August 22, 2009

Pythagorean tuning:: is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2

3:2

Twelve tone scale

There are several ways to create a just tuning of the twelve tone scale.

The oldest known form of tuning, Pythagorean tuning, can produce a twelve tone scale, but it does so by involving ratios of very large numbers, corresponding to natural harmonics very high in the harmonic series that do not occur widely in physical phenomena. This tuning uses ratios involving only powers of 3 and 2, creating a cycle of just perfect fifths, as follows:

Note	G♭	D♭	A♭	E♭	B♭	F	C	G	D	A	E	B	F♯
Ratio	$\frac{1024}{729}$	$\frac{256}{243}$	$\frac{128}{81}$	$\frac{32}{27}$	$\frac{16}{9}$	$\frac{4}{3}$	$\frac{1}{1}$	$\frac{3}{2}$	$\frac{9}{8}$	$\frac{27}{16}$	$\frac{81}{64}$	$\frac{243}{128}$	$\frac{729}{512}$
Cents	588.27	90.22	792.18	294.13	996.09	498.04	0.00	701.96	203.91	905.87	407.82	1109.78	611.73

Indian scales

In Indian music, the just diatonic scale described above is used, though there are different possibilities for the 6th pitch (Dha), and further modifications may be made to all pitches excepting Sa and Pa.^[5]

Note	Sa	Re	Ga	Ma	Pa	Dha	Ni	Sa
Ratio	1/1	9/8	5/4	4/3	3/2	5/3 or 27/16	15/8	2/1
Cents	0.00	203.91	386.31	498.04	701.96	884.36 or 905.87	1088.27	1200.00

The major scale based on C, obtained from this tuning is:

Note	C		D		E		F		G		A		B		C
Ratio	1/1		9/8		81/64		4/3		3/2		27/16		243/128		2/1
Step		9/8		9/8		256/243		9/8		9/8		9/8		256/243

1 to 2 ... stepup whole note per 100%

Note	C	D♭	D♭	D	D	E♭	E♭	E	E	F	F	F♯
Ratio	$\frac{1}{1}$	$\frac{256}{243}$	$\frac{16}{15}$	$\frac{10}{9}$	$\frac{9}{8}$	$\frac{32}{27}$	$\frac{6}{5}$	$\frac{5}{4}$	$\frac{81}{64}$	$\frac{27}{20}$	$\frac{4}{3}$	$\frac{45}{32}$
Cents	0.00	90.22	111.73	182.40	203.91	294.13	315.64	386.31	407.82	519.55	498.04	590.22

Note	F♯	G	A♭	A♭	A	A	B♭	B♭	B	B	C
Ratio	$\frac{64}{45}$	$\frac{3}{2}$	$\frac{128}{81}$	$\frac{8}{5}$	$\frac{5}{3}$	$\frac{27}{16}$	$\frac{16}{9}$	$\frac{9}{5}$	$\frac{15}{8}$	$\frac{243}{128}$	$\frac{2}{1}$
Cents	609.78	701.96	792.18	813.69	884.36	905.87	996.09	1017.60	1088.27	1109.78	1200.00