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A musical scale is a discrete set of pitches used in making or describing music. Typically a scale has an interval of repetition, which is normally the octave. This means that for any pitch in the scale, we have also an equivalent pitch an octave above and an octave below it. While the limits of human hearing are finite, matters are somewhat simplified if we ignore that fact, as is usually done in discussions of theory though of course never in practice. Because we are often interested in the relations or ratios between the pitches rather than the precise pitches themselves in describing a scale, it is usual to refer all the scale pitches in terms of their ratio from a particular pitch, which is given the value of one (often written 1/1 when discussing just intonation.) This note can be, but is not necessarily, a note which functions as the tonic of the scale. For comparison with the current standard tuning cents are often used. See also logarithmic scale.
The most important scale in the Western tradition is the diatonic scale, but the scales used and proposed in various historical eras and parts of the world have been many and varied. Scales may broadly be classed as scales of just intonation, tempered scales, and practice-based scales. A scale is in just intonation if the ratios between the frequencies for all degrees of the scale are either ratios of small integers, or obtained by a succession of such ratios. It is tempered if it represents an adjustment, or tempering, of just intonation. It is practice-based if it simply reflects musical practice, as for instance various measurements of the tuning of a gamelan might do.
Pythagorean tuning is tuning based only on the perfect consonances, the (perfect) octave, perfect fifth, and perfect fourth. Thus the major third is considered not a third but a ditone, literally "two tones", and is 81:64 = (9:8)2, rather than the independent and harmonic just 5:4, directly below. A whole tone is a secondary interval, being derived from two perfect fifths, (3:2)2/2 = 9:8.
If we take the ratios constituting a scale in just intonation, there will be a largest prime number to be found among their prime factorizations. This is called the prime limit of the scale; a scale which uses only the primes 2, 3 and 5 is called a 5-limit scale. Below is a typical example of a 5-limit justly tuned scale, one of the scales Johannes KeplerThis article is about Johannes Kepler the astronomer. For the planned planet-finding space telescope, see Kepler Space Mission. Johannes Kepler ( December 27, 1571 November 15, 1630), a key figure in the scientific revolution, was a German astronomer, mat presents in his Harmonice mundi or Harmonics of the World of 1619, in connection with planetary motion. The same scale was given in transposed form by Alexander Malcolm in 1721 and theorist Jose Wuerschmidt in the last century and is used in an inverted form in the music of northern India. American composer Terry RileyTerry Riley (born 1935), is an American minimalist composer. A pupil of the late Pandit Pran Nath, as were La Monte Young and Marian Zazeela, he has composed in just intonation and much of his music is based in improvisation. He has collaborated with Paul also made use of the inverted form of it in his ‘‘Harp of New Albion’’. Despite this impressive pedigree, it is only one out of large number of somewhat similar scales.
| 0 | 1:1 | unisonIn music, a unison is an interval, the ratio of 1:1 or 0 halfsteps and zero cents. Two tones in unison are considered to be the same pitch, but are still perceivable as coming from separate sources. The unison is considered the most consonant interval whi |
| 1 | 135:128 | major chroma |
| 2 | 9:8 | major secondIntervals The musical interval of a major second — also called a whole-tone — is the relationship between the first note (the root or tonic) and the second note in a major scale (and also a minor scale). It is the inversion of the minor seventh. It is abb |
| 3 | minor thirdThe musical interval of a minor third is the relationship between the first note (the root or tonic) and the third note in a minor scale. It is the inversion of the Major sixth. It can be produced by starting on a high note and playing the third below or | |
| 4 | major thirdIntervals The musical interval of a Major third is the relationship between the first note (the root or tonic) and the third note in a major scale. It is the inversion of the minor sixth. It is abbreviated as M3 . It can be produced by starting on a high | |
| 5 | perfect fourth | |
| 6 | 45:32 | diatonic tritone |
| 7 | perfect fifth | |
| 8 | 8:5 | minor sixth |
| 9 | 27:16 | Pythagorean major sixth |
| 10 | 9:5 | minor seventh |
| 11 | 15:8 | major seventh |
| 12 | 2:1 | octave |
(In theory unisons and octaves and their multiples are also "perfect" but this terminology is rarely used.)
To calculate the frequency of a note in a scale given in terms of ratios, the frequency ratio is multiplied by the frequency we associate to the unison, which will often be the tonic frequency. For instance, with a tonic of A4 (A natural above middle C), the frequency is 440 Hz, and a justly tuned fifth above it (E5) is simply
440*(3/2) = 660 Hz.
The just major third, 5:4 and minor third, 6:5, are a syntonic comma, 81:80, apart from their pythagorean equivalents 81:64 and 32:27 respectively. According to Carl Dahlhaus (1990, p.187), "the dependent third conforms to the Pythagorean, the independent third to the harmonic tuning of intervals."