Diatonic scale

The modern keyboard is based on the intervallic patterns of the diatonic scale.

In music theory, a diatonic scale (from the Greek διατονικος, meaning "[progressing] through tones", also known as the heptatonia prima and set form 7-35) is a seven-note musical scale comprising five whole-tone and two half-tone steps, in which the half tones are maximally separated. Thus between each of the two half steps there are either two or three whole steps, with the pattern repeating at the octave. The term diatonic originally referred to the diatonic genus, one of the three genera of the ancient Greeks.

These scales are the foundation of the European musical tradition. The modern major and minor scales are diatonic, as are all of the so-called 'church' modes. What we now call major and minor were, during the medieval and Renaissance periods, only two of many different modes formed by taking the diatonic scale to begin on different degrees. By the start of the Baroque period, the notion of musical key was established, and major and minor scales came to dominate until at least the start of the 20th century. Some church modes survived into the early 18th century, as well as appearing occasionally in classical and 20th century music, and later in modal jazz.

Within the 12 notes of the chromatic scale, there are twelve distinct diatonic scales. The white keys on a piano map out the seven notes of one such diatonic scale, repeated in each octave. The modern musical keyboard, with its black notes grouped in twos and threes, is essentially diatonic; this arrangement not only helps musicians to find their bearings on the keyboard, but simplifies the system of key signatures compared with what would be necessary for a continuous alternation of black and white notes.

Theory of diatonic scales

Technically speaking, diatonic scales are obtained from a chain of six successive fifths in some version of meantone temperament, and resulting in two tetrachords separated by intervals of a whole tone. If our version of meantone is the twelve tone equal temperament the pattern of intervals in semitones will be 2-2-1-2-2-2-1; these numbers stand for whole tones (2 semitones) and half tones (1 semitone). The major scale starts on the first note and proceeds by steps to the first octave. In solfege, the syllables for each scale degree are "Do-Re-Mi-Fa-So-La-Ti-Do".

The natural minor scale can be thought of in two ways, the first is as the relative minor of the major scale, beginning on the sixth degree of the scale and proceeding step by step through the same tetrachords to the first octave of the sixth degree. In solfege "La-Ti-Do-Re-Mi-Fa-So-La." Alternately, the natural minor can be seen as a composite of two different tetrachords of the pattern 2-1-2-2-1-2-2. In solfege "Do-Re-Mé-Fa-So-Lé-Té-Do."

Western harmony from the Renaissance until the late 19th century is based on the diatonic scale and the unique hierarchical relationships, or diatonic functionality, created by this system of organizing seven notes. Most longer pieces of common practice music change key, which leads to a hierarchical relationship of diatonic scales in one key with those in another.

The diatonic scale has some specific properties which mark it out among seven-note scales. David Rothenberg conceived of a property of scales he called propriety, and around the same time Gerald Balzano independently came up with the same definition in the more limited context of equal temperaments, calling it coherence. Rothenberg distinguished proper from a slightly stronger characteristic he called strictly proper. In this vocabulary, there are five proper seven-note scales in 12 equal temperament. None of these are strictly proper, which means none are coherent in the sense of Balzano, but in any system of meantone tuning with the fifth flatter than 700 cents, they are strictly proper. The scales are the diatonic scale, the ascending minor scale, the harmonic minor scale, the harmonic major scale, and the locrian major scale; of these, all but the last are well-known and in fact constitute the backbone of diatonic practice when taken together.

Among these four well-known variants of the diatonic scale, the diatonic scale itself has additional properties of what has been called simplicity, because it is produced by iterations of a single generator, the meantone fifth. The scale, in the vocabulary of Erv Wilson, who seems to have been the first to consider the notion, is what is sometimes called a MOS scale.

The diatonic collection contains each interval class a unique number of times (Browne 1981 cited in Stein 2005, p.49, 49n12). Diatonic set theory describes the following properties, aside from propriety: maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity.

The following table shows the approximate ratio of the frequencies of each note.

C	C#	D	D#	E	F	F#	G	A#	A	A#	B	C
1	18/17	9/8	6/5	5/4	4/3	7/5	3/2	8/5	5/3	16/9	17/9	2
1.00	1.06	1.12	1.19	1.26	1.33	1.41	1.50	1.59	1.68	1.78	1.89	2.00

The earliest diatonic scales

The earliest claimed occurrence of diatonic tuning is in the 45,000 year-old so-called "Neanderthal flute" found at Divje Babe. Although there is no consensus that this is a musical instrument, there has been one claim that it played a diatonic scale.¹ (See Divje Babe.)

There is other evidence that the Sumerians and Babylonians used some version of the diatonic scale. This derives from surviving inscriptions which contain a tuning system and musical composition. Despite the conjectural nature of reconstructions of the piece known as the Hurrian hymn from the surviving score, the evidence that it used the diatonic scale is much more soundly based. This is because instructions for tuning the scale involve tuning a chain of six fifths so that the corresponding circle of seven major and minor thirds are all consonant-sounding, and this is a recipe for tuning a diatonic scale. See Music of Mesopotamia.

9,000 year old flutes found in China indicate the evolution, over a period of 1,200 years, of flutes having 4, 5 and 6 holes to having 7 and 8 holes, the latter exhibiting striking similarity to diatonic hole spacings and sounds .

See also

• Pitch
• Piano key frequencies
• History of music
• Prehistoric music
• Musical Acoustics
• Jiahu Site of oldest still-playable flute -- Neolithic
• Diatonic and chromatic

External links

• Diatonic Scale on Eric Weisstein's Treasure trove of Music
• Tonalsoft Encyclopedia of Microtonal Music-theory

References

• Browne, Richmond (1981). "Tonal Implications of the Diatonic Set", In Theory Only 5, nos. 1 and 2: 3-21
• Stein, Deborah (2005). Engaging Music: Essays in Music Analysis. New York: Oxford University Press. ISBN 0-19-517010-5.
• Ellen Hickmann, Anne D. Kilmer and Ricardo Eichmann, (ed.) Studies in Music Archaeology III, 2001, VML Verlag Marie Leidorf GmbH., Germany ISBN 3-89646-640-2
• Kilmer, Crocket, Brown: Sounds From Silence 1976, Bit Enki Publications, Berkeley, Calif. LC# 76-16729.
• ^1 Fink, Bob. Random Samples, Science April 1997, vol 276 no 5310 pp 203-205 (available online).

Further reading

• Balzano, Gerald J. (1980). "The Group Theoretic Description of 12-fold and Microtonal Pitch Systems", Computer Music Journal 4 66-84.
• Balzano, Gerald J. (1982). "The Pitch Set as a Level of Description for Studying Musical Pitch Perception", Music, Mind, and Brain, Manfred Clynes, ed., Plenum press.
• Clough, John (1979). "Aspects of Diatonic Sets", Journal of Music Theory 23: 45-61.
• Franklin, John C. (2002). "Diatonic Music in Greece: a Reassessment of its Antiquity", Mnemosyne 56.1, 669-702 [1]
• Gould, Mark (2000). "Balzano and Zweifel: Another Look at Generalised Diatonic Scales", "Perspectives Of New Music" 38/2, 88-105
• Johnson, Timothy (2003). Foundations Of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1-930190-80-8.
• Kilmer, A.D. (1971) "The Discovery of an Ancient Mesopotamian Theory of Music'". Proceedings of the American Philosophical Society 115, 131-149.
• David Rothenberg (1978). "A Model for Pattern Perception with Musical Applications Part I: Pitch Structures as order-preserving maps", Mathematical Systems Theory 11 199-234 [2]