thumb|300px|12 tone equal temperament chromatic scale on , one full octave ascending, notated only with sharps.
An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.
In classical music and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament (also known as 12-tone equal temperament, ' or ', informally abbreviated as 12 equal), which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2, (<math display=inline>\sqrt[12]{2}</math> ≈ 1.05946). That resulting smallest interval, the width of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means '.
In modern times, is usually tuned relative to a standard pitch of 440 Hz, called A 440, meaning one note, A (musical note)|, is tuned to 440 hertz and all other notes are defined as some multiple of semitones away from it, either higher or lower in frequency. The standard pitch has not always been 440 Hz; it has varied considerably and generally risen over the past few hundred years.
Other equal temperaments divide the octave differently. For example, some music has been written in 19 equal temperament| and 31 equal temperament|, while the Arab tone system uses
Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the Bohlen–Pierce scale, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "pseudo-octave" in that system, into 13 equal parts.
For tuning systems that divide the octave equally, but are not approximations of just intervals, the term equal division of the octave, or ' can be used.
Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.
Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups.
General properties
In an equal temperament, the distance between two adjacent steps of the scale is the same interval. Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly spaced and would not permit transposition to different keys.) Specifically, the smallest interval in an equal-tempered scale is the ratio:
:<math>\ r^n = p\ </math>
:<math>\ r = \sqrt[n]{p\ }\ </math>
where the ratio divides the ratio (typically the octave, which is 2:1) into equal parts. (See Twelve-tone equal temperament below.)
Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent). This logarithmic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in ethnomusicology. The basic step in cents for any equal temperament can be found by taking the width of above in cents (usually the octave, which is 1200 cents wide), called below , and dividing it into parts:
:<math>\ c = \frac{\ w\ }{ n }\ </math>
In musical analysis, material belonging to an equal temperament is often given an integer notation, meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the logarithm of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced to pitch classes, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g., is 0 regardless of octave register. The MIDI encoding standard uses integer note designations.
General formulas for the equal-tempered interval
Twelve-tone equal temperament
12-tone equal temperament, which divides the octave into 12 intervals of equal size, is the musical system most widely used today, especially in Western music.
History
The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: ) in 1584 and Simon Stevin in 1585. According to F. A. Kuttner, a critic of giving credit to Zhu,
Kenneth Robinson credits the invention of equal temperament to Zhu</math> (78.0 cents)
- beta: <math display=inline>\sqrt[11]{\frac{3}{2</math> (63.8 cents)
- gamma: <math display=inline>\sqrt[20]{\frac{3}{2</math> (35.1 cents)
Alpha and beta may be heard on the title track of Carlos's 1986 album Beauty in the Beast.
Equal temperament with a non-integral number of notes per octave
While traditional equal temperaments—such as 12‑TET, 19‑TET, or 31‑TET—divide the octave into an integral number of equal parts, it is also possible to explore systems that divide the octave into a non-integral (often irrational) number. In such temperaments, the interval between successive pitches is defined by the ratio 2^(1/N), where N is not an integer. This results in irrational step sizes, meaning their multiples never exactly equal an octave.
Such tunings are of interest because, by deliberately sacrificing the octave (i.e., the second harmonic), they can yield a system that offers an improved overall approximation of other intervals in the harmonic series.
For example, in a tuning system based on 18.911‑EDO, the step size is 1200⁄18.911 ≈ 63.45 cents. Approximating the just perfect fifth (with a ratio of 3:2, or about 701.96 cents) requires about 11 steps:
- 11 steps × 63.45 cents ≈ 698.95 cents,
yielding an error of roughly 3 cents.
Similarly, for the just major third (with a ratio of 5:4, or about 386.31 cents), 6 steps are used:
- 6 steps × 63.45 cents ≈ 380.70 cents,
resulting in an error of approximately 5.61 cents.
Thus, although a perfect octave is absent, the consonance of many other intervals in these systems can be significantly higher than in integer-based equal temperaments.
Proportions between semitone and whole tone
In this section, semitone and whole tone may not have their usual 12 EDO meanings, as it discusses how they may be tempered in different ways from their just versions to produce desired relationships. Let the number of steps in a semitone be , and the number of steps in a tone be .
There is exactly one family of equal temperaments that fixes the semitone to any proper fraction of a whole tone, while keeping the notes in the right order (meaning that, for example, , , , , and are in ascending order if they preserve their usual relationships to ). That is, fixing to a proper fraction in the relationship also defines a unique family of one equal temperament and its multiples that fulfil this relationship.
For example, where is an integer, sets sets and sets The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside the circle of fifths. (This is not true in general; in 24 , the half-sharps and half-flats are not in the circle of fifths generated starting from .) The extreme cases are where and the semitone becomes a unison, and , where and the semitone and tone are the same interval.
Once one knows how many steps a semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament with the above properties (including having no notes outside the circle of fifths) divides the octave into and the perfect fifth into If there are notes outside the circle of fifths, one must then multiply these results by , the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24 , six in 72 ). (One must take the small semitone for this purpose: 19 has two semitones, one being tone and the other being . Similarly, 31 has two semitones, one being tone and the other being .)
The smallest of these families is and in particular, 12 is the smallest equal temperament with the above properties. Additionally, it makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons 12 has become the most commonly used equal temperament. (Another reason is that 12 EDO is the smallest equal temperament to closely approximate 5 limit harmony, the next-smallest being 19 EDO.)
Each choice of fraction for the relationship results in exactly one equal temperament family, but the converse is not true: 47 has two different semitones, where one is tone and the other is , which are not complements of each other like in 19 ( and ). Taking each semitone results in a different choice of perfect fifth.
Related tuning systems
Equal temperament systems can be thought of in terms of the spacing of three intervals found in just intonation, most of whose chords are harmonically perfectly in tune—a good property not quite achieved between almost all pitches in almost all equal temperaments. Most just chords sound amazingly consonant, and most equal-tempered chords sound at least slightly dissonant. In C major those three intervals are:]]
The diatonic tuning in 12-tone equal temperament can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps (or some circular shift or "rotation" of it). To be called a regular diatonic tuning, each of the two semitones () must be smaller than either of the tones (greater tone, , and lesser tone, ). The comma is implicit as the size ratio between the greater and lesser tones: Expressed as frequencies or as cents .
The notes in a regular diatonic tuning are connected in a "spiral of fifths" that does not close (unlike the circle of fifths in Starting on the subdominant (in the key of C) there are three perfect fifths in a row—–, –, and –—each a composite of some permutation of the smaller intervals The three in-tune fifths are interrupted by the grave fifth – (grave means "flat by a comma"), followed by another perfect fifth, –, and another grave fifth, –, and then restarting in the sharps with –; the same pattern repeats through the sharp notes, then the double-sharps, and so on, indefinitely. But each octave of all-natural or all-sharp or all-double-sharp notes flattens by two commas with every transition from naturals to sharps, or single sharps to double sharps, etc. The pattern is also reverse-symmetric in the flats: Descending by fourths the pattern reciprocally sharpens notes by two commas with every transition from natural notes to flattened notes, or flats to double flats, etc. If left unmodified, the two grave fifths in each block of all-natural notes, or all-sharps, or all-flat notes, are "wolf" intervals: Each of the grave fifths out of tune by a diatonic comma.
Since the comma, , expands the lesser tone into the greater tone, a just octave can be broken up into a sequence (or a circular shift of it) of 7 diatonic semitones , 5 chromatic semitones , and 3 commas Various equal temperaments alter the interval sizes, usually breaking apart the three commas and then redistributing their parts into the seven diatonic semitones , or into the five chromatic semitones , or into both and , with some fixed proportion for each type of semitone.
The sequence of intervals , , and can be repeatedly appended to itself into a greater spiral of 12 fifths, and made to connect at its far ends by slight adjustments to the size of one or several of the intervals, or left unmodified with occasional less-than-perfect fifths, flat by a comma.
Morphing diatonic tunings into EDO
Various equal temperaments can be understood and analyzed as having made adjustments to the sizes of and subdividing the three intervals—, , and , or at finer resolution, their constituents , , and . An equal temperament can be created by making the sizes of the major and minor tones (, ) the same (say, by setting , with the others expanded to still fill out the octave), and both semitones (diatonic semitone| and ) the same, then 12 equal semitones, two per tone, result. In , the semitone, , is exactly half the size of the same-size whole tones = .
Some of the intermediate sizes of tones and semitones can also be generated in equal temperament systems, by modifying the sizes of the comma and semitones. One obtains in the limit as the size of and tend to zero, with the octave kept fixed, and in the limit as and tend to zero; is of course, the case and For instance:
; and : There are two extreme cases that bracket this framework: When and reduce to zero with the octave size kept fixed, the result is a 5-tone equal temperament. As the gets larger (and absorbs the space formerly used for the comma ), eventually the steps are all the same size, and the result is seven-tone equal temperament. These two extremes are not included as "regular" diatonic tunings.
;: If the diatonic semitone is set double the size of the chromatic semitone, i.e. (in cents) and the result is with one step for the chromatic semitone , two steps for the diatonic semitone , three steps for the tones = , and the total number of steps 19 steps. The imbedded 12-tone sub-system closely approximates the historically important meantone system.
;: If the chromatic semitone is two-thirds the size of the diatonic semitone, i.e. with the result is 31 , with two steps for the chromatic semitone, three steps for the diatonic semitone, and five steps for the tone, where 31 steps. The imbedded 12-tone sub-system closely approximates the historically important meantone.
;: If the chromatic semitone is three-fourths the size of the diatonic semitone, i.e. with the result is 43 , with three steps for the chromatic semitone, four steps for the diatonic semitone, and seven steps for the tone, where 43. The imbedded 12-tone sub-system closely approximates meantone.
;: If the chromatic semitone is made the same size as three commas, (in cents, in frequency ) the diatonic the same as five commas, that makes the lesser tone eight commas and the greater tone nine, Hence for 53 steps of one comma each. The comma size / step size is exactly, or the syntonic comma. It is an exceedingly close approximation to 5-limit just intonation and Pythagorean tuning, and is the basis for Turkish music theory.
See also
- Just intonation
- Musical acoustics<br/>(the physics of music)
- Music and mathematics
- Microtuner
- Microtonal music
- Piano tuning
- List of meantone intervals
- Diatonic and chromatic
- Electronic tuner
- Musical tuning
Footnotes
References
Sources
- As cited by
Further reading
- <br/> — A foundational work on acoustics and the perception of sound. Especially the material in Appendix XX: Additions by the translator, pages 430–556, (pdf pages 451–577) (see also wiki article On Sensations of Tone)
External links
- An Introduction to Historical Tunings by Kyle Gann
- Xenharmonic wiki on EDOs vs. Equal Temperaments
- Huygens-Fokker Foundation Centre for Microtonal Music
- A.Orlandini: Music Acoustics
- "Temperament" from A supplement to Mr. Chambers's cyclopædia (1753)
- Barbieri, Patrizio. Enharmonic instruments and music, 1470–1900. (2008) Latina, Il Levante Libreria Editrice
- Fractal Microtonal Music, Jim Kukula.
- All existing 18th century quotes on J.S. Bach and temperament
- Dominic Eckersley: "Rosetta Revisited: Bach's Very Ordinary Temperament"
- Well Temperaments, based on the Werckmeister Definition
- F<small>AVORED</small> C<small>ARDINALITIES</small> O<small>F</small> S<small>CALES</small> by P<small>ETER</small> B<small>UCH</small>