RE: 22-tone equal & chromaticism

>}Gregg Gibson said:
>}> >}but 22-tone
>}> >}equal is not a temperament at all, but a mere tuning artefact that
>}> >}reproduces the worst defects of just intonation.
>
>}Paul Ehrlich replied;
>}> >But these "defects" (the non-vanishing of the syntonic comma) can only be
>}> >seen as such in the context of diatonicism! Since we are speaking of
>escaping
>}> >diatonicism, the relevant properties of a tuning system will be quite
>}> >different.
>
>}The failure of 22-tone equal to provide both consonant thirds and fifths
>}has nothing to do with diatonicism; the presence of consonant melody and
>}harmony is no less vital for the use of chromatic modes than for the use
>}of diatonic ones; I assume you will agree if you consider the matter. If
>}anything, because chromatic modes have but four (at most), not six
>}degrees (as in the diatonic modes) on which consonant triads are
>}erigible, chromatic modes in 22-tone equal would be even more
>}disorganized than diatonic ones. For enharmonic modes the matter is less
>}clear-cut, I concede.
>
>OK folks, switch to a constant-width font -- it's ASCII diagram time!
>
>We will represent the 3:2 ratio ("perfect fifths") by lines running left to
>right.
>We will represent the 5:4 ratio ("major thirds") by lines running from lower
>left to upper right.
>We will represent the 6:5 ratio ("minor thirds") by lines running from upper
>left to lower right.
>
>Consonant triads will therefore be represented by isosceles triangles. Major
>triads have the base of the triangle on the bottom; minor triads have the
>base on top.
>
>First of all, let's look at the diatonic scale -- the white keys on a piano.
>It has six consonant triads:
>
>
>D---------A---------E---------B
> \ / \ / \ / \
> \ / \ / \ / \
> \ / \ / \ / \
> \ / \ / \ / \
> F---------C---------G---------D
>
>Notice that "D" occurs twice.
>
>In 19-equal, the 3:2 is 11 steps, the 5:4 is 6 steps, and the 6:5 is 5 steps.
>Setting the leftmost D=0, the diatonic scale is then
>
>0---------11--------3---------14
> \ / \ / \ / \
> \ / \ / \ / \
> \ / \ / \ / \
> \ / \ / \ / \
> 5---------16--------8---------0
>
>Both instances of "D" are represented by "0." The diatonic scale "works" in
>19-equal.
>
>In 22-equal, the 3:2 is 13 steps, the 5:4 is 7 steps, and the 6:5 is 6 steps.
>Again setting the leftmost D=0, the diatonic scale becomes
>
>0---------13--------4---------17
> \ / \ / \ / \
> \ / \ / \ / \
> \ / \ / \ / \
> \ / \ / \ / \
> 6---------19--------10--------1
>
>"D" is now represented by both "0" and "1". As most classical composers made
>use of the fact that "D" or its equivalent in keys other that C major is one
>and only one pitch, 22-equal is unusable for most classical music. Other
>tunings unusable for most classical music are 15, 27, 34, 41, 53TET, and JI
>-- these are the tunings where the syntonic comma (representing 81:80) does
>not vanish, causing two different "D"s to arise at a very small interval from
>one another. Tunings that are usable at least for the early stuff are 12, 19,
>26, 31, 43, 50, and 55TET, as well as meantone tunings such as Kornerup's
>Golden Tuning, LucyTuning, 1/4-comma meantone, 2/7-comma meantone, 3/14-comma
>meantone, etc.
>
>Now let's look at Gregg Gibson's four "new" scales. They each contain four
>consonant triads:
>
>D melodic minor ascending:
>
> C#
> / \
> / \
> / \
> / \
>A---------E---------B
> \ / \
> \ / \
> \ / \
> \ / \
> G---------D---------A
> \ /
> \ /
> \ /
> \ /
> F
>
>This scale has the same step sizes as the diatonic scale. Gregg has made a
>big deal about 12-equal not really representing the "chromatic" scales
>adequately, since the minor third is conflated with the augmented second in
>12-equal. This scale has no augmented second or any other non-diatonic steps,
>so I presume the term "chromatic" applies to the other three:
>
>A harmonic minor:
>
> G#
> / \
> / \
> / \
> / \
>D---------A---------E---------B
> \ / \ /
> \ / \ /
> \ / \ /
> \ / \ /
> F---------C
>
>
>C harmonic major:
>
> E---------B
> / \ / \
> / \ / \
> / \ / \
> / \ / \
>F---------C---------G---------D
> \ /
> \ /
> \ /
> \ /
> Ab
>
>E major Gypsy:
>
> G#--------D#
> / \ /
> / \ /
> / \ /
> / \ /
> A---------E---------B
> / \ /
> / \ /
> / \ /
> / \ /
>F---------C
>
>The reader is encouraged to work out these diagrams in 19- and 22-equal. Note
>that in these last three scales, there are no notes that appear twice in the
>diagram, and thus no potential for the scales not to "work" in any tuning
>that has good 5-limit approximations, including 22-equal. The harmonic major
>and harmonic minor scales have four step sizes in 22-equal but three in 12-
>and 19-equal; if any tuning is avoiding "conflations" within these scales it
>is 22-equal, though I prefer the greater uniformity of step sizes in 12- and
>19-equal. For the last of these scales, the number of different step sizes is
>the same in all three tunings (12, 19, 22) so it is really, really hard to
>claim that it "works" in some but not all of these tunings.
>
>My interest in 22-equal has nothing to do with these scales. I use these
>scales all the time in 12-equal (the last one is great for surf music) and
>see no great advantage to justify the expense of moving to 19-equal (though
>I'd be the first to pick up a 19-tone guitar, keyboard, and bass if they were
>commercially available). In fact, two of my favorite scales do not work in
>19-equal at all:
>
>C augmented (six consonant triads):
>
> G#--------D#
> / \ /
> / \ /
> / \ /
> / \ /
> E---------B
> / \ /
> / \ /
> / \ /
> / \ /
> C---------G
> / \ /
> / \ /
> / \ /
> / \ /
>Ab--------Eb
>
>This is like the diatonic scale turned on its side. In 12-equal Ab and G# are
>the same and Eb and D# are the same. In almost any other tuning, this scale
>doesn't work; you get eight instead of six tones and some really small
>intervals. 27-equal is a rare counterexample, but 27-equal does not support
>the diatonic scale.
>
>C diminished (eight consonant triads):
>
>F#--------C#
> \ / \
> \ / \
> \ / \
> \ / \
> A---------E
> \ / \
> \ / \
> \ / \
> \ / \
> C---------G
> \ / \
> \ / \
> \ / \
> \ / \
> Eb--------Bb
> \ / \
> \ / \
> \ / \
> \ / \
> Gb--------Db
>
>This scale doesn't work unless F#=Gb and C#=Db, i.e., in almost any tuning
>other than 12-equal. 28-equal is a rare counterexample which doesn't,
>however, support the diatonic scale.
>
>So far all the scales discussed are well-represented in 12-equal. No other
>tuning can represent them all, expect maybe one with a vast number of notes.
>However, my interest was to see if one could expand consonant harmony to
>include the 7-limit (Partch seemed confident that at least the 11-limit could
>possess some measure of consonance) while maintaining a diatonic-like
>structure. This entails throwing out the diatonic scale and all concepts
>associated with it, such as classifying the consonances as "major thirds",
>"perfect fifths", etc. -- the 3:2 is now a "perfect seventh" and the 5:4 and
>6:5 are major and minor "fourths."
>
>To add the 7-limit to these diagrams, a note in the center of a major triad
>will signify a 7 added to the 4:5:6 major triad to yield a 4:5:6:7 otonal
>tetrad, and a note in the center of a minor triad will signify a 1/7 added to
>the 1/6:1/5:1/4 minor triad to yield a 1/7:1/6:1/5:1/4 utonal tetrad. In
>22-equal, the 7:4 is 18 steps, the 7:5 is 11 steps, and the 7:6 is 5 steps.
>Here are the two decatonic scales in 22-equal:
>
>Symmetrical decatonic scale (8 consonant tetrads):
>
> 13--------4---------17
> / \ / \ /
> / \ 8 / \ 21 /
> / 2---------15--------6
> / / \ \ / / \ \ / /
>6--/---\--19-/---\--10 /
> / 13 \ / 4 \ /
> / \ / \ /
>17--------8---------21
>
>
>Pentachordal decatonic scale (6 consonant tetrads):
>
> 2
> / \
> / \
>0---------13--------4
> \ / / \ \ / \
> \ 17-/---\--8 / \
> \ / 2---------15 \
> \ / \ \ / / \
> 6------\--19-/------10
> \ /
> \ /
> 8
>
>These scales have 10 tones and 2 step sizes, and many, many important
>properties in common with the diatonic scale (see my paper). In most any
>other tuning with good 7-limit approximations, such as 27-, 31-equal, and JI,
>there will be more than 10 tones and more than 2 step sizes, including some
>really small steps. These steps represent what are known in JI as the
>septimal sixth-tone (49:48) and, in the case of the symmetrical decatonic
>scale, the septimal comma (64:63). That these intervals vanish in 22-equal is
>as essential for the functioning of the decatonic scales as the vanishing of
>the syntonic comma is for the functioning of the diatonic scale.
>
>In 26-equal the septimal comma does not vanish but the septimal sixth-tone
>does. So the pentachordal decatonic scale does have 10 tones:
>
> 2
> / \
> / \
>0---------15--------4
> \ / / \ \ / \
> \ 20-/---\--9 / \
> \ / 2---------17 \
> \ / \ \ / / \
> 7------\--22-/------11
> \ /
> \ /
> 9
>
>However, there are three step sizes instead of two. Besides making it
>melodically awkward, this fact makes modulation difficult. An interval which
>is a "second" in one key can become a "third" in a neigboring key (3:2 away)
>-- which is disturbing when one considers that the "thirds" are supposed to
>represent consonances: a major "third" is 7:6, and a minor "third" is 8:7.
>The scalar grammar falls apart. So even for the pentachordal decatonic scale,
>both the septimal sixth-tone and the septimal comma must vanish. For neither
>of the decatonic scales must the syntonic comma vanish, however.


SMTPOriginator: tuning@eartha.mills.edu
From: "Paul H. Erlich"
Subject: Error in "Stretching the 19-tone Equal"
PostedDate: 19-12-97 19:08:15
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