Re: XH 17: David Finnamore's response

Margo Schulter wrote:

>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

Yes, that could be one of them, though I haven't explored that possibility
yet. Interestingly, you've split the 1:1-to-9:8 and 3:2-to-27:16 in this
example 15:14 first, whereas the others are split 21:20 first (going from low
to high pitch). That would put some like intervals next to each other (eg.,
the two adjacent 21:20s on either side of the 9:8). Is there a historical
and/or logical reason for doing it that way? One "problem" with it is that it
would yeild intervals involving 7^2 in some modes, and those are pretty
difficult for the ear to digest IMHO (21^2:20^2 = 7^2*3^2:5*2^2). I've been
splitting all of them one direction in any given tuning; that keeps all
diatonic intervals in all modes down to one 7 apiece. Here's an example; left
column is the intervals between each member of the tuning, middle column is
the intervals from 1:1, right colum is cents:

1:1 1:1 0.00
21:20 21:20 84.47
15:14 9:8 203.91
21:20 189:160 288.38
15:14 81:64 407.82
256:243 4:3 498.05
21:20 7:5 582.51
15:14 3:2 701.96
21:20 63:40 786.42
15:14 27:16 905.87
21:20 567:320 990.33
15:14 243:128 1109.78
256:243 2:1 1200.00

Its sister tuning would swap all the septimal intervals in the left column. I
also have generated versions in which the 256:243s are a m3 and a m2 apart
instead of the usual P4, as above.

>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.

Ah, wonderful. I'm delighted to know that there's an actual connection to
"early" music theory.

>However, these schemes tend either to minimize the contrast between
>the usual Pythagorean diatonic and chromatic semitones [snip]
>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).

Yes, deeper colors are what I'm after. Please forgive this elementary
question: What makes one diatonic and the other chromatic?

>Apart from adding a bit of extra
>contrast, this tuning has the advantage of not drastically
>compromising the fifths Bb-F and B-F#'[snip]
>...The actual compromise of 5.76 cents or
>so, a bit less than 1/4 Pythagorean comma, should be much less
>disadvantageous.

Good point. That would be an advantage in a static tuning. Since my music is
fantastical rather than historical, I allow myself to shift the tuning center
as needed to get pure fifths, and to get seconds and thirds with colors
appropriate to the mood of the piece. Now if only I can find a real Celtic
harp player who's willing to retune for every piece! :-)

Thanks, Margo! You've done an excellent job analyzing the idea behind my
"new" tunings, and given me some more insight into their appropriateness and
usefulness.

Since you seem interested in these ideas, here's a related 11-limit tuning
concept that uses all 4 primes: Take a cycle of 3:2s (4 or so) and split them
each into 7:6 and 9:7; then split each 7:6 into 11:10 and 35:33, and each 9:7
into 8:7 and 9:8; then octave-reduce. Eg.,

Ratio Cents
1:1 0.00
11:10 165.00
9:8 203.91
7:6 266.87
99:80 368.91
81:64 407.82
21:16 470.78
4:3 498.05
3:2 701.96
33:20 866.96
27:16 905.87
7:4 968.83
297:160 1070.87
63:32 1172.74
2:1 1200.00


David J. Finnamore
Just tune it! (It only makes cents.)

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End of TUNING Digest 1522
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