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 |