Another criterion

To tuning yahoogroups

I was reviewing the way how the frequency tones are calculated when designing a new scale and thougth that I could explain the criterion I applied in a JI scale design to serve as an alternate guide to get the appropriate tone frequency values.

In this explanation I will use semitone factors which are frequency ratios between two consecutive tones.

If the twelve scale frequencies are seeing as a whole, that is, from C up to 2C, we do not get a complete conception of its interior relations.

A well calculated scale is formed by three groups of semitone factors; this is a meaningful statement. The first and third group contain five semitone factors or five ratios of six tones.

The first group comprises the tones C, C#, D, Eb, E. The second group comprises the tones F# and G and the third one is formed by Ab, A, Bb, B, 2C.

The product of the five semitone factors that work in the first and third group equals 4/3 while the product of the two semitone factors of the second group equals 9/8.

If we call e1 to C#/C semitone factor; e2 to D/C#, e3 to ratio Eb/D....etc., the octave can be given as follows:

C#/C .......e1
D/C# ...... e2
Eb/D ...... e3
E/Eb ...... e4
F/E .........e5 .........(e1 x e2 x e3 x e4 x e5) = (4/3)

F#/F ...... e6
G/F# ...... e7 ...........(e6 x e7) = (9/8)

Ab/G ...... e1
A/Ab ...... e2
Bb/A ...... e3
B/Bb ...... e4
2C/B ...... e5.........(e1 x e2 x e3 x e4 x e5) = (4/3)

(4/3) (9/8) (4/3) = 2

Three scales were calculated by using a different criterion and obtained 6 and 7 perfect fifths. On the other hand, three scales were designed by using the explained method and showed 8 and 9 perfect fifths. Similarly, the number of perfect major thirds (1.25) increased from 2 to 3.

Thanks

Mario Pizarro

piagui@ec-red.com

Lima, July 21, 2009

02:00 p.m.