Intervals

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 represent a fraction using a decimal expansion, i.e., we will divide the number by the 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.

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, unison, and scale in the Xenharmonic Wiki.

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.

Interval Examples

1. Harmonic series {WP}
• 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, ...
• 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 Perfect unison (PU) {XW}
• 2/1: Perfect octave (P8) {XW}
• 3/2 Perfect fifth (P5) {XW}
• 4/3: Perfect fourth(P4) {XW}
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 PU
• 9/8 M2 {XW}
• 81/64 M3 {XW}
• 4/3 P4
• 3/2 P5
• 27/16 M6 {XW}
• 243/128: M7 {XW}
• 2/1 P8
• Other intervals in the system:
• m7: (4/3) * (4/3) = 16/9 {XW}
• m3: (4/3) / (9/8) = (4/3) * (8/9) = 32/27 {XW}
• m6: (2/1) / (81/64) = (2/1) * (64/81) = 128/81 {XW}
• m2: (4/3) / (81/64) = (4/3) * (64/81) = 256/243 {XW}
• 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
• 1/1 PU
• 9/8 Large M2
• 5/4 M3
• 4/3 P4
• 3/2 P5
• 5/3 M6
• 15/8 M7
• 2/1 P8
• Other intervals in the system:
• m3: (3/2) / (5/4) = (3/2) * (4/5) = 6/5 {XW}
• m6: (2/1) / (5/4) = 2/1 * 4/5 = 8/5 {XW}
• m7: ((9/8)*(2/1)) / (5/4) = 18/8 * 4/5 = 9/5 {XW}
• Small M2: (5/4) / (9/8) = (5/4) * (8/9) = 10/9 {XW}
• m2: (4/3) / (5/4) = (4/3) * (4/5) = 16/15 {XW}
• A4: (9/8) * (5/4) = 45/32 {XW}
6. Twelve-Tone Equal Temperament (12-tet/12-edo) {WP; XW}
   The octave (2/1) is divided into 12 equal-sized intervals called the semitones. The size of the semitone is equal to the twelfth root of two:

2^1/12 ≈ 1.059

The diatonic scale may  be represented as 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

it may be represented as:

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

Cents

To compare the size of two interval frequency ratios, we will 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 twelve-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 (rounding to three decimal places, or 1/1000 cent):

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 that will play any interval, 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}
   
• (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}
• (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}
• (2/1) / (5/4)^3 = (128/64) / (125/64) = (128/64) x (64/125) = 128/125 ≈ 41.1 ¢
2. Schisma {WP; XW}
• PC/SC = (531441/524288) / (81/80) = 32805/32768 ≈ 1.95 ¢
3. Septimal comma {WP; XW}
• (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}
5. Matthew Yacavone's 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

Reference

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

Updated: April 13, 2024