RE: [MMM] ? How to use log for Pythagorean note frequency derivations -observations, questions

Let's assume you require an octave to be a doubling in pitch. f2=2*f1,
f2/f1 = 2.

Now you can take the logarithm of both sides, You'll see why later:
log (f2/f1) = log 2

But, let's use base 10 logarithms because calculators have those
operations built in: log (f2/f1) = 0.3010299...

Now, FOR 12 EQUALLY TEMPERED STEPS IN AN OCTAVE each step can be divided
into 100 cents and an octave is made up of 12 steps or 1200 cents.

Now cents per octave would be such that (Constant)(log 2) = 1200 or
(Constant) = 3986.3137...

Now we can convert any frequency ratio into cents: (Cents in ratio) =
(3986.3137)*log (f2/f1).

The ratio 5/4 would be 3986.3137 log (5/4)= 386.313712521 cents

You could do the same for any number of steps in an octave.

-----Original Message-----
From: MakeMicroMusic@yahoogroups.com
[mailto:MakeMicroMusic@yahoogroups.com] On Behalf Of Dan Amateur
Sent: Wednesday, October 04, 2006 3:31 AM
To: MakeMicroMusic@yahoogroups.com
Subject: [MMM] ? 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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