Table of Pythogorean and Just Intonation Ratios
    in cycles of fourths and fifths format with the corresponding Carnatic notes identified.


In the main table below, going down the rows takes you through "cycles of fourths" and going up the rows engages in "cycles of fifths."

A cycle of fourth involves taking a particular note and producing its perfect 4th or its "M1" using the note as Sa or the tonic.
In ratio terms, you multiply by a factor of (4/3).

A cycle of fifth is similar except you produce a perfect 5th or "Pa" by multiplying by (3/2) instead.

To bring a ratio in the range 1/1 to 2/1, we can multiply or divide by a factor of 2 as many times as we wish, essentially shifting up or down octaves.

The ratios in the 1st and 4th columns, involving only powers of 2 and 3 are the Pythogorean ratios. The ratios in the 2nd and 3rd columns are found in "Just Intonation" tables and have an extra factor of 5 in them.

The ratios/rationals shown in the larger, bold font are the ones typically found in "22-sruthi" tables (which are nonsense).

NOTE:  Pi / 1.3 is a ratio, for example. Frequency ratios are important in the perception of music. But often, people mean "rationals" when they talk about "ratios." p/q where p and q are integers is a rational number. Such numbers also represent ratios, p:q, and the reader should keep in mind the distinction.

Formula for converting from ratios to cents:

    cents = 1200 * log (ratio) / log (2) = 1200 log2(ratio)

In the following tables, two decimal places are given for the cent values for amusement only.
They are practically or musically insignificant.

Some Famous Mathematical Intervals
Name
 Origin Cents
Tonic, Unison
 1/1 0
Western Semitone
 2^(1/12) 100
Mathematical Octave
 2/1 1200
Schisma
 32805/32768 1.95
Comma of Didymos or Synotic Comma 81/80 21.51
Comma of Pythagoras 531441/524288 23.46
Just Major 3rd
 5/4
 386.31
Perfect / Just 4th
 4/3
 498.04
Perfect / Just 5th
 3/2
 701.96


Note
 Rational
 Cents
 Rational
 Cents
 Note
 Rational
 Cents
 Rational
 Cents
 Note
S
 1
 /
 1
 0
 32805
 /
 32768
 1.95
 S
 81
 /
 80
 21.51
 531441
 /
 524288
 23.46
 S
M1
 4
 /
 3
 498.04
 10935
 /
 8192
 500.00
 M1
 27
 /
 20
 519.55
 177147
 /
 131072
 521.51
 M1
N2
 16
 /
 9
 996.09
 3645
 /
 2048
 998.04
 N2
 9
 /
 5
 1017.60
 59049
 /
 32768
 1019.55
 N2
G2
 32
 /
 27
 294.13
 1215
 /
 1024
 296.09
 G2
 6
 /
 5
 315.64
 19683
 /
 16384
 317.60
 G2
D1
 128
 /
 81
 792.18
 405
 /
 256
 794.13
 D1
 8
 /
 5
 813.69
 6561
 /
 4096
 815.64
 D1
R1
 256
 /
 243
 90.22
 135
 /
 128
 92.18
 R1
 16
 /
 15
 111.73
 2187
 /
 2048
 113.69
 R1
M2
 1024
 /
 729
 588.27
 45
 /
 32
 590.22
 M2
 64
 /
 45
 609.78
 729
 /
 512
 611.73
 M2
N3
 4096
 /
 2187
 1086.31
 15
 /
 8
 1088.27
 N3
 256
 /
 135
 1107.82
 243
 /
 128
 1109.78
 N3
G3
 8192
 /
 6561
 384.36
 5
 /
 4
 386.31
 G3
 512
 /
 405
 405.87
 81
 /
 64
 407.82
 G3
D2
 32768
 /
 19683
 882.40
 5
 /
 3
 884.36
 D2
 2048
 /
 1215
 903.91
 27
 /
 16
 905.87
 D2
R2
 65536
 /
 59049
 180.45
 10
 /
 9
 182.40
 R2
 4096
 /
 3645
 201.96
 9
 /
 8
 203.91
 R2
P
 262144
 /
 177147
 678.49
 40
 /
 27
 680.45
 P
 16384
 /
 10935
 700.00
 3
 /
 2
 701.96
 P
S
 1048576
 /
 531441
 1200 - 23.46
 160
 /
 81
 1200 - 21.51
 S
 65536
 /
 32805
 1200 - 1.95
 1
 /
 1
 0
 S
Note
 Rational
 Cents
 Rational
 Cents
 Note
 Rational
 Cents
 Rational
 Cents
 Note