Re: RE: The world according to Rothenberg (re Kraig Grady); my contri butions to scale theory

"Paul H. Erlich" wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Carl Lumma wrote,
>
> >> >> Proper scales are not necessarily MOS, and MOS scales are not
> necessarily
> >> >> proper (take the recent 11-tone chain-of-minor-thirds scale, for
> >>>> example).
>
> Kraig Grady wrote
>
> >> >I believe it is an MOS!
>
> I wrote,
>
> >> Yes it is, but it's not proper.
>
> Kraig wrote,
>
> >then I am confused about what is Proper still. Why isn't it proper.
>
> In 19-tET, the 11-tone chain-of-minor-thirds MOS is 0 1 2 5 6 7 10 11 12 15
> 16 (19). It's improper because the largest "second" (3/19 oct.: 2 to 5, 7 to
> 10, 12 to 15, 16 to 19) is larger than the smallest "third" (2/19 oct.: 0 to
> 2, 5 to 7, 10 to 12).

Thanks for clearing this up for me! Us MOSers tend to be attracted to those
points right before the hub of the next zig or zag. So in 19 et we would
gravitate to the 15 tone subset. Hanson, Wilson and Haverstick I believed
relished in this one.

>
>
> One reason propriety may be important in the diatonic scale (in any meantone
> tuning including 12-tET) is that a triad formed by stacking two thirds is a
> consonant 5-limit triad in 6 out of 7 positions. That these triads form the
> harmonic basis of common-practice diatonic music may rely on the fact that
> thirds can be melodically "measured" without any ambiguity. Evidence of this
> effect is provided by the fact that, in 12-tET, the augmented second in the
> harmonic minor scale sounds dissonant even though is acoustically identical
> to the consonant minor thirds in the scale.

I like it!

>
>
> My two decatonic scales in 22-tET are examples of proper scales that are not
> MOS. One of the scales, 0 2 4 6 9 11 13 15 17 20 (22), is distributionally
> even, and the other, 0 2 4 7 9 11 13 16 18 20 (22) is "pentachordal" (kind
> of like Chalmers' "tetrachordal" but with more notes) in every octave
> species. Propriety may be important here because the tetrad constructed by
> stacking two "fourths" and a "third" is a consonant 7-limit tetrad in most
> positions. If one wishes to project a full 7-limit harmonic style that
> originates from melody, this aspect of propriety may be necessary.

Thanks! Examples work better than anything. This example makes Lumma original
statement very clear.

>
>
> Carl Lumma just visited me for a few days and he agreed with my feeling of
> amazement that no one had come up with the decatonic scales before me. The
> task of combining three harmonic functions (1,3,5 and 7) into a single,
> melodically coherent scale may have seemed impossible because, from the MOS
> standpoint, no single generator can create all these functions within a
> small number of notes. The trick to the decatonic scales is that each is
> composed of _two_ interleaved MOS pentatonic scales, _each_ of which is
> generated by a sharpened fifth and contains the harmonic functions 1, 3, and
> 7 in two otonal and two utonal formations. These are not unlike the slendro
> scales some JI composers have formulated (but temperament allows a greater
> quantity of consonant 1:3:7 triads). The 5-function that completes all four
> (in the DE case) or three (in the pentachordal case) of these formations in
> one pentatonic is then found in the other pentatonic, for a total of eight
> or six consonant tetrads, respectively.
>
> A similar trick can be used with two interleaved meantone diatonic scales.
> Each contains harmonic functions 1, 3, and 5 in three otonal and three
> utonal formations. If each is generated by a sufficiently _flattened_ fifth
> (such as that of 19-tET or 26-tET), then the 7-function that completes all
> six (in the DE case) or five (in the "heptachordal" case) of these
> formations in one diatonic is found in the other diatonic, for a total of
> twelve or ten consonant tetrads, respectively. The resulting 14-tone scales
> may be too complex, however, for propriety to work its magic on our ears
> (for one thing, intervals that would fall in the same equivalence class in
> the over-familiar diatonic scale are in different equivalence classes in
> this scale -- while the "extra" intervals in the decatonic scale tend to
> fall in the gaps between the conventional diatonic equivalence classes).

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com