Intervals

Web apps: Bain, Ratio to Cents; Matthew Yacavone, Xen-calc

See also: Mathematical Terms & Concepts

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. For more information about the relationships between pitch and frequency see the Wikipedia article pitch (music).

Reciprocal relationship
Be sure to notice the inverse 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 Reduced fraction in Wolfram's MathWorld, or Simplifying (reducing) fractions in the Wikipedia article Fraction.

Decimal expansion
From time to time, we will need to represent a fraction as decimal expansion, i.e., we will divide the fraction's numerator (top number) by its denominator (bottom number) as shown in the following examples:

2/1 = 2.0, 3/2 = 1.5, 4/3 = 1.333..., 5/4 = 1.25, etc.

For more information, see Decimal expansion in Wolfram MathWorld.

Unity
In tuning theory, the ratio 1:1 serves as a point of reference. It is used to represent the unison {XW} interval and tonic scale degree. Therefore, it is sometimes referred to as unity.

For more information, see the Xenharmonic Wiki articles unison and scale, respectively.

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

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

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

For more information, see the Xenharmonic Wiki article octave reduction.

Interval Ratio Examples

1. Harmonic series {WP}
• String lengths:
• 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, ...
  
• Pitches:
• 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, ...
• Pitches under octave reduction:
• 1/1, 2/1, 3/2, 2/1, 5/4, 3/2, 7/4, 2/1, 9/8, 5/4, ...
  
  
2. Tetractys {WP}: 1, 2, 3, 4
• 1/1 {XW} – Perfect unison (PU)
• 2/1 {XW} – Perfect octave (P8)
• 3/2 {XW} – Perfect fifth (P5)
• 4/3 {XW} – Perfect fourth(P4)
  
  For more information about traditional tonal interval names, see BAIN MUSC 115 Intervals (pdf).
  
  
3. Octave complement, or complementary interval {XW}
   e.g., (3/2) * x = 2/1; x = (2/1) * (2/3) = 4/3
   
   
4. Pythagorean (3-limit) {WP; XW}
   2^p * 3^q, where p and q are integers
   Diatonic scale: 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1
• 1/1
  
• 9/8 {XW} – Large major second
• 81/64 {XW} – Pythagorean major third
  
• 4/3
  
• 3/2
  
• 27/16 {XW} – Pythagorean major sixth
• 243/128: {XW} – Pythagorean major seventh
• 2/1
  
• Other intervals in the system:
• 16/9 {XW}: (4/3) * (4/3) = Pythagorean minor seventh
  
• 32/27 {XW}: (4/3) / (9/8) = (4/3) * (8/9) – Pythagorean minor third
• 128/81 {XW}: (2/1) / (81/64) = (2/1) * (64/81) – Pythagorean minor sixth
  
• 256/243 {XW}: (4/3) / (81/64) = (4/3) * (64/81) – Pythagorean minor second
  
• etc.
  
  
5. Just (5-limit) {WP; XW}
   2^p * 3^q * 5^r, where p, q, and r are integers
   Diatonic scale: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1
   Senario: 1, 2, 3, 4, 5, 6 (Duffin 2006)
   
• 1/1
• 9/8
• 5/4 M3 {XW} – Just major third
  
• 4/3 P4
• 3/2 P5
• 5/3 M6 {XW} – Just major sixth
• 15/8 M7 {XW} – Just major seventh
  
• 2/1 P8
• Other intervals in the system:
• 6/5 {XW}: (3/2) / (5/4) = (3/2) * (4/5) = 6/5 – Just minor third
  
• 8/5 {XW}: (2/1) / (5/4) = 2/1 * 4/5 = 8/5 – Just minor sixth
  
• 9/5 {XW}: (9/8) * (2/1) / (5/4) = 18/8 * 4/5 – Just minor seventh (2/1 * 8/9
  
• 10/9 {XW}: (5/4) / (9/8) = (5/4) * (8/9) = 10/9  – Small major second
  
• 16/15: {XW} (4/3) / (5/4) = (4/3) * (4/5) = 16/15  – Just diatonic semitone
  
• 25/24 {XW}: (5/4) / (6/5) = (5/4) * (5/6) = 25/24 – Just chromatic semitone
• 45/32: {XW}: (9/8) * (5/4) – Just diatonic tritone
• Chromatic scale:
• 1/1, 25/24, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1
  
  
6. Twelve-Tone Equal Temperament (12-tet/12-edo) {WP; XW}
   The octave (2/1) is divided into 12 equal-sized intervals called semitones. The size of each semitone is equal to the twelfth root of two:

2^1/12 ≈ 1.059

Example: The diatonic scale may be represented using fractional powers of 2:

2^0/12, 2^2/12, 2^4/12, 2^5/12, 2^7/12, 2^9/12, 2^11/12, 2^12/12

or using EDO-step notation (n\n-EDO), where n is the nth step in n-EDO.

Example: Using EDO-step notation, the diatonic scale may be represented as:

0\12, 2\12, 4\12, 5\12, 7\12, 9\12, 11\12, 12\12

Interval Size Comparison (Cents)

To compare the size of two interval frequency ratios, we will convert ratios to cents (¢). The cent 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 twelve-tone equal tempered (12-tet) semitone. Using EDO-step notation, the cent may be represented as:

1\1200 octave

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

c = 1200 log₂ (f₁/f₂)

For example, the ratio 3/2 is equivalent to 702 cents (rounding to the nearest cent), or more precisely (rounding to three decimal places, or 1/1000 cent):

1200 log₂ (3/2) ≈ 701.995 ¢

The following mobile app will perform this calculation:

Calculator: Bain, Ratio to Cents

For a multi-purpose interval calculator/converter that will also play the interval, see Matthew Yacavone's Xen-calc. For other interval size units, see the Xenharmonic Wiki article Interval size measure.

Comma {WP; XW}

1. Pythagorean comma (PC) {WP; XW}
   
• Difference between 12 just perfect fifths and seven octaves
  (3/2)^12 / (2/1)^7 = 531441/524288 ≈ 23.5¢
• Also:
• (9/8)^6 / (2/1) ≈ 23.5¢
• 1.5^12 / 2^7 ≈ 129.746/128 ≈ 1.014
2. Syntonic comma (SC) {WP; XW}
• Difference between the Pythagorean major third and just major third
  
• (81/64) / (5/4) = (81/64) * (4/5) = 81/80 ≈ 21.5¢
• 81/80 ≈ 1.013

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

1. Diesis {WP; XW}
• Difference between the octave and three just major thirds
  (2/1) / (5/4)^3 = (128/64) / (125/64) = (128/64) x (64/125) = 128/125 ≈ 41.1¢
2. Schisma {WP; XW}
• Difference between the Pythagorean comma and syntonic comma
  (531441/524288) / (81/80) = 32805/32768 ≈ 1.95¢
3. Septimal comma {WP; XW}
• Difference between the Pythagorean minor seventh and just minor seventh
  (16/9) / (7/4) = (16/9) * (4/7) = 64/63 ≈ 27.3¢
4. Limma {WP}
5. Kleisma {WP; XW}
   

Interval Galleries

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. Wikipedia, List of pitch intervals {WP}
4. Xenharmonic Wiki
• Gallery of Just Intervals {XW}
• List of Superparticular Intervals {XW}
• List of Overtones {XW}

Calculator: Matthew Yacavone, Xen-calc
This web app can play any interval frequency ratio

Links

Huygens-Fokker Foundation: Centre for Microtonal Music {HFF} – https://mathworld.wolfram.com

Wikipedia {WP} – https://www.wikipedia.org

Wolfram MathWorld {MathWorld} – https://mathworld.wolfram.com

Xenharmonic Wiki {XW} – https://en.xen.wiki

References

Duffin, Ross W. 2006. "Just Intonation in Renaissance Theory and Practice." Music Theory Online 12/3. {MTO}

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

Updated: May 1, 2025