Why were those frequencies chosen for the 8 notes in an octave?

if you study the ratio of frequencies in the notes that make up an octave, you will have your answer.

use the term overtone in your searches :-)

That's true - please click here

Mathmatics.

Dear Jon,

You may find this thread of interest:

Tuning in Baroque Times.

It does not particularly restrict itself to a consideration of the scheme of semitones used in modern chromatic tuning, but relates to some extent, how we arrived at modern keyboard tuning ...

ATB from George

Why 12 notes to the Octave?

Any moderately curious person will ask themselves at some point why, in western music, is the octave divided into 12 'semi-tones'. From a mathematical point of view, we can easily explain why 12 works nicely.

The Greeks realized that sounds which have frequencies in rational proportion are perceived as harmonious. For example, a doubling of frequency gives an octave. A tripling of frequency gives a perfect fifth one octave higher. They didn't know this in terms of frequencies, but in terms of lengths of vibrating strings.

Pythagoras, who experimented with a mono-chord, noticed that subdividing a vibrating string into rational proportions produces consonant sounds. This translates into frequencies when you know that the fundamental frequency of the string is inversely proportional to its length, and that its other frequencies are just whole number multiples of the fundamental.

First, we should examine what ratios are "meant" to exist in the western scale. The prominence of the major triad in western music reflects the Greek discoveries mentioned above. Starting with the note C as a fundamental, we get the major triad from the 3rd and 5th overtones, dropping down one and two octaves respectively, obtaining ratios of 3/2 (G:C) and 5/4 (E:C) respectively. Two other prominent features in western music include the V I cadence, and the I,IV,V triads. Both reflect the importance of the 3/2 ratio, with the IV further taking into account the reciprocal of 3/2, namely 2/3 aka 4/3. Musically, the reciprocal ratio corresponds to going down rather than up. While 3/2 corresponds to going up a fifth, 2/3 corresponds to going down a fifth, and 4/3 corresponds to going down a fifth and up an octave. Together, 3/2 and 4/3 divide the octave, so that going up by 3/2 followed by 4/3 gives an octave.

The IV and V triads give us the four new notes, B and D of G,B,D, and F and A of F,A,C. Their ratios, relative to C are 15/8 for B, 9/8 for D, 4/3 for F, and 5/3 for A. The notes formed from the I,IV, and V major triads produce the C major scale: C D E F G A B C. Throwing in reciprocals for each of these intervals yields all the intervals that made up western music until the rise of chromaticism.

1/1 unison C
2/1 octave C

3/2 perfect fifth G
4/3 fourth F

5/4 major third E
8/5 minor 6th Ab

6/5 minor 3rd Eb
5/3 major 6th A

9/8 major 2nd D
16/9 minor 7th Bb

15/8 major 7th B
16/15 minor 2nd C#


While this list of intervals does include a few of the most basic intervals and their reciprocals: unison, perfect 5th, major 3rd, major 6th = 3rd above a 4th (or also a 4th above a 3rd), major 2nd = a 5th above a 5th, and major 7th = a 3rd above a 5th (or also a 5th above a 3rd), some obvious ones are missing (such as 7/4, 25/16 = a 3rd above a 3rd, or 9/5 = a fifth above a minor 3rd).

The tritone (such as C to F#) is also omitted from this list, an interval that did not affect the evolution of the western scale as it was not used in western music until twelve note chromaticism had become firmly established. Actually, a tritone refers to two different possible intervals:

7/5 tritone
10/7 also called a tritone.

Read 'The Rest is Noise' by Alex Ross on 20th C music to understand the personalities of composers that pulled apart and rearranged music From Mahler and Strauss to today. Some really interesting characters. And he goes into some of the theories and physics of music.