Intervals

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

See also: Mathematical Terms & Concepts

String Length and Pitch Interval

If we divide a string of length L and fundamental frequency f into segments; e.g.,

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

plucking those segments will produce 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 relationship between pitch and frequency see the Wikipedia article pitch (music).

Reciprocal relationship
Be sure to notice the inverse relationship between L and f above; e.g., 1/2 & 2/1, 1/3 & 3/1, 1/4 & 4/1, 1/5 & 5/1, etc. In mathematics, the reciprocal of x is defined to be 1/x; e.g.,

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

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

Decimal expansion
We can also represent a fraction using a decimal expansion, where we divide the fraction's numerator by its denominator 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.

1:1
In tuning theory, the ratio 1:1 serves as a very important point of reference. For example, it may be used to represent the unison interval {XW} or the fundamental of a harmonic series. Most importantly, it is used to represent the tonic degree in a scale, where every over pitch in the scale is expressed as a precise frequency ratio in relation to the prime unity 1:1 (Partch 1947).  For more information, see the Xenharmonic Wiki articles unison and scale, respectively.

Octave reduction
In tuning theory, pitches and intervals are often expressed as ratios 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

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

Basic Interval Models

1. Harmonic series {WP; XW}
   
• as a sequence of string lengths:
• 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, ...
  
• as a sequence of pitches:
• 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, ...
• as a sequence of pitches under octave reduction
• 1/1, 2/1, 3/2, 2/1, 5/4, 3/2, 7/4, 2/1, 9/8, 5/4, ...
• as one axis of the Lambdoma (Hero and Foulkrod 1999)
• Wilson Archives, Glossary, Lambdoma
• Telfer, Cymatic Music, Lambdoma & String Concords
  
  
  
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. Pythagorean (3-limit) tuning {WP; XW}
   Pitch formula: 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: 1/1
  
• 2: 9/8 {XW} – Large major second
• 3: 81/64 {XW} – Pythagorean major third
  
• 4: 4/3
  
• 5: 3/2
  
• 6: 27/16 {XW} – Pythagorean major sixth
• 7: 243/128: {XW} – Pythagorean major seventh
• 8: 2/1
  
  
• Other intervals in the system:
• 16/9 {XW} – Pythagorean minor seventh
  (4/3) * (4/3) = 16/9
  
• 32/27 {XW} – Pythagorean minor third
  (4/3) / (9/8)  (4/3) * (8/9) = 32/27
  
• 128/81 {XW} – Pythagorean minor sixth
  (2/1) / (81/64) = (2/1) * (64/81) = 128/81
  
• 256/243 {XW} – Pythagorean diatonic semitone
  (4/3) / (81/64) = (4/3) * (64/81) = 256/243
  
• 2187/2048 {XW} – Pythagorean chromatic semitone, or apotome; e.g., F3–C4–G4–D5–A5–E6–B6–F#7
  (3/2)^7 / (2/1)^4 = (2187/128) / 16 = 2187/2048
  
  
• Chromatic scale:
• 1/1, 256/243, 9/8, 32/27, 81/64, 4/3, 729/512, 3/2, 128/81, 27/16, 16/9, 243/128, 2/1
  
  
  
4. Just (5-limit) tuning {WP; XW}
   Pitch formula: 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

• 1: 1/1
• 2: 9/8
• 3: 5/4 M3 {XW} – Just major third
  
• 4: 4/3 P4
• 5: 3/2 P5
• 6: 5/3 M6 {XW} – Just major sixth
• 7: 15/8 M7 {XW} – Just major seventh
  
• 8: 2/1 P8
  
  
• Other intervals in the system:
• 6/5 {XW} = (3/2) / (5/4) = (3/2) * (4/5) – Just minor third
  
• 8/5 {XW} = (2/1) / (5/4) = (2/1) * (4/5) – Just minor sixth
  
• 9/5 {XW} = (18/8) / (5/4) = (9/4) * 4/5  – Just minor seventh
  
• 10/9 {XW} = (5/4) / (9/8) = (5/4) * (8/9)  – Small major second
  
• 16/15: {XW} = (4/3) / (5/4) = (4/3) * (4/5) – Just diatonic semitone
  
• 25/24 {XW} = (5/4) / (6/5) = (5/4) * (5/6) – 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

Twelve-tone equal temperament (12-tet)

In twelve-tone equal temperament (abbr. 12-tet; See also12-edo, or 12ed2) {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

In 12-tet, the chromatic scale may be represented using fractional powers of 2:

2^0/12, 2^1/12, 2^2/12, 2^3/12, 2^4/12, 2^5/12, 2^6/12, 2^7/12, 2^8/12, 2^9/12, 2^10/12, 2^11/12, 2^12/12

which may be simplified to:

1/1, 2^1/12, 2^1/6, 2^1/4, 2^1/3, 2^5/12, 2^1/2, 2^7/12, 2^2/3, 2^3/4, 2^5/6, 2^11/12, 2/1

Here is the complete chromatic scale using EDO-step notation (n\n-EDO), where n is the nth step in n-EDO, with the diatonic scale highlighted using bold type:

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 12-tet semitone. To compare the size of two interval frequency ratios, we will typically convert ratios to cents (¢). The following formula may be used to convert an interval frequency ratio (f₁/f₂) to cents (c):

c = 1200 log₂ (f₁/f₂)

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

1200 log₂ (3/2) ≈ 701.995¢

when rounding to three decimal places, or nearest 1/1000 cent. The following mobile app will perform this calculation:

Calculator: Bain, Ratio to Cents

Here is the complete chromatic scale using cents, with the diatonic scale highlighted in bold type:

0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200

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

Commas

A comma is a small difference that arises when a note is tuned in different ways. Here are some of the commas we will encounter in our study of tuning theory. For more information, see the Xenharmonic Wiki article comma.

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 (81/64) and just major third (5/4)
  
• (81/64) / (5/4) = (81/64) * (4/5) = 81/80 ≈ 21.5¢

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}

Some important small EDO intervals

1. 1/4 tone = 200/4 = 50¢; see 24
   
   
2. 1/5 tone = 200/5 ≈ 40¢
   
   
3. 1/6 tone = 200/6 ≈ 33¢
   
   
4. 1/8 tone = 200/8 = 25¢

etc.

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}

What is more, this web app can play any interval frequency ratio:

Matthew Yacavone, Xen-calc

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

See: BAIN MUSC 726T Bibliography

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}

Hero, Barbara and R. M. Foulkrod, 1999. "The Lambdoma matrix and harmonic intervals." IEEE Engineering in Medicine and Biology Magazine 18/2 (March-April 1999): 61–73. {Lambdoma.com}

Partch, Harry. 1974/1949. Genesis of a Music, 2nd ed. New York: Da Capo Press. {GB}

Updated: January 19, 2026