Intervals

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

See also: Mathematical Terms & Concepts

Types of Numbers

In mathematics, the natural numbers (ℕ) are the set of whole numbers:

1, 2, 3, 4, 5, …

The ellipsis symbol (...) means 'and so on'. The integers (ℤ) are the set of positive and negative whole numbers plus 0:

…, –3, –2, –1, 0, 1, 2, 3, …

Geometrically, the integers are often represented as discrete points on the integer number line. Given two integers p and q, the rational numbers (ℚ) are the set of all numbers that may be expressed as the quotient:

p/q,

where q ≠ 0. An irrational number (e.g., √2 = 1.41..., π = 3.14..., e = 2.71..., etc.) is a number that cannot be expressed as the quotient p/q. The real numbers (ℝ ) are the set of rational and irrational numbers, which are usually represented by a decimal expansion. As shown below, the real numbers form a continuous set of numbers that may be represented as quantities on a number line.

Credit: Wikimedia

For more information, see the Wikipedia article Lists of types of numbers.

Credit: Wikipedia

String Length and Pitch Interval

If we divide a string of length L, whose fundamental frequency is f, into parts; e.g.,

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

plucking those parts of the string will sound the following frequencies, respectively:

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

Comparing the quotients 1/2 &  2/1, 1/3 &  3/1, 1/4 &  4/1, 1/5 & 5/1, ... we notice they are multiplicative inverses, or reciprocals, of one another. In mathematics, the multiplicative inverse of q/r is defined to be r/q; i.e., q/r * r/q = 1. The reciprocal of x is defined to be 1/x. For example, the reciprocal of 1/2 is:

1 / (1/2) = 1 * (2/1) = 2/1

For more information, see the Wikipedia article multiplicative inverse.

The Law of Pythagoras
In physics, Mersenne's first law states: if the tension and linear mass density of a stretched string are constant, the fundamental frequency (f) of the string is inversely proportional to its length (L) which may be written:

The English physicist James Jeans (1877-1946) called this relationship the Law of Pythagoras (Jeans 1968). For more information about the relationship between string length and frequency, see the Wikipedia article Mersenne's Laws.

* * *

Pitch interval
In music, an interval is the distance between two pitches, which we will describe as a ratio between two frequencies.

For more information, see the Xenharmonic Wiki article interval.


Ratio
In mathematics, a ratio expresses the relationship between two or more quantities; e.g., the ratio between the frequencies 880 Hz and 440 Hz is 2:1. All ratios may be expressed as fractions; e.g. 2/1.

For more information, see the Wikipedia article ratio.


Fraction
A fraction describes the number of parts in a whole. The top number in a fraction is called the numerator (n), and the bottom number is called the denominator (d). A fraction is typically represented as the quotient q = n/d, where n and d are integers. Non-integer values for n and d are also possible.

For more information, see the Wikipedia article fraction.


String-length divisions
If we divide a string of length L into parts as shown below,

Credit: Wikipedia

we can abstractly represent the integer string-length divisions using fractions as shown below; e.g.,

L

1/2 L + 1/2 L = L

1/3 L + 2/3 L = L

1/4 L + 3/4 L = L

1/5 L + 4/5 L = L

etc.

In physics and music, the integer divisions of the string are called harmonics. These are the nodes of the string.

For more information see the following articles in Wikipedia: standing wave, node, harmonic series, and harmonic.

Simplest form
In mathematics, fractions are usually reduced to their simplest form, or reduced form.

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

Decimal expansion
A fraction (n/d) may also be represented as a decimal expansion where we divide the fraction's numerator (n) by its denominator (d).

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

For more information, see the Wolfram MathWorld article  Decimal expansion.

Unity (1:1)
In tuning theory, the ratio 1:1 serves as a point of reference. For example, it may represent the unison interval or the fundamental of a harmonic series. Most often, it is used to represent the tonic degree in a scale, and the other degrees in the scale are expressed as a 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 usually expressed as ratios within the 1/1 to 2/1 octave; i.e., as fractions under simplification and octave reduction.

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.

Interval System Models

1.  The 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. The 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)
  
• The 6:8:9:12 framework, or nexus
  (1/1, 4/3, 3/2, 2/1) = (6/6, 8/6, 9/6, 12/6) = 6:8:9:12
  
• See: Pythagorean hammers {WP}
  
  
  
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} – Pythagorean 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 (Gann 2019, p. 57)
  
  
  
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 – Large whole tone
  
• 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} – Just minor third
  (3/2) / (5/4) = (3/2) * (4/5) = 6/5
  
• 8/5 {XW} – Just minor sixth
  (2/1) / (5/4) = (2/1) * (4/5) = 8/5
  
• 9/5 {XW} – Just minor seventh
  (18/8) / (5/4) = (9/4) * 4/5  = 9/5
  
• 10/9 {XW} – Small whole tone
  (5/4) / (9/8) = (5/4) * (8/9) = 10/9
  
• 16/15: {XW} – Just diatonic semitone
  (4/3) / (5/4) = (4/3) * (4/5) = 16/15
  
• 25/24 {XW}  – Just chromatic semitone
  (5/4) / (6/5) = (5/4) * (5/6) = 25/24
  
• 45/32: {XW} – Just diatonic tritone
  (9/8) * (5/4) = 45/32
  
  
• Chromatic scale:
• 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1 (Gann 2019, p. 57)
  

Twelve-tone equal temperament (12-tet)

In twelve-tone equal temperament (abbr. 12-tet; see also 12edo, or 12ed2) {WP; XW}, the 2/1 octave 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

As such, the chromatic scale may be represented as fractional powers of 2; i.e.,

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

Simplifying the fractional exponents we get:

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

Equivalently, we may use decimal expansions:

1.0, 1.059, 1.122, 1.189, 1.260, 1.335, 1.414, 1.498, 1.587, 1.682, 1.782, 1.889, 2.0

Finally, we can also use EDO-step notation (n\n-EDO), where n is the nth step in n-EDO:

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 a 2/1 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 will 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¢

The following mobile Web app will perform this calculation.

Calculator: Bain, Ratio to Cents

It rounding all cent values to 3 decimal places, or nearest 1/1000 cent. Here is the 12-tet chromatic scale using cents, with the diatonic scale degrees highlighted using 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 small differentials {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 (PC/SC) {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}

EDO intervals

1. Semitone = 100¢
   
   
2. 1/4 tone = 200/4 = 50¢
   
   
3. 1/5 tone = 200/5 ≈ 40¢
   
   
4. 1/6 tone = 200/6 ≈ 33¢

etc.

In n-EDO, the step size is 2^1/n. For example, the step size in 31edo is:

2^1/31 ≈ 1.02261144 ≈ 38.71¢

For more information, see the Xenharmonic Wiki article EDO.

Interval Names

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}

The Web app Xen-calc 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}

Jeans, James. 1968/1937. Science & Music. Mineola, NY: Dover. {GB}

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}

Rossing, Thomas, F. Richard Moore and Paul Wheeler. 2002. The Science of Sound, Third Edition. New York: Addison Wesley. {GB}

Updated: April 14, 2026