Pythagorean tuning is derived from the 3/2 frequency ratio, and has been believed by some to be the basis of many historical scales.
Pythagorean tuning uses steps of the frequency ratios 3/2 (702
cents) to generate steps of fifths;
and 4/3 (498 cents) to generate steps of fourths. This results
in a spiral tuning yet the resultant harmony is generally agreed to be
unsatisfactory.
Some authorities claim that this is the basis of all Indian tunings, and that North Indian scales are derived from steps of fifths in the sa - ma pattern; whilst the South Indian tuning uses steps of fourths in the sa - pa pattern. As Indian musical practice tends to contain many ornamentations, much vibrato, and glissando, this is doubtful.
Indian
Name |
Position | Western
Equivalent |
Ratio | Cents | |
Sa |
I |
C | 2/1 | 1200.0 | |
Ma |
V |
G | 3/2 | 702.0 | |
Kai. N |
II |
D | 9/8 | 203.9 | |
Sadh G |
VI |
A | 27/16 | 905.9 | |
Sudh D |
III |
E | 81/64 | 407.8 | |
Sudh R |
VII |
B | 243/128 | 1109.8 | |
Pra M |
#IV |
F# | 729/512 | 611.7 | |
Kak N |
#I |
C# | 2187/2048 | 113.7 | |
Antara G |
#V |
G# | 6561/4096 | 815.6 | |
Sadh G |
#II |
D# | 19683/16384 | 317.6 | |
Ch D |
#VI |
A# | 59049/32768 | 1019.6 | |
Ch R |
#III |
E# | 177147/131072 | 521.5 | |
Pa (a comma
flatter) |
#VII |
B# | 531441/524288 | 23.5 |
Indian
Name |
Position | Western
Equivalent |
Ratio | Cents | |
Sa |
I |
C | 2/1 | 1200.0 | |
Pa |
IV |
F | 4/3 | 498.0 | |
Ch.R |
bVII |
Bb | 16/9 | 996.1 | |
Ch. D |
bIII |
Eb | 32/27 | 294.1 | |
Ant G |
bVI |
Ab | 128/81 | 792.2 | |
Kakali N |
bII |
Db | 256/243 | 90.2 | |
Pra M |
bV |
Gb | 1024/729 | 588.3 | |
Sud R |
bVIII |
Cb | 4096/2187 | 1086.3 | |
Sud D |
bIV |
Fb | 8192/6561 | 384.4 | |
Sadh G | bbVII | Bbb | 32768/19683 | 882.4 | |
Kaisiki N | bbIII | Ebb | 65536/59049 | 180.5 | |
Sud M | bbVI | Abb | 262144/177147 | 678.5 | |
Pa (a comma
sharper) |
bbII | Dbb | 1048576/531441 | 1176.5 |
Compare to LucyTuned values (Pythagorean flats are flatter and sharps are sharper)
Hybrid tunings mapped using L & s
Download Microsoft Works 4.0 Spreadsheet (includes adjustable cent values and fretting positions for Pythagorean Tuning)- worksfret.zipInteger frequency ratio and Just Intonation tunings
back to Pitch, Pi,..... Chapter One