RE: Tuning of decatonic modes

>}I've worked out that Paul Erlich's Pentachordal modes can be
>}written as 10 JI notes plus three commas of 50/49. This
>}interval is larger than a syntonic comma, so some tempering is
>}advisable, but it should be small enough for the scales to
>}function as 10 pitch classes.
>
>In my paper, I mention that in JI, two of the ten notes of the pentachordal
>modes would each require two versions a 50:49 apart. An earlier version of
>the paper (such as the one on John Starrett's web page) had 49:48 instead of
>50:49. That was an error. However, the number two is correct. You don't need
>three commas, just two, and the two are located at scale steps a 6:5 from one
>another.
>
>The symmetrical decatonic modes, however, require four 50:49 commas, and two
>64:63 commas.
>
>
>For those confused but interested, the pentachordal scale is
>
>s s s L s s s L s s
>
>and the symmetrical scale is
>
>s s s s L s s s s L
>
>and the template (1,4,7,9) is supposed to produce complete 7-limit tetrads on
>most scale degrees, just like, in the diatonic scale, the pattern (1,3,5)
>produces complete 5-limit triads on most scale degrees. Unlike Partch, I
>think the most stable root of these chords is the lower note of the 3:2
>interval found within them, and this root will always be the (1) in these
>templates.
>
>}As 10 fixed pitches, the comma of 50/49 means that the tritones
>}10/7 and 7/5 must be equal. With just octaves, they must both
>}be an exact half-octave. As these are the worst 7-limit
>}intervals in 22-equal, that means there is no better way of
>}tuning decatonic modes unless you allow more than 10 notes.
>
>Yuppers. This is all in the paper.


SMTPOriginator: tuning@eartha.mills.edu
From: "Paul H. Erlich"
Subject: Re: 60 vs 45 / GUITAR
PostedDate: 29-01-98 20:35:33
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