System Tuning

A tuning of system is the system used to define which tones, or pitches, to use when playing music. In other words, it is the choice of number and spacing of frequency values used.

Due to the psychoacoustic interaction of tones and timbres, various tone combinations sound more or less "natural" in combination with various timbres. For example, using harmonic timbres:
  • A tone caused by a vibration twice the frequency of another (the ratio of 1:2) forms the natural sounding octave.
  • A tone caused by a vibration three times the frequency of another (the ratio of 1:3) forms the natural sounding perfect twelfth, or perfect fifth (ratio of 2:3) when octave-reduced.

More complex musical effects can be created through other relationships.[2]
The creation of a tuning system is complicated because musicians want to make music with more than just a few differing tones. As the number of tones is increased, conflicts arise in how each tone combines with every other. Finding a successful combination of tunings has been the cause of debate, and has led to the creation of many different tuning systems across the world. Each tuning system has its own characteristics, strengths and weaknesses.

Theoretical comparison

There are many techniques for theoretical comparison of tunings, usually utilizing mathematical tools such as those of linear algebra, topology and group theory.

Systems for the twelve-note chromatic scale

It is impossible to tune the twelve-note chromatic scale so that all intervals are pure. For instance, three pure major thirds stack up to 125/64, which at 1159 cents is nearly a quarter tone away from the octave (1200 cents). So there is no way to have both the octave and the major third in just intonation for all the intervals in the same twelve-tone system. Similar issues arise with the fifth 3/2, and the minor third 6/5 or any other choice of harmonic-series based pure intervals.
Many different compromise methods are used to deal with this, each with its own characteristics, and advantages and disadvantages. The main ones are:
  • Just Intonation: In just intonation, the frequencies of the scale notes are related to one another by simple numeric ratios, a common example of this being 1:1, 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, 2:1 to define the ratios for the seven notes in a C major scale. In this example, though many intervals are pure, the interval from D to A (5:3 to 9:8) is 40/27 instead of the expected 3/2. The same issue occurs with most just intonation tunings. This can be dealt with to some extent using alternative pitches for the notes. Even that, however, is only a partial solution, as an example makes clear: If one plays the sequence C G D A E C in just intonation, using the intervals 3/2, 3/4 and 4/5, then the second C in the sequence is higher than the first by a syntonic comma of 81/80. This is the infamous "comma pump". Each time around the comma pump, the pitch continues to spiral upwards. This shows that it is impossible to keep to any small fixed system of pitches if one wants to stack musical intervals this way. So, even with adaptive tuning, the musical context may sometimes require playing musical intervals that are not pure. Instrumentalists with the ability to vary the pitch of their instrument may micro-adjust some of the intervals naturally; there are also systems for adaptive tuning in software (microtuners). Harmonic fragment scales form a rare exception to this issue. In tunings such as 1:1 9:8 5:4 3:2 7:4 2:1, all the pitches are chosen from the harmonic series (divided by powers of 2 to reduce them to the same octave), so all the intervals are related to each other by simple numeric ratios.
  • Pythagorean Tuning: A Pythagorean tuning is technically a type of just intonation, in which the frequency ratios of the notes are all derived from the number ratio 3:2. Using this approach for example, the 12 notes of the Western chromatic scale would be tuned to the following ratios: 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. Also called "3-limit" because there are no prime factors other than 2 and 3, this Pythagorean system was of primary importance in Western musical development in the Medieval and Renaissance periods. As with nearly all just intonation systems, it has a wolf interval. In the example given, it is the interval between the 729:512 and the 256:243 (F♯ to D♭, if one tunes the 1/1 to C). The major and minor thirds are also impure, but at the time when this system was at its zenith, the third was considered a dissonance, so this was of no concern. See also: Shí-èr-lǜ.
  • Meantone Temperament: A system of tuning that averages out pairs of ratios used for the same interval (such as 9:8 and 10:9). The best known form of this temperament is quarter-comma meantone, which tunes major thirds justly in the ratio of 5:4 and divides them into two whole tones of equal size – this is achieved by flattening the fifths of the Pythagorean system slightly (by a quarter of a syntonic comma). However, the fifth may be flattened to a greater or lesser degree than this and the tuning system retains the essential qualities of meantone temperament. Historical examples include 1/3-comma and 2/7-comma meantone.
  • Well Temperament: Any one of a number of systems where the ratios between intervals are unequal, but approximate to ratios used in just intonation. Unlike meantone temperament, the amount of divergence from just ratios varies according to the exact notes being tuned, so that C-E is probably tuned closer to a 5:4 ratio than, say, D♭-F. Because of this, well temperaments have no wolf intervals.
  • Equal Temperament: The standard twelve-tone equal temperament is a special case of meantone temperament (extended eleventh-comma), in which the twelve notes are separated by logarithmically equal distances (100 cents): (.ogg format, 96.9KB). This is the most common tuning system used in Western music, and is the standard system used as a basis for tuning a piano. Since this scale divides an octave into twelve equal-ratio steps and an octave has a frequency ratio of two, the frequency ratio between adjacent notes is then the twelfth root of two, 21/12, or ~1.05946309.... However, the octave can be divided into other than 12 equal divisions, some of which may be more harmonically pleasing as far as thirds and sixths are concerned, such as 19 equal temperament (extended third-comma meantone), 31 equal temperament (extended quarter-comma meantone) and 53 equal temperament (extended Pythagorean tuning).
  • Tempered timbres: A timbre's partials (also known as harmonics or overtones) can be tempered such that each of the timbre's partials aligns with a note of a given tempered tuning. This alignment of tuning and timbre is a key component in the perception of consonance,[3] of which one notable example is the alignment between the partials of a harmonic timbre and a just intonation tuning. Hence, using tempered timbres, one can achieve a degree of consonance, in any tempered tuning, that is comparable to the consonance achieved by the combination of just intonation tuning and harmonic timbres. Tempering timbres in real time, to match a tuning that can change smoothly in real time, using the tuning-invariant fingering of an isomorphic keyboard, is a central component of dynamic tonality.[4]
Tuning systems that are not produced with exclusively just intervals are usually referred to as temperaments.