Limma

Diatonic intervals
Prime second third fourth fifth sixth seventh octave none decime undezime duodecime tredezime semitone / whole tone
Special intervals
Microinterval Comma Diësis Limma Apotome Ditone Tritone Wolf fifth Natural septime
units
Cent Millioctave Octave Savart

The Limma or Leimma ( Greek  λεῖμμα , remnant ') denotes the Pythagorean semitone in music since Euclid (around 300 BC) . In Euclid's "division of the canon" one finds the following definitions with the "proportions" (=  aspect ratios , corresponds to our current frequency ratios ):

• Octave: ${\displaystyle {\frac {2}{1}}}$
• Fifth: ${\displaystyle {\frac {3}{2}}}$
• Fourth = octave - fifth: ${\displaystyle {\frac {4}{3}}}$
• Whole tone = fifth - fourth: ${\displaystyle {\frac {9}{8}}}$
• Ditonos = 2 whole tones: ${\displaystyle {\frac {81}{64}}}$
• Limma = fourth - ditonos: cent . ${\displaystyle {\frac {256}{243}}\,\,{\widehat {\approx }}\,\,90{,}22}$

The Limma is the diatonic semitone of the Pythagorean tuning.

As a supplement, the

• Apotome = whole tone - Limma:, the chromatic Pythagorean semitone. ${\displaystyle {\frac {2187}{2048}}}$

The apotone (113.7 cents) is a Pythagorean comma (23.5 cents) larger than the limma (90.2 cents).

See also

• Calculations in the Pythagorean tuning

Remarks

1. ↑ When adding or subtracting intervals, the frequency ratios are multiplied or divided.