> <http://groups.yahoo.com/group/tuning-math/message/44>
> From: "monz" <joemonz@y...>
> Date: Sun May 27, 2001 5:55 pm
> Subject: Re: Fwd: optimizing octaves in MIRACLE scale.
>
> (My quotes of Schoenberg are from the English translation
> of _Harmonielehre_ by Roy Carter, and the page numbers
> refer to that edition.)
>
>
> Schoenberg [p 23] posits the existences of two "forces", one
> pulling downward and one pulling upward around the tonic,
> which he illustrates as: F <- C -> G and likens to resistance
> against gravity. In mathematical terms, he is referring to
> the harmonic relationships of 3^-1 and 3^1, respectively.
>
>> [Schoenberg, p 24:]
>>
>> ...thus it is explained how the scale that finally emerged
>> is put together from the most important components of a
>> fundamental tone and its nearest relatives. These nearest
>> relatives are just what gives the fundamental tone stability;
>> for it represents the point of balance between their opposing
>> tendencies. This scale appears as the residue of the properties
>> of the three factors, as a vertical projection, as addition:
>
>
> Schoenberg then presents a diagram of the overtones and the
> resulting scale, which I have adaptated, adding the partial-numbers
> which relate all the overtones together as a single set:
>
> b-45
> g-36
> e-30
> d-27
> c-24
> a-20
> g-18 g-18
> f-16
> c-12 c-12
> f-8
>
>
> f c g a d e b
> 8 12 18 20 27 30 45
>


I will now lattice these pitches, using as nomenclature
for the notes my ASCII 72-EDO notation; legend:

- + ~cents alteration from 12-EDO

b # 100 [i.e., 12-EDO]
v ^ 50
< > 33&1/3
- + 16&2/3
no accidental 0 [12-EDO]



Here is a "standard triangular" 5-limit lattice
of this diatonic scale:


20--- 30--- 45
A- E- B-
/ \ / \ / \
/ \ / \ / \
8 ---12--- 18--- 27
F C G D

Look familiar? It should.



An ASCII representation of a Monzo lattice
of this scale looks like this:


D
/27
B- /
/45'-._ /
/ G
/ /18
E- /
30'-._ /
/ ' C
/ /12
A- /
20'-._ /
' F
8





>
>> [Schoenberg:]
>>
>> Adding up the overtones (omitting repetitions) we get the seven
>> tones of our scale. Here they are not yet arranged consecutively.
>> But even the scalar order can be obtained if we assume that the
>> further overtones are also in effect. And that assumption is
>> in fact not optional; we must assume the presence of the other
>> overtones. The ear could also have defined the relative pitch
>> of the tones discovered by comparing them with taut strings,
>> which of course become longer or shorter as the tone is lowered
>> or raised. But the more distant overtones were also a
>> dependable guide. Adding these we get the following:
>
>
>
> Schoenberg then extends the diagram to include the
> following overtones:
>
> fundamental partials
>
> F 2...12, 16
> C 2...11
> G 2...12
>
> (Note, therefore, that he is not systematic in his employment
> of the various partials.)
>
>
> Again, I adapt the diagram by adding partial-numbers:
>
> d-108
> c-99
> b-90
> a-81
> g-72
> f-66
> f-64
> (f-63)
> e-60
> d-54 d-54
> c-48 c-48
> b-45
> b-44
> (bb-42)
> a-40
> g-36 g-36 g-36
> f-32
> e-30
> (eb-28)
> d-27
> c-24 c-24
> a-20
> g-18 g-18
> f-16
> c-12 c-12
> f-8
>
>
> (eb) (bb)
> c d e f g a b c d e f g a b c d
> [44] [64]
> (28) (42) [66]
> 24 27 30 32 36 40 45 48 54 60 63 72 81 90 99 108
>
>
> (Note also that Schoenberg was unsystematic in his naming
> of the nearly-1/4-tone 11th partials, calling 11th/F by the
> higher of its nearest 12-EDO relatives, "b", while calling
> 11th/C and 11th/G by the lower, "f" and "c" respectively.
> This, ironically, is the reverse of the actual proximity
> of these overtones to 12-EDO: ~10.49362941, ~5.513179424,
> and ~0.532729432 Semitones, respectively).
>
>
> The partial-numbers are also given for the resulting scale
> at the bottom of the diagram, showing that 7th/F (= eb-28)
> is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker
> than 5th/G (= b-45).
>
> Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C
> (= f-66) are all weaker still, thus I have included them in
> square brackets. These overtones are not even mentioned by
> Schoenberg.


Here is a triangular 5-limit lattice of this expanded
scale, showing 7- and 11-limit ratios in () and [] which
are near in pitch to the 5-limit ones as physically close
to them on the diagram. * indicates notes which are not
exact 8ve-equivalents even tho Schoenberg implied that
they are.


*81*----- ------90
40------60--(42)[44]45
20--(28)30------
A- E- B-
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
63[64][66]-*99*------72-----108
32------48-------36------54
16------24-------18------27
8------12------- ------
F C G D


And here is an ASCII-fied 11-limit Monzo lattice
showing where all of these pitches actually fall
according to my lattice formula, when prime-factored:


C^
/ \99
/ \
/ \ A
F^ \ /81
/ \[66] \ /
/ \ \ /
/ \ \ D
Bb^ \ \ /27 54 108
\[44] \ B- \ /
\ \/90-._ \ /
\ /\45 ' G ---------------F<
\ / \ /18 36 72 /63
\ E- \ / /
\/60-._ \ / /
/\30 ' C ---------------Bb<
/ \ /12 24 48 /(42)
A- \ / /
40'-._ \ / /
20 ' F ---------------Eb<
8 16 32 [64] (28)



Of course, this description of the scale is only valid
where "C" is the "tonic". Given Schoenberg's ideas about
"pantonality", it will be difficult if not impossible
to determine what the "tonic" is. Perhaps an interval
analysis of the composition can reveal something. Of
course, the tonic will be dynamically shifting all the
time.

Also, this explanation was intended to describe the
rational implications of the *diatonic* scale, allowing
only a few of the chromatic pitches. For the full
chromatic scale, Schoenberg's later 13-limit system
presented in "Problems of Harmony" must be consulted.

Could anyone out there do some periodicity-block
calculations on this theory and say something about that?
Altho I analyze this tuning exactly according to Schoenberg's
analysis (i.e., 8ve-specific), he himself considered it
to represent 8ve-invariant pitches. His unison-vectors
are 33:32, 45:44 64:63, and 81:80. His unusual explanation
of Eb< (28) and Bb< (42) also makes 16:15 some type of
special interval, if not a unison-vector.




-monz
http://www.monz.org
"All roads lead to n^0"





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