? How to use log for Pythagorean note frequency derivations -observations, questions

Saw this online, hoped it would answer the question I
had about Pythagorean note frequencies, but not quite
there.... can anyone shed light on this?

">>Re: from Cents to Hertz and back again
>>From: <kleinebre_at_hotmail.com>
>>Date: 24 Oct 2005 11:00:55 -0700
There is only one way to answer this question:
Thoroughly ;) So
A sharp=440* (2^(1/12)) =440* 1.05946309
=466.163762 Hz.
in an octave there arent 12 cents but 1200, so the
multiplier= 2^(1/1200)=1.00057779
>>Pythagoras' time, note frequencies calculated ;
Frequency of next octave(A4->A5)
=base frequency*2/1 e.g.440*2/1=880 Hz
Frequency a fifth higher (A4->E5) =base
frequency*3/2 e.g. 440*3/2=660 Hz "
-------------------------------------------

This all made sense and I understand much better,
except for the Pythagorean part.

If I multiply the first note in a 12 note Pythagorean
scale, its frequency being;

261.6256, by 3/2, as you noted above, I get 397.7925.
However, the next proper frequency in the scale
is 279.3824, not 397.7925. In fact, 397.7925 doesn't
even appear at all in the scale. So I'm left
wondering...
what part of the math I'm not getting. I'm guessing
something simple.....

Also, wondering about the cents multiplier mentioned
above (1.00057779), am guessing this applies only
to edo and not a Pythagorean scale?

i.e. :19683/16384 317.595 Pythagorean augmented
second
81/64 407.820 Pythagorean major third

Scala reports the associated cents above as, 317.595
and 407.820, can I use the formula you supplied to
figure this?

Regarding the frequency question I have I went to the
freeware application called Scala;

In Scala, to get a twelve note Pythagorean tuning
based on 3/2
I type the following commands;

Pythag
Size = 12
Enter formal octave 2/1 = (enter key)
Enter fifth degree (0 for monotonic scale) = (enter
key)
Enter formal fifth = 3/2
Enter Count downwards = (enter key)

Then I type "show"

It lists the intervals;

0: 1/1 0.000 unison,
perfect prime
1: 2187/2048 113.685 apotome
2: 9/8 203.910 major whole
tone
3: 19683/16384 317.595 Pythagorean
augmented second
4: 81/64 407.820 Pythagorean
major third
5: 177147/131072 521.505 Pythagorean
augmented third
6: 729/512 611.730 Pythagorean
tritone
7: 3/2 701.955 perfect
fifth
8: 6561/4096 815.640 Pythagorean
augmented fifth
9: 27/16 905.865 Pythagorean
major sixth
10: 59049/32768 1019.550 Pythagorean
augmented sixth
11: 243/128 1109.775 Pythagorean
major seventh
12: 2/1 1200.000 octave

Then, show scale by frequencies;

0: 261.6256 Hertz 8.00000 oct MIDI nr:
60.00000
1: 279.3824 Hertz 8.09474 oct MIDI nr:
61.13685
2: 294.3288 Hertz 8.16993 oct MIDI nr:
62.03910
3: 314.3052 Hertz 8.26466 oct MIDI nr:
63.17595
4: 331.1199 Hertz 8.33985 oct MIDI nr:
64.07820
5: 353.5933 Hertz 8.43459 oct MIDI nr:
65.21505
6: 372.5098 Hertz 8.50978 oct MIDI nr:
66.11730
7: 392.4383 Hertz 8.58496 oct MIDI nr:
67.01955
8: 419.0736 Hertz 8.67970 oct MIDI nr:
68.15640
9: 441.4931 Hertz 8.75489 oct MIDI nr:
69.05865
10: 471.4578 Hertz 8.84963 oct MIDI nr:
70.19550
11: 496.6798 Hertz 8.92481 oct MIDI nr:
71.09775
12: 523.2511 Hertz 9.00000 oct MIDI nr:
72.00000

So lets say that I have a base frequency of 419.0736,
the 7th note in the scale, not counting the octave (12
@ 523.2511)
. I want to, via a math formula derive what the next
frequency (note) is, which is (9) 441.4931. How would
I do this?

Or, lets say that I have a frequency of 392.4383 (# 6
in the scale), and from that I want to determine its
position in the scale and derive all the other notes
by frequency,
notes 0 through 11, how might that be done?

If I had a Pythagorean scale that was greater than 12
notes, how would I adjust mathematically the formula
to do the same calculations?

Dan

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