Clarification

Sorry, just to be totally clear:

1) A chord can be linked to its transposition by voice leading in which only one note moves if and only if the chord is generated.

2) A chord can be linked to its transposition by efficient voice leading if and only if the chord divides the octave nearly evenly.

3) A chord can be linked to its transposition by voice leading in which only one note moves, and it moves by only one scale step, if and only if the chord is maximally even but not transpositionally symmetrical (relative to that scale). (Which is, I am guessing, what "MOS" means.)

My personal feeling is that property 2 is the most significant, musically.

DT
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Dmitri Tymoczko
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--- In tuning-math@yahoogroups.com, Dmitri Tymoczko <dmitri@...> wrote:
>
> Sorry, just to be totally clear:
>
> 1) A chord can be linked to its transposition by voice leading in
> which only one note moves if and only if the chord is generated.
>
> 2) A chord can be linked to its transposition by efficient voice
> leading if and only if the chord divides the octave nearly evenly.
>
> 3) A chord can be linked to its transposition by voice leading in
> which only one note moves, and it moves by only one scale step, if
> and only if the chord is maximally even but not transpositionally
> symmetrical (relative to that scale). (Which is, I am guessing, what
> "MOS" means.)

MOS refers to scales; what MOS means is that you can produce a
(quasi)periodic scale by interating a generator and reducing it with
respect to a scale period, and then requiring that the result have two
sizes of steps. These will occur when the number of steps is the
denominator of a semiconvergent for generator/period.

> My personal feeling is that property 2 is the most significant,
musically.

Have you ever composed music when dividing the octave nearly evenly is
pushed to an extreme? I have, and it is pretty strange; the chords
ooze gelatinously. Enough of a good thing is plenty.

>Have you ever composed music when dividing the octave nearly evenly is
>pushed to an extreme? I have, and it is pretty strange; the chords
>ooze gelatinously. Enough of a good thing is plenty.

Yeah, I tried playing around with this once, as a reductio of Richard Cohn's suggestion that parsimonious voice leading would be really cool in other tuning systems. The more notes you have per octave, the *less* interesting parsimonious voice leading becomes -- at least after a certain point.

DT
--
WARNING: Princeton Email is currently very unreliable. If you need to reach me quickly, you should call me.

Dmitri Tymoczko
Assistant Professor of Music, Princeton University
Radcliffe Institute for Advanced Study
34 Concord Ave.
Cambridge, MA 02138
FAX: (617) 495 8136
http://music.princeton.edu/~dmitri

>2) A chord can be linked to its transposition by efficient voice
>leading if and only if the chord divides the octave nearly evenly.

I suppose the test is progressions involving consonant chords that
do not evenly divide the octave, like maybe 8:9:10.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >2) A chord can be linked to its transposition by efficient voice
> >leading if and only if the chord divides the octave nearly evenly.
>
> I suppose the test is progressions involving consonant chords that
> do not evenly divide the octave, like maybe 8:9:10.

There's a question of what "evenly" means. Tetrads do not divide the
octave very evenly, since the intervals range from 5/4 down to 8/7.
But they divide it evenly enough. Like triads and quintads, they have
the nice property that they are broken symmetries; <4 6 9 11| converts
tetrads into 4-et, or in other words 12edo diminished seventh chords.

> > >2) A chord can be linked to its transposition by efficient voice
> > >leading if and only if the chord divides the octave nearly evenly.
> >
> > I suppose the test is progressions involving consonant chords that
> > do not evenly divide the octave, like maybe 8:9:10.
>
> There's a question of what "evenly" means. Tetrads do not divide the
> octave very evenly, since the intervals range from 5/4 down to 8/7.
> But they divide it evenly enough. Like triads and quintads, they have
> the nice property that they are broken symmetries; <4 6 9 11|
> converts tetrads into 4-et, or in other words 12edo diminished
> seventh chords.

It's true that consonant chords will tend to divide the octave
fairly evenly, because of the critical band. But even though
chords like 8:9:10 contain intervals smaller than a critical
band in some registers, they tend to be moderately to very
consonant (depending on the timbre).

-Carl

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
> >>>2) A chord can be linked to its transposition by efficient voice >>>leading if and only if the chord divides the octave nearly evenly.
>>
>>I suppose the test is progressions involving consonant chords that
>>do not evenly divide the octave, like maybe 8:9:10.
> > > There's a question of what "evenly" means. Tetrads do not divide the
> octave very evenly, since the intervals range from 5/4 down to 8/7.
> But they divide it evenly enough. Like triads and quintads, they have
> the nice property that they are broken symmetries; <4 6 9 11| converts
> tetrads into 4-et, or in other words 12edo diminished seventh chords. A 4:5:6:7 chord isn't maximally even in 10-equal. That may be important for miracle and pajara.

Graham