Deep scale property
Encyclopedia
In diatonic set theory
, the deep scale property is the quality of pitch class
collections or scales containing each interval class
a unique number of times. Examples include the diatonic scale
(including major
, natural minor, and the modes
). (Johnson 2003, p.41)
The common tone theorem describes that scales possessing the deep scale property share a different number of common tones for every different transposition
of the scale, suggesting an explanation for the use and usefulness of the diatonic collection. (ibid, p.42)
Diatonic set theory
Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and...
, the deep scale property is the quality of pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
collections or scales containing each interval class
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...
a unique number of times. Examples include the diatonic scale
Diatonic scale
In music theory, a diatonic scale is a seven note, octave-repeating musical scale comprising five whole steps and two half steps for each octave, in which the two half steps are separated from each other by either two or three whole steps...
(including major
Major scale
In music theory, the major scale or Ionian scale is one of the diatonic scales. It is made up of seven distinct notes, plus an eighth which duplicates the first an octave higher. In solfege these notes correspond to the syllables "Do, Re, Mi, Fa, Sol, La, Ti/Si, ", the "Do" in the parenthesis at...
, natural minor, and the modes
Musical mode
In the theory of Western music since the ninth century, mode generally refers to a type of scale. This usage, still the most common in recent years, reflects a tradition dating to the middle ages, itself inspired by the theory of ancient Greek music.The word encompasses several additional...
). (Johnson 2003, p.41)
The common tone theorem describes that scales possessing the deep scale property share a different number of common tones for every different transposition
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
of the scale, suggesting an explanation for the use and usefulness of the diatonic collection. (ibid, p.42)
Further reading
- Winograd, Terry. "An Analysis of the Properties of 'Deep Scales' in a T-Tone System", unpublished.
- Gamer, Carlton (1967). "Deep Scales and Difference Sets in Equal-Tempered Systems", American Society of University Composers: Proceedings of the Second Annual Conference: 113-22 and "Some Combinational Resources of Equal-Tempered Systems", Journal of Music Theory 11: 32-59.
- Browne, Richmond (1981). "Tonal Implications of the Diatonic Set" In Theory Only 5, no. 6-7: 6-10.
Source
- Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1-930190-80-8.