Concerto_ (1935) >are masterpieces. // >but that said, the volume of literature with an analytical >focus on Webern's _Symphonie, op. 21_ (1927-28) and >_Piano Variations, op. 27_ (1935-36) does IMO do justice >to the quality of those two pieces. in addition, his >_String Quartet, op. 28_ (1936-38)is a marvel. // >Boulez's _Le Marteau Sans Maitre_ (19654-55) is a stunning >virtuosic display of total serialism, and a piece which i >absolutely love, and see live every time i find a performance >of it. > >Babbitt's // _Three Compositions for Piano_ (1947) and >_2nd String Quartet_ (1954). Babbitt is a huge fan of >Broadway shows, and i think that somehow that populist >lyricism breaks thru his convoluted theoretical apparatus. Thanks monz!!! -Carl

Message: 8742 - Contents - Hide Contents Date: Tue, 09 Dec 2003 00:30:19 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

>>> Another possibility would be a variant on the Euclidean reduced >>> scales I did once--minimal geometric complexity. >>

>> Do you want to try answering the question before changing it? >

> Har har har--little do you know, my friend! > > I first Tenney reduced in the range -35 to 36 steps, and shifted > anything less than 1 up an octave. I then reduced this scale for > minimum geometric complexity. Finally reduced it your way. The result > in all three cases turns out to be *exactly the same*! > > Take that. :)

I knew Tenney would agree with "my way" if you shifted to -1/2 to 1/2 octaves. Didn't know about geo. complexity.

> > I think under the circumstances I am justified in calling this the > canonical 11-limit reduced 72-epimorphic JI scale, in whatever order > you prefer those words in. > > ! red72_11.scl > Canonical 11-limit reduced scale > 72 > ! > 81/80 > 45/44 > 33/32 > 25/24 > 21/20 > 35/33 > 77/72 > 175/162 > 35/32 > 54/49 > 49/44 > 55/49 > 198/175 > 8/7 > 81/70 > 64/55 > 33/28 > 25/21 > 6/5 > 40/33 > 11/9 > 99/80 > 5/4 > 63/50 > 14/11 > 77/60 > 35/27 > 21/16 > 175/132 > 147/110 > 66/49 > 49/36 > 48/35 > 242/175 > 88/63 > 99/70 > 63/44 > 175/121 > 35/24 > 72/49 > 49/33 > 220/147 > 264/175 > 32/21 > 54/35 > 120/77 > 11/7 > 100/63 > 8/5 > 160/99 > 18/11 > 33/20 > 5/3 > 42/25 > 56/33 > 55/32 > 140/81 > 7/4 > 175/99 > 98/55 > 88/49 > 49/27 > 64/35 > 324/175 > 144/77 > 66/35 > 40/21 > 48/25 > 64/33 > 88/45 > 160/81 > 2/1

Thanks, Gene. I appreciate it.

Message: 8743 - Contents - Hide Contents Date: Tue, 09 Dec 2003 00:31:12 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >

>> I think under the circumstances I am justified in calling this the >> canonical 11-limit reduced 72-epimorphic JI scale, in whatever > order

>> you prefer those words in. >

> I see however it doesn't contain either a 9/8 or a 10/9, so I'd > better check to see if I've totally goofed again.

Yes, it would seem so . . .

Message: 8744 - Contents - Hide Contents Date: Tue, 9 Dec 2003 12:22:58 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul What's the 11-limit TM-reduced basis of 72-tET again? Manuel

Message: 8747 - Contents - Hide Contents Date: Wed, 10 Dec 2003 16:30:04 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Manuel Op de Coul

>Its <225/224, 243/242, 385/384, 4000/3993>

I tried that one indeed, but it's very uneven. Also the triangular lattice size is larger than the scale I posted has. I wouldn't know how to assemble a PB from Kees' list. Manuel

Message: 8748 - Contents - Hide Contents Date: Wed, 10 Dec 2003 23:32:13 Subject: Re: Digest Number 862 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:

> I'm sure this seems awfully stubborn, but for me all of this still > does not reconcile the semantic discrepancy as I outlined it > originally. > > Without invoking physics, is there a way to find a better > agreement of terms? > > A

In case the two uses of the term appear to come into conflict with one another, it's best to go over the historical explanation of where both usages come from -- and then you're done. There is no real conflict anyway because one definition applies to pure number and the other to physical phenomena, so at worst you'll have to explain which you're talking about (though it's usually obvious from the context). This strategy is going to lead to far better communication than attempting to alter the accepted definitions in either set of fields - - in the latter case you're going to be misunderstood by everyone in one of the sets of fields who comes across your writing without being aware that you've altered the definitions, because your context will indicate one definition but in fact you'll be using another. If you want to spend your time re-writing the entire dictionary so that no word has multiple meanings, that's noble. But you shouldn't then *expect* the entire world to suddenly change the language they've been speaking for hundreds of years. If I have something to say that I feel is important, I say it in terms that I feel will be understood correctly by the majority of likely listeners. If I have time, I define lots of terms to avoid any possible ambiguity. But we have to pick our battles. Changing language, especially when its origin can be traced historically and its development makes perfect sense and did not involve any errors, is simply too vast a battle for me to contemplate, even if I know I'll have you on my side. That's my opinion and to me it seems like the *least* arrogant approach, even though I'm sensing you have the opposite opinion of me right now.

Message: 8749 - Contents - Hide Contents Date: Wed, 10 Dec 2003 15:48:06 Subject: Re: Question for Manuel, Gene, Kees, or whomever . . . From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" <manuel.op.de.coul@e...> wrote:

> > Here's an attempt, but it's possible that Gene finds a > still better one. It's also strictly proper. > > 0: 1/1 0.000 unison, perfect prime > 1: 126/125 13.795 small septimal comma

Obviously 81/80 is simpler already. Sorry guys I'm behind.