Re Meantone: "Linear" is an historical term (Ellis?) contrasting with "cyclic" or closed. I agree that in general Meantone tone systems are 2-D, but under the common assumption of octave equivalence, the dimensionality may be reduced to 1. In which case, both ET's and meantone tunings are generated by one interval, the tempered fifth. Tempered systems are closed and have a finite number N of points on the line of fifths, while meantone are open and infinite. In this sense they are "linear" and open as opposed to closed and cyclic.
BTW, Fokker used the idea of defining intervals in 2 and 3-D tones spaces in a somewhat different way. He chose sets of "unison vectors," which he set to 0, in 3x5 and 3x5x7 tone spaces to define temperaments as "periodicity blocks" in the space. For example, the diesis and the syntonic comma define a repeating block of 12 pitches as the area defined by the absolute value of the cross product of the vectors describing these intervals. (The box product is used for 3 intervals in 3-D space.) I believe the concept can be extended to higher dimensions by computing the absolute values of the determinants of the square matrices whose row vectors are the defining intervals.
Paul Rapoport has also done much work in this area.
Wu"rschmidt, I believe, was the first to identify "defining intervals" for tunings and to distinguish them from "constructing intervals," the successive intervals of the JI scales corresponding to each of the ET's as defined by this method (there are many possible sets of defining intervals for any given ET and there are many possible sets of 5 limit intervals interpretable as 12-tet.)
Re Partch: HP's dual use of ratios to label scale degrees and to define functions can be somewhat confusing at first as Marion has pointed out. Partch assumed octave equivalence in deriving his theories, but did recognize that inversions have different sounds and inn his music used a variety of voicings. Hence, sometimes 1/1 appears in the middle of a chord where the voicing would indicate that the ratio should be 1/2, 2/1, 4/1 etc.
As for odd numbers (or factors) in the denominators, they determine the roots of his otonal or harmonic series chords, though his term "numerary nexus," has not caught on. They also indicate the harmonic distances from the common tonic, the 1/1 (tempered G). When HP writes 8/5 1/1 6/5, the notation is a shorthand for 8/5 10/5 12/5 a major triad on the root 8/5, a minor sixth above his 1/1 G. This notation makes the harmonic relations clear, while somewhat obscuring the melodic movement and actual voicing. Usually, he's clear enough as to the actual inversion and voicing. Otherwise, he'd have to make the register explicit and increase the number of symbols, e.g., 1/3, 2/3, 4/3, 8/3 etc. for 4/3 (roughly C) in various registers.. Such schematic spellings are not unknown in conventional music theory.
For the subharmonic (utonal) chords, the odd factors in numerator serve the same function. The Diamond diagram may clarify his meaning.
One really has to read Partch's theoretical sections as an introduction and outline, not as a detailed composition manual.
--John
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''22tET can be considered 5-dimensional because it consistently represents all ratios of 3, 5, 7, 9, and 11.''
Interestingly, Erv Wilson discovered that the 3,5,7,9,11 Eikosany cannot be mapped without conflict onto 22 tones while the 3,7,9,11,15 can - and makes a quite convenient keyboarding.
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>Interestingly, Erv Wilson discovered that the 3,5,7,9,11 Eikosany cannot >be >mapped without conflict onto 22 tones while the 3,7,9,11,15 can - and mak>es >a quite convenient keyboarding.
Is "the" 3,5,7,9,11 Eikosany 2)5 or 3)5? Same question for "the" 3,7,9,11,15.
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 9 Jul 1997 14:58 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA27302; Wed, 9 Jul 1997 14:58:39 +0200 Date: Wed, 9 Jul 1997 14:58:39 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA27169 Received: (qmail 23485 invoked from network); 9 Jul 1997 12:57:25 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 9 Jul 1997 12:57:25 -0000 Message-Id: <199707090856_MC2-1A84-4515@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu
''Is "the" 3,5,7,9,11 Eikosany 2)5 or 3)5? Same question for "the" 3,7,9,11,15.''
Mea culpa! I dropped the ''1'' from both eikosanies. I meant to compare the 3)6 (1,3,5,7,9,11) with the 3)6 (1,3,7,9,11,15)
2)5 or 3)5 will give dekanies (incidentally, I find dekanies to be the most useful subsets of Eikosanies, and often modulate a dekany by a hexany).
For those not up on their CPS (combination-product set) terminology, for a given set of 5 generating intervals (A,B,C,D,E), a dekany will have all combinations of two elements (AB, AC, AD, AE, BC, BD, BE, CD, CE, DE), while an eikosany will have all combinations of three elements from a setof six,
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On Wed, 9 Jul 1997, Daniel Wolf wrote: > (incidentally, I find dekanies to be the most >useful subsets of Eikosanies, and often modulate a dekany by a hexany).
This sounds interesting but I'm not sure what it means. How do you modulate a dekany by a hexany? Does this mean to transpose a dekany so that the transposed and untransposed dekanies have the hexany in common?
--pH http://library.wustl.edu/~manynote <*> O /\ "Foul? What the hell for?" -\-\-- o "Because you are chalking your cue with the 3-ball."
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How do you modulate a dekany by a hexany? Taken any dekany, treat one degree as 1/1. Take any hexany, treat one degree as 1/1. Now transpose (multiply) the entire dekany by each remaining pitch in the hexany. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 9 Jul 1997 20:32 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05531; Wed, 9 Jul 1997 20:33:12 +0200 Date: Wed, 9 Jul 1997 20:33:12 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA05447 Received: (qmail 17828 invoked from network); 9 Jul 1997 18:32:34 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 9 Jul 1997 18:32:34 -0000 Message-Id: <199707091428_MC2-1A85-EB31@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu
>Topologically, meantone tunings wrap the 5-limit plane onto an infinite >cylinder. I like to see ETs as having as many dimensions as odd factors >with which they are consistent. ETs consistent with the 5-limit wrap >said plane onto a torus.
A 2-D scale is linear with octave invariance, because each point will lie on a line drawn on Paul's cylinder, at the intersections with the original lattice. I generally don't like this idea of "linear temperaments" because it places too strong an emphasis on octave invariance. However, if you define a temperament as "a formula that produces a scale" that would make the difference.
Applying octave invariance would wrap 3-D pitch space into a hypercylinder. Note that there would be two straight lines connecting any two points in this hypercylinder, so this space would also imply inversional invariance. A meantone would then wrap this into a toroidal hyperprism, and an ET a hypertorus. Quite something, given that an integer CET or TET would be a finite set of points in this hypertorus, having a Hausdorff dimension of zero. Anyway, back to Earth:
>22tET can be considered 5-dimensional because >it consistently represents all ratios of 3, 5, 7, 9, and 11.
This is a 1-D scale approximating 5-D harmony in my terminology, including 2 and implying 9. Call it an ET approximating the 11-limit if you prefer. However, I would certainly not say that the scale is 5 dimensional, rather that you are mapping it to a 5-dimensional space. The number of points in the full 5-D real pitch space depends linearly upon the number of octaves you look at, hence the scale has a Hausdorff dimension of 1. I wouldn't say that adding a direction for 9 alters the dimension. A definition, then:
The dimension of a scale is the smallest number of fundamental intervals (FIs) such that every interval in the scale is a linear combination of integer multiples (phew!) of those FIs. This implies that none of the FIs can be a linear combination of integer multiples of the others. The FIs would be real numbers denoting the interval size in octaves, c*nts, or some such unit. Distinct prime numbers always constitute a set of FIs. Note the analogy reals->vectors, integers->scalars.
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