XH 17: David Finnamore's response

> That's helped me make more sense of a set of 7-(prime)limit,
> 12-tone/2:1 tunings, the heptatonic subsets of which I've been
> exploring for use with what might be thought of as
> neo-/Xeno-Medieval/Gothic/Rennaissance compositions (for lack of
> more concise term :-). Those tunings are built of five alternating
> 21:20s and 15:14s, with two 256:243s sandwiched in here and there to
> fill out the octave. Note that 9:8 can be divided into 21:20 and
> 15:14.

Hello, there, and I guess that such a tuning might look like this:

15:14 189:160 10:7 45:28 567:320
119.44 288.38 617.49 821.40 990.33
c#' eb' f#' g#' bb'
_119.4|84.5_84.5|119.4_ _119.4|84.5_119.4|84.5_84.5|119.4_
c' d' e' f' g' a' b' c''
1:1 9:8 81:64 4:3 3:2 27:16 243:128 2:1
0 203.91 407.82 498.04 701.96 905.87 1109.78 1200
203.91 203.91 90.22 203.91 203.91 203.91 90.22

Here are some quick comments, hoping that the above interpretation of
your scheme is (at least mostly) right.

The idea of leaving the Pythagorean whole-tones (9:8) and diatonic
semitones at e-f and b-c' (256:243) unaltered, but changing the
divisions of the whole-tones into semitones, occurs in some medieval
and Renaissance sources.

However, these schemes tend either to minimize the contrast between
the usual Pythagorean diatonic and chromatic semitones -- around 90.22
cents and 113.69 cents -- or indeed to make them precisely equal. A
14th-century organ-pipe plan dividing the tone by pipe ratios of
18:17:16 (roughly 99.95 cents-104.96 cents), and the scheme of
Henricus Grammateus (1518) to achieve a true equal division of the 9:8
tone by Euclidean methods (about 101.955 cents) are examples.

In contrast, your scheme slightly accentuates the difference between
the diatonic and chromatic semitones, respectively 21:20 (about 84.5
cents) and 15:14 (about 119.4 cents). Apart from adding a bit of extra
contrast, this tuning has the advantage of not drastically
compromising the fifths Bb-F and B-F#' in the way that either the
18:17:16 or Grammateus tunings do, making one or both intervals more
than 10 cents from a just 3:2. The actual compromise of 5.76 cents or
so, a bit less than 1/4 Pythagorean comma, should be much less
disadvantageous.

Your arrangement has the interesting property of producing augmented
fourths and diminished fifths of precisely 10:7 and 7:5.

Another consequence, quite consonant (pun intended) with the desire of
Marchettus of Padua (1318) for _very_ wide cadential major thirds and
sixths (and narrow cadential semitones), is to make these intervals
about 5.76 cents larger than usual Pythagorean size when they involve
accidentals (e.g. e-g#, eb-c'). In these cadences M6 is 320:189, or
about 911.62 cents, while M3 is 80:63, or about 413.58 cents.

These last intervals, interestingly, vary from 5:3 and 5:4 by a
septimal comma (64:63), a neat 7-limit connection. Again, the tuning
stays fairly close to Pythagorean but tends to give it a bit of an
extra edge.

Having shared these first impressions, please let me return to some of
your other comments.

> Is it accepted practice to refer to primes as "complex"? Aren't
> primes, by definition, the simplest whole numbers? Perhaps "highest
> prime number" would be a better way to state it?

This sounds like an excellent point: "highest" or "greatest" or
"largest" prime indeed seems clearer to me.

>> the septimal minor seventh (7:4) also becomes literally and
>> figuratively a "prime factor" in defining the scale,

> Figuratively? How do you mean?

What I meant was a kind of play on words: in such a tuning, 7:4 is a
"prime factor" not only in the mathematical sense of 7 being a prime,
but in the figurative sense of being a "principal" or "very important"
factor in defining the scale and some of the possible sonorities.

> David J. Finnamore

Thank you again both for your helpful suggestions on my review of
articles by Paul Erlich and Brian McLaren in XH 17, and for your
intriguing scale -- hoping I got it mostly right.

Most appreciatively,

Margo Schulter
mschulter@value.net

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End of TUNING Digest 1521
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