TUNING digest 1389, Daniel Wolf's query

Daniel Wolf writes:

===============
I am in the process of revising an old article. It
concerns a scale with the following properties:

(1) There are seven pitch classes in the system,
which I will number ascending from 1 to 7. Octave
equivalence is assumed, but may actually vary to
create beating for coloristic or perhaps a
vibrato-like effect to cover the difference of the
tuning from a small-number-ratio just intonation.

(2) There are three distinct pentatonic 'keys',
beginning at degrees 1 (12 456), 2 (23 567) and
5 (56 123) of the seven tone system, that is to
say, at 'fifths' apart.

(3) the intervals 1-5, 2-6 and 3-7 are recognizable
as 'fifths' (let's say +/- 25 cents of a 3/2) and
are used as dyads.

(4) 5-2', 6-3', and 4-1' _may_ be recognizeable as
fifths. The interval 7-4' is not used as a dyad.

(5) the intervals 1-2, 2-3 and 5-6 are between 80
and 190 cents apart.

(6) the intervals 1-2-3, 4-5-6, and 5-6-7
encompass an interval between an 8/7 and a 6/5.

(7) the interval 1-2 is smaller than 2-3 and the
interval 5-6 is smaller than 6-7. (The relationship
between the intervals 4-5 and 5-6 is not so fixed).

I am interested in what kinds of tunings others can
come up with that will meet the above specifications
in order to find out if the set of specifications
above is sufficient and necessary. Anyone game to try?
===============

Daniel,

This sounds like a generalized definition of Javanese pelog. If so, I
would question condition (1). In some if not most pelogs I've studied, 2-6
can be more than 50 cents narrower than a 3/2. It seems generally to be
slightly flatter than the 1-5 and 3-7 dyads, even though it is, in Javanese
theory, the same interval. I also question whether 1-2 is always smaller
than 2-3, and whether 4-6 is always or even usually larger than an 8/7.
The other conditions seem to me accurate constraints on pelog.

Parenthetically, re condition (2): as I'm sure you know, 4 is a subsidiary
tone in pelog lima, an alternate to 3; and some gendhing use all 7 tones;
so the boundaries between the three "keys" are not strict. In KNOWING
MUSIC, MAKING MUSIC, Ben Brinner writes, "The difference between pieces in
pelog lima and pelog nem is so obscure that some musicians ignore the fuzzy
boundary and group them together as pathet bem."

But to your question. Only conditions (3), (5), (6), & (7) put hard
constraints on the scale construction. Satisfying them keeps the pitches
in these ranges:

pitch cents range conditions met
1 0
2 80-157 (80-190 cts above 1 and less than half 1-3)
3 231-316 (231-316 cts above 1)
4 439-684 (231-316 cts below 6)
5 675-725 (675-725 cts above 1)
6 755-875 (675-725 cts above 2 and less than half 5-7)
7 906-1041 (231-316 cts above 5)

This is not to say that any and all pitches in these ranges will satisfy
the conditions. Condition (7) must be met for pitches 2 and 6 after fixing
pitches 3 and 7.

Are you looking specifically for small-number ratios?

To guarantee a functional pelog, I wonder if you might also need to
constrain the 6-7 interval (already >80 cents, but usually a good deal
larger), and possibly the 3-4 (usually at least twice as large as 4-5).

--Carter

>I have an older Mac 660AV using only Quicktime 2.5. Is there a program for
>making the Mac play a variety of tunings and temperaments?

I think that part of the answer may be JICalc, but I'd better defer to
somebody else to describe it.