Re: [tuning] Harmonic scales and a Thought

Sorry about the name on my last post (that's my dad's). Our computer has
this User Identification System that works by the average time it takes to
press different keys... :)

If it ever happens again, just play 12TET at me in punishment. :)

Stay Tuned,
Keenan Pepper

Keenan Pepper wrote,

> The eighth sounds like a badly out of tune major scale with an extra
note, the sixth sounds like a messed-up (but in tune) blues scale, and
the seventh sounds downright peculiar. The ninth and twelfth also
sound interesting.

About ten years ago I wrote a set of overtone series etudes for the
Alesis HR-16:B drum machine, as the it (somewhat mysteriously as no
one at Alesis could either confirm or deny this) had voices tuned in
groups of overtone octaves. The incremental pitch numbers on the
Alesis HR-16:B drum machine are arranged in a number line from -16
through +15, and the "pipe" voice for example corresponds to harmonics
16 through 47, giving eight complete octaves at 16 through 23, and a
ninth that abruptly stops at 47/24. I think I'll see if I can't dig
any of these up and try and post 'em up at the TuningPunk site
someday... I do remember that the step writing and editing process was
a mind numbingly tedious experience, but that the results were
peculiar enough (to me) to (almost) justify the labor intensive
interface.

Dan Stearns

BTW, awhile back I was scribbling away on the back of an envelope when
I noticed a simple little something that I thought was very
interesting - well to me anyway. If you take an overtone sequence as
say h ... 2h-1, the undertone sequence, i.e., 2h-1 ... h (or 2h ...
h+1 and h+1 ... 2h) will also be the superparticular ratios in the
guise of a consecutive integer sequence where their sum is the equal
temperament:

series 2-4

(4) 3 2
0 3 5
0 720 1200

4 3 (2)
0 4 7
0 686 1200

series 3-6

(6) 5 4 3
0 5 9 12
0 500 900 1200

6 5 4 (3)
0 6 11 15
0 480 880 1200

series 4-8

(8) 7 6 5 4
0 7 13 18 22
0 382 709 982 1200

8 7 6 5 (4)
0 8 15 21 26
0 369 692 969 1200

(etc.)

So in other words I'm letting "h" be the variable that represents any
series number, and say h = 5, then 2h (i.e., 2*5) is 10, and 2h-1 is
9. So if you then take the 2h-1...h, you have 9, 8, 7, 6, 5, and if
you now let these numbers represent fractions of their sum, i.e.,
9/35, 17/35, 24/35, 30/35, 35/35ths of (a 1200�) octave, you can see
that this is an ET mimicking a 5:6:7:8:9:10 series where the
superparticular ratios and the uniquely articulated fractions of an
octave both increase by consecutive increments of 1, i.e.,

9 = 6/5
8 = 7/6
7 = 8/7
6 = 9/8
5 = 10/9

Using the parenthetical example "(or 2h ... h+1 and h+1 ... 2h)" where
h = 5, we'd have 10, 9, 8, 7, 6, at 10/40, 19/40, 27/40, 34/40, and
40/40, where 40 is the ET mimicking the 5:6:7:8:9:10 series, and the
superparticular ratios and the fractions of an octave both increase by
consecutive increments of 1 starting at 2h (as opposed to 2h-1):

10 = 6/5
9 = 7/6
8 = 8/7
7 = 9/8
6 = 10/9

Anyway, I found this little numerical curiosity (sort of a sideways
analogue to something like Pythagorean perfect numbers in a way) to be
very interesting even if I haven't found any practical utility for it.