Re: Math Music Scale(s)?

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From: SFPhysics@aol.com
Date: Thu Feb 10 2000 - 23:04:37 PST 

From: SFPhysics@aol.com
Message-ID: <92.155454e.25d50e85@aol.com>
Date: Fri, 11 Feb 2000 02:04:37 EST
Subject: Re: Math Music Scale(s)?

> hello pinholers!
> during the summer institute last summer, we learned of
> the mathmatical origin of the 8 (12) note scale. i
> forgot most of what i learned, but it seemed like it
> would fit well into my current unit on sound. the
> only thing i remember is something about the square
> root of 12? also, i remember something about a 3:2
> ratio in frequencies, so maybe it was the square root
> of one and a half which is 1.2 which looks kinda like
> 12? i tried an unsuccessful web search. anyone?
> keep the funk alive
> --eric friedman
>>

Hello Eric:

There are many musical scales. I believe the first one you refer to is the
Scientific or Just Scale and not the one we use today for modern music which
is the equal temperament scale first devised by Johann Sebastian Bach.

How this works out in more easily understood terms is that these ratios that
show the SCIENTIFIC or JUST Scale: (based on 8 notes, no sharps/flats)

8:8 or 1:1 for C 256 Hz
9:8 for D 288 Hz
10:8 or 5:4 for E 320 Hz
                      F 341.33 Hz
12:8 or 3:2 for G 384 Hz
                      A 426.67 Hz
15:8 for B 480 Hz
16:8 or 2:1 for C 512 Hz the next octave above C

Various temperaments were developed through the ages until we reach the
modern scale developed by Bach. Bach started with the basic octave which is
the doubling of frequencies say middle C at 256Hz to the next c1 at 512Hz.
Since there were 11 more notes between them Bach arranged to have them placed
as the mathematical *normal mensor of the 12th root of 2.* So we have ended
up with octaves of 12 equal steps called semitones. Each note or semitone is
1.059463xxxxx..... times the frequency of the note before. So, using modern
C of 262Hz: (rounded off)

C 262 Hz times 1.059463xxxxx = 277 Hz or C# (and so on)
C# 277 Hz
D 294 Hz
D# 311 Hz
E 330 Hz
F 349 Hz
F# 370 Hz
G 392 Hz
G# 415 Hz
A 440 Hz
A# 466 Hz
B 493 Hz
C 523 Hz

Note that this is the modern A440Hz concert pitch that was adopted as the
American Standard Pitch in 1936 but was in use from about 1920 and on.

Sorry to take so long in answering your post but I put the above in the
outgoing and forgot to send it! Senility.............

Al Sefl
Glad not to have a Sn ear!

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