So I had to CALCULATE

>> >Re Mr. Tsuji's question "Can we meet the fifth somewhere?" If I understand
>> >this to mean "will some number of 7:4's = some number of 3:2's," the answe
r
>> >is "not in JI."
>>
>> Surely seven of the one is half-a-dozen of the other in any context?
> I think that David Doty's point is that a stack of 7:4s atop one another
>never produces any composite interval exactly the same as any interval produc
e
>d by a stack of 3:2s, no matter how high you pile up either stack.

Sorry everyone. I should have calculated before asking questions. What I had
in mind was the potentiality of using the circle of 7:4s instead of 3:2s. And
if the double 88s could mean something similar to that.....ah! have to calcul
ate again.....anyway...it wasn't circle at last! So it means getting rid of th
e octaves will bring in a good chance for each equal tempered notes becomming
a root of a well tuned chord, doen't it? Is this just a FAQ? Maybe it is. Sorr
y again!

By the way, in Japan, a book called "MAGICAL MAX TOUR" came out this week. I
s someone trying retuned music on MAX? Is that kind of thing possible?

Thanx for your reply.

BUYO-BUYO-IGOR=Masaaki Tsuji>>>cxl03253@niftyserve.or.jp

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With the current interest in Pythagorean scales, it's a good time
to explain how the command in Scala works that creates a Pythagorean
scale.
Shameless plug alert: for those who missed it, Scala is a microtuning
program written by me which can be downloaded from:
ftp://ella.mills.edu/ccm/tuning/software/pc/scala/scala.zip
The best $0 you will ever spend!

Of course there's nothing against calculating a scale by hand, it's
a good exercise. A program is usually quicker though.

The command is called PYTHAGOREAN. It can calculate any Pythagorean
scale. First it wants to know how many notes the scale must get.
Type for instance: 26. Then, what the formal octave must be. I use the
word "formal" to distinguish it from the octave of 2/1. It may be any
value namely. But type for instance: 2/1. This will be the value of
the last note in the scale and the interval that is subtracted every
time the cycle of fifths passes over the last note. Instead of "formal
octave", "interval of equivalence" is sometimes used. Then the scale
degree for the formal fifth must be given (again to distinguish it
from the perfect fifth). In 12-tET the degree of the fifth is 7. Here
one can also enter 0, which means "I don't know" and then the program
takes the nearest degree depending on the formal fifth to be given. So
let's enter 0. Then enter the formal fifth, for instance: 7/4. Last,
it is asked how many of the notes must result from fifths stacked in
the downward direction. In this example they will be: 8/7, 64/49, etc.
Going upwards you get 7/4, 49/32, etc. Let's enter: 10. That's all.
The scale that results is this one:

1: 16807/16384 44.12955
2: 282475249/268435456 88.25910
3: 132.389 cents 132.3886
4: 131072/117649 187.0446
5: 8/7 231.1741 septimal whole tone
6: 2401/2048 275.3037
7: 40353607/33554432 319.4333
8: 363.563 cents 363.5628
9: 1048576/823543 418.2188
10: 64/49 462.3483
11: 343/256 506.4779
12: 5764801/4194304 550.6074
13: 594.737 cents 594.7370
14: 8388608/5764801 649.3930
15: 512/343 693.5225
16: 49/32 737.6521
17: 823543/524288 781.7817
18: 825.911 cents 825.9112
19: 67108864/40353607 880.5672
20: 4096/2401 924.6967
21: 7/4 968.8264 harmonic seventh
22: 117649/65536 1012.955
23: 1977326743/1073741824 1057.085
24: 536870912/282475249 1111.741
25: 32768/16807 1155.871
26: 2/1 1200.000 octave

With the command SHOW CYCLE we can find out where this leaves the
"wolf" 7/4 and see that it is placed between degrees 3 and 24.
We can also see that the scale is fairly close to 26-tET. The largest
difference is 6 cents.


Masaaki Tsuji asks:
> By the way, in Japan, a book called "MAGICAL MAX TOUR" came out this week. Is
> someone trying retuned music on MAX? Is that kind of thing possible?

I haven't used MAX but have read somewhere you have to use a pitch-bend
object called "xbendout" and it can have a value of 0 .. 16383.

Manuel Op de Coul coul@ezh.nl

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