Some terminology used in list:
- In music, the prime limit
In music theory, limit is any of a variety of methods used to characterize the harmonies found in a piece of music, genre of music, or by extension, the harmonies that can be made with a particular scale or class of scales. The term was introduced by Harry Partch, who used it to give an upper...
(henceforth referred to simply as the limit) is a number measuring the harmony of an intervalIn music theory, the term interval describes the relationship between the pitches of two notes.Intervals may be described as:* vertical if the two notes sound simultaneously* linear , if the notes sound successively....
. The lower the number, the more consonantIn music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance — considered unstable...
the interval is considered to be. It is defined as the largest prime numberIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
occurring in the factorizationsIn number theory, integer factorization or prime factorization is the breaking down of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
of the numerator and denominator of the frequency ratio. The limit of the just perfect fourthThe perfect fourth is a musical interval which spans four scale degrees. It consists of the note and the note five semitones above it on the musical scale. For example, the interval between a C and the next F above it is a perfect fourth; similarly the interval between a G and the next C above...
(4 : 3) is 3, but the just minor tone (10 : 9) has a limit of 5, because 9 can be factorized into 3×3, and 10 into 2×5.
Discussion
Ask a question about 'List of musical intervals' Start a new discussion about 'List of musical intervals' Answer questions from other users |
| Code |
Legend |
| E |
12 tone equal temperament. |
| Q |
24 tone equal temperament, or Arab tone system The modern Arab tone system, or system of musical tuning, is based upon the theoretical division of the octave into twenty-four equal divisions or 24-tone equal temperament , the distance between each successive note being a quarter tone . Each tone has its own name not repeated in different...
. |
| 2 |
2-limit tones (only fundamentalThe fundamental tone, often referred to simply as the fundamental and abbreviated f 0 or F 0, is the lowest frequency in a harmonic series....
and octaveIn music, an octave , is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon which has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." It may be derived from the...
s). |
| 2 3 |
3-limit just intonation, or Pythagorean. |
| 2 3 5 |
5-limit (not 3-limit) just intonation, or just. |
| 2 3 5 7 |
7-limit (not 5-limit) just intonation, or septimal. |
| 2 3 5 7 11 |
11-limit (not 7-limit) just intonation, or undecimal. |
| 2 3 5 7 11 13 |
13-limit (not 11-limit) just intonation, or tridecimal. |
| U |
A unit of measurement. |
Some terminology used in list:
- In music, the prime limit
In music theory, limit is any of a variety of methods used to characterize the harmonies found in a piece of music, genre of music, or by extension, the harmonies that can be made with a particular scale or class of scales. The term was introduced by Harry Partch, who used it to give an upper...
(henceforth referred to simply as the limit) is a number measuring the harmony of an intervalIn music theory, the term interval describes the relationship between the pitches of two notes.Intervals may be described as:* vertical if the two notes sound simultaneously* linear , if the notes sound successively....
. The lower the number, the more consonantIn music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance — considered unstable...
the interval is considered to be. It is defined as the largest prime numberIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
occurring in the factorizationsIn number theory, integer factorization or prime factorization is the breaking down of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
of the numerator and denominator of the frequency ratio. The limit of the just perfect fourthThe perfect fourth is a musical interval which spans four scale degrees. It consists of the note and the note five semitones above it on the musical scale. For example, the interval between a C and the next F above it is a perfect fourth; similarly the interval between a G and the next C above...
(4 : 3) is 3, but the just minor tone (10 : 9) has a limit of 5, because 9 can be factorized into 3×3, and 10 into 2×5. There exists another type of limit, the odd limit, which differs slightly from the prime limit, but is not used here.
- Equal-tempered refers to 12-tone equal temperament
Equal temperament is a musical temperament, or a system of tuning in which every pair of adjacent notes has an identical frequency ratio. In equal temperament tunings, an interval — usually the octave — is divided into a series of equal steps...
with intervals corresponding to 100 centThe cent is a logarithmic unit of measure used for musical intervals. Typically cents are used to measure extremely small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is much too small to be heard between successive...
multiples (e.g., 100, 200, 300, etc.).
- Pythagorean
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. Its name comes from medieval texts which attribute its discovery to Pythagoras, but its use has been documented as long ago as 3500 B.C. in Babylonian texts...
means 3-limit just intonationIn music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series....
—a ratio of numbers with prime factors no higher than three.
- Just means 5-limit just intonation—a ratio of numbers with prime factor
In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization....
s no higher than five.
- Similarly, septimal, undecimal, tridecimal, and septendecimal mean, respectively, 7, 11, 13, and 17-limit just intonation.
- By definition every tone in a 3-limit unit can also be part of a 5-limit tuning and so on. By sorting the limit column all tones of that limit can be brought together (tip: sort backwards by clicking the button twice).
- Since the table is sortable, you can also sort the table by frequency ratio, by cents or alphabetically.
List
List of musical intervals
| Cents |
Freq. Ratio |
Factors |
Interval Name |
E |
Q |
2 |
3 |
5 |
7 |
11 |
13 |
U |
|
| 1 : 1 |
1 : 1 |
|
E |
Q |
2 |
3 |
5 |
7 |
11 |
13 |
|
|
| 4375 : 4374 |
| |
|
|
|
|
|
7 |
11 |
13 |
|
|
| 2401 : 2400 |
| |
|
|
|
|
|
7 |
11 |
13 |
|
|
| 21/1200 : 1 |
|
| |
|
|
|
|
|
|
|
U |
|
| 2 1/1000 : 1 |
|
| |
|
|
|
|
|
|
|
U |
|
|
| 38·5 : 215 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 101/1000 : 1 |
|
| |
|
|
|
|
|
|
|
U |
|
| 225 : 224 |
| |
|
|
|
|
|
7 |
11 |
13 |
|
|
| 15625 : 15552 |
56 : 26·35 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 1728:1715 |
26·33 : 5·73 |
|
|
|
|
|
|
|
|
|
|
|
| 126 : 125 |
2·32·7 : 53 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 121 : 120 |
112 : 23·3·5 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 21/72 : 1 |
|
| |
|
|
|
|
|
|
|
U |
|
| 2048 : 2025 |
| |
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 81 : 80 |
34 : 24·5 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 21/53 : 1 |
|
| |
|
|
|
|
|
|
|
U |
|
| 312 : 219 |
312 : 219 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 64 : 63 |
26 : 32·7 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 21/41 : 1 |
|
| |
|
|
|
|
|
|
|
U |
|
| 56 : 55 |
23·7 : 5·11 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 50 : 49 |
2·52 : 72 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 49 : 48 |
72 : 24·3 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 21/31 : 1 |
|
| |
|
|
|
|
|
|
|
U |
|
| 128 : 125 |
27 : 53 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 36 : 35 |
22·32 : 5·7 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 21/24 : 1 |
|
|
|
Q |
|
|
|
|
|
|
|
|
| 25 : 24 |
52 : 23·3 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 21 : 20 |
3·7 : 22·5 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 256 : 243 |
28 : 35 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 135 : 128 |
33·5 : 27 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 21/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
U |
|
| 16 : 15 |
24 : 3·5 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 2187 : 2048 |
37 : 211 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 181/19 : 51/19 |
|
|
|
|
|
|
|
|
|
|
|
|
| 15 : 14 |
3·5 : 2·7 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 27 : 25 |
33 : 52 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 23/24 : 1 |
|
|
|
Q |
|
|
|
|
|
|
|
|
| 12 : 11 |
22·3 : 11 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 11 : 10 |
11 : 2·5 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 21/7 : 1 |
|
| |
|
|
|
|
|
|
|
|
|
| 65536 : 59049 |
216 : 310 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 10 : 9 |
2·5 : 32 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 22/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 9 : 8 |
32 : 23 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 8 : 7 |
23 : 7 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 21/5 : 1 |
|
| |
|
|
|
|
|
|
|
|
|
| 7 : 6 |
7 : 2·3 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 32 : 27 |
25 : 33 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 23/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 6 : 5 |
2·3 : 5 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 11 : 9 |
11 : 32 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 27/24 : 1 |
|
|
|
Q |
|
|
|
|
|
|
|
|
| 16 : 13 |
24 : 13 |
|
|
|
|
|
|
|
|
13 |
|
|
| 5 : 4 |
5 : 22 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 24/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 81 : 64 |
34 : 26 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 14 : 11 |
2·7 : 11 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 9 : 7 |
32 : 7 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 4 : 3 |
22 : 3 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 25/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 27 : 20 |
33 : 22·5 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 11 : 8 |
11 : 23 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 7 : 5 |
7 : 5 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 26/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 10 : 7 |
2·5 : 7 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 16 : 11 |
24 : 11 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 40 : 27 |
23·5 : 33 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 27/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 3 : 2 |
3 : 2 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 14 : 9 |
2·7 : 32 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 11 : 7 |
11 : 7 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 128 : 81 |
27 : 34 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 28/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 8 : 5 |
23 : 5 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 13 : 8 |
13 : 23 |
|
|
|
|
|
|
|
|
13 |
|
|
| 217/24 : 1 |
|
|
|
Q |
|
|
|
|
|
|
|
|
| 18 : 11 |
2·32 : 11 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 5 : 3 |
5 : 3 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 29/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 27 : 16 |
33 : 24 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 12 : 7 |
22·3 : 7 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 7 : 4 |
7 : 22 |
|
|
|
|
|
|
7 |
11 |
13 |
|
|
| 16 : 9 |
24 : 32 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 210/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 9 : 5 |
32 : 5 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 20 : 11 |
22·5 : 11 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 11 : 6 |
11 : 2·3 |
|
|
|
|
|
|
|
11 |
13 |
|
|
| 221/24 : 1 |
|
|
|
Q |
|
|
|
|
|
|
|
|
| 15 : 8 |
3·5 : 23 |
|
|
|
|
|
5 |
7 |
11 |
13 |
|
|
| 211/12 : 1 |
|
|
E |
Q |
|
|
|
|
|
|
|
|
| 243 : 128 |
35 : 27 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 2 : 1 |
2 : 1 |
|
E |
Q |
2 |
3 |
5 |
7 |
11 |
13 |
|
|
| 3 : 1 |
3 : 1 |
|
|
|
|
3 |
5 |
7 |
11 |
13 |
|
|
| 4 : 1 |
22 : 1 |
|
E |
Q |
2 |
3 |
5 |
7 |
11 |
13 |
|
External links