BAIN MUSC 726T
Tuning Theory

Intervals

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

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., the reciprocal of 4/3 is calculated:

1 / (4/3) = 1 * (3/4) = 3/4

For more information, see the Wikipedia article multiplicative inverse.

Interval
In tuning theory, an interval is the distance between two pitches, which we will describe as a ratio (see below) between two frequencies.

For more information, see the Xenharmonic Wiki article interval.

Fraction
In mathematics, a fraction describes the number of parts in a whole. For example, a string may be divided into two equal parts: 1/2 + 1/2 = 1. The top number in a fraction is called the numerator (n). The bottom number is called the denominator (d). A fraction is usually represented as the quotient (q), where q = n/d, and n and d are integers. Non-integer values for n and d are also possible.

Ratio
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.

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

Examples. 880/440 may be reduced to 2/1, 660/440 to 3/2, 550/440 to 5/4, 600/500 to 6/5, 4/2 to 2/1, 6/4 to 3/2, 12/9 to 4/3, etc.

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

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) 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 point of reference. For example, it may be used to represent the unison interval {XW} 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 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 usually expressed as ratios within the 1/1 to 2/1 octave; i.e., as fractions under simplification and 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.

Complementary intervals
The octave complement of an interval is the difference between that interval and an octave. For example, using simple algebra, the octave complements of 3/2, 5/4, and 9/8, respectively:

(3/2) * x = 2/1; x = (2/1) * (2/3) = 4/3

(5/4) * x = 2/1; x = (2/1) * (4/5) = 8/5

(9/8) * x = 2/1; x = (2/1) * (8/9) = 16/9

For more information, see the Xenharmonic Wiki article octave complement. See also: fifth complement.

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.

Examples. Superparticular ratios include:

2/1, 3/2, 4/3, 5/4, 6/5, 7/6, 8/7, 9/8, 10/9, etc.

For more information, see the Wikipedia article Superparticular ratio.

Pythagorean means
In mathematics, the three classical Pythagorean means are the: arithmetic mean, geometric mean, and harmonic mean. According to Boethius (c. 480-524), the ancient Greek mathematician Archytus (4th c. BCE) proved that a superparticular ratio cannot be divided into two equal rational parts by a geometric mean. That is, superparticular intervals are inherently indivisible in half. As such, the arithmetic and harmonic means were justified as a harmonious way to fill gaps in musical scales.

For more information, see the Wikipedia article Pythagorean means.

Basic Interval Models

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

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}

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)

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 12-edo, 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 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

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

Or we may use decimal expansions to specify the 12-tet chromatic scale:

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, here is the chromatic scale notated using EDO-step notation (n\n-EDO), where n is the nth step in n-EDO.

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

EDO is an abbreviation for equal division of the octave. For more information, see the Xenharmonic Wiki article EDO. For other important EDOs, see Marc Sabat's article Introduction to EDO tunings on the Ableton website.

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.

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

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)

Lesser 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¢

Schisma (PC/SC) {WP; XW}

Difference between the Pythagorean comma and syntonic comma
(531441/524288) / (81/80) = 32805/32768 ≈ 1.95¢

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¢

Limma {WP}
Kleisma {WP; XW}

Some important EDO intervals

1/4 tone = 200/4 = 50¢
1/5 tone = 200/5 ≈ 40¢
1/6 tone = 200/6 ≈ 33¢

etc.

Interval Names

The following interval galleries will be helpful when we are learning historical tunings and temperaments:

Kyle Gann, Anatomy of an Octave – A list of over 1000 intervals within the octave {Gann 2019} (Gann 2019)
Manuel Op de Coul, Scala: List of intervals {HFF} (See also: Scala)
Wikipedia, List of pitch intervals {WP}
Xenharmonic Wiki

Gallery of Just Intervals {XW}
List of Superparticular Intervals {XW}
List of Overtones {XW}

Again, 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}

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: Marcy 5, 2026

Reginald Bain | University of South Carolina | School of Music
https://reginaldbain.com/vc/musc726t/