Tuning Theory


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String Length and Pitch Interval

Dividing a string of length L and fundamental frequency f into parts such as

1/2 L, 1/3 L, 1/4 L, 1/5 L, etc.,

produces the following frequencies, respectively:

2/1 f, 3/1 f, 4/1 f, 5/1 f, etc.

For more information about the relationship between string length and pitch interval, see the Wikipedia article Mersenne's Laws.

Reciprocal relationship
Be sure to notice the reciprocal relationship between L and f, where the reciprocal of x is defined to be 1/x.

Example. The reciprocal of 4/3 is calculated: 1 / (4/3) = 1 * (3/4) = 3/4

For more information, see Wikipedia: Multiplicative Inverse.

Simplified, or reduced, fraction
In mathematics, ratios are typically reduced to their simplest form.

Examples. 4/2 may be reduced to 2/1, 6/4 may be reduced to 3/2, 12/9 may be reduced to 4/3, 20/16 may be reduced to 5/4, 60/50 may be reduced to 6/5, etc.

For more information, see Reduced fraction in MathWorld, or Simplifying (reducing) fractions in the Wikipedia article Fraction.

Octave reduction
In tuning theory, pitches and intervals in scales are typically expressed under octave reduction {XW}; i.e., as ratios within the 1/1 to 2/1 octave, inclusive.

Examples. The frequency ratios 2/1, 3/1, 4/1 & 5/1 are equivalent to the following ratios under octave reduction, respectively:

2/1 f, 3/2 f, 2/1 f, 5/4 f, etc.

Superparticular ratios
Superparticular ratios are ratios of the form (n+1)/n, where n is a positive integer. Such ratios correspond to intervals formed by adjacent members of the harmonic series.

Interval Examples

  1. Harmonic series {WP}
  2. Tetractys {WP}: 1, 2, 3, 4
  3. Complementary intervals, also called octave complement {XW}
    e.g., (3/2) * x = 2/1; x = (2/1) * (2/3) = 4/3
  4. Pythagorean (3-limit) {WP; XW}
    2p * 3q, where p and q are integers
    Diatonic scale: 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128
  5. Just (5-limit) {WP; XW}
    2p * 3q * 5r, where p, q, and r are integers
    Diatonic scale: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8


To compare the size of two interval frequency ratios, we will typically convert ratios to cents (). The cent (c) is a logarithmic unit of pitch interval proposed by the nineteenth-century English mathematician A.J. Ellis (1814-90). By definition, there are 1200 cents in an octave and 100 cents in a 12-tone equal-tempered (12-tet) semitone, which may be notated:

c = 1\1200 octave

The following formula may be used to convert an interval frequency ratio (f1/f2) to cents:

c = 1200 log2 (f1/f2)

For example, the ratio 3/2 is equivalent to 702 cents (rounding to the nearest cent), or more precisely:

1200 log2 (3/2) ≈ 701.995

The following mobile app can perform this calculation:

Calculator: Bain, Ratio to Cents

For a multi-purpose interval calculator/converter, see Matthew Yacavone's Xen-calc. For other interval size units, see Interval size measure {XW}.

Comma {WP; XW}

  1. Pythagorean comma (PC) {WP; XW}
  2. Syntonic comma (SC) {WP; XW}

Other commas {WP} (Gann 2019, pp. 35-36)

  1. Diesis {WP; XW}
  2. Schisma {WP; XW}
  3. Septimal comma {WP; XW}
  4. Limma {WP}
  5. Kleisma {WP; XW}

Interval Galleries

The following interval galleries will be useful when we compare various tuning systems:
  1. Kyle Gann, Anatomy of an Octave A list of over 1000 intervals within the octave {Gann 2019} (Gann 2019)
  2. Manuel Op de Coul, Scala: List of intervals {HFF} (See also: Scala)
  3. Xenharmonic Wiki


Huygens-Fokker Foundation {HFF}
Wolfram Mathworld {Mathworld}
Xenharmonic Wiki {XW}
Wikipedia {WP}


Gann, Kyle. 2019. The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician. Urbana: University of Illinois Press. {GB}

Updated: February 27, 2024

Reginald Bain | University of South Carolina | School of Music