Musical scales and mathematics

Summary: Musical scales have distinct mathematical properties and patterns.

Western music is based on a system of 12 pitches within each octave. The interval between adjacent pitches in this 12-tone system is called a “half step” or “semitone.” Pitches separated by two successive semitones are said to be at the interval of a “whole step,” or a “tone.” Based on a variety of theoretical underpinnings, the concept and sound of tones and semitones have evolved throughout the history of Western music. In modern music practice, a uniform division of the octave into 12 equally spaced pitches, known as “equal temperament,” holds sway. Scales are arrangements of half and whole step intervals in the octave. Denoting a half step as h and a whole step as w, the familiar diatonic major scale is defined by the sequence wwhwwwh. The diatonic natural minor scale is whwwhww. Beginning these patterns from each of the 12 pitches results in 24 distinct diatonic scales. This suggests a set-theoretic description by which each major scale can be represented as a transposition (in algebra this would be called a “translation”) of the set of pitches C, D, E, F, G, A, B, and C. In the twentieth century, such mathematical formalisms have led to the conceptualization of non-diatonic scales with special transposition properties.

Octave Equivalence

The concept of octave (the musical interval between notes with frequencies that differ by a factor of two) is fundamental to understanding musical scales. In Western music notation, pitches separated by an octave are given the same note name. The piano keyboard provides a visual representation of this phenomenon. Counting up the white keys from middle C as “1,” the eighth key in the sequence is again called C. This eight-note distance explains the etymology of the word “octave.” The perception and conceptualization of such pairs of pitches as higher or lower versions of the same essential pitch is called “octave equivalence.” Octave equivalence is thought to be common to all systematic musical cultures. Evidence of octave equivalence is found in ancient Greek and Chinese music. Recent psycho-acoustic research suggests a neurological basis for octave equivalence in auditory perception.

The mathematical explanation of octave equivalence comes from the fact that the sound of a musical pitch is a combination of periodic waveforms that can be modeled as sinusoidal functions of time. In the two periodic functions, f(t)=sin(t) and g(t)=sin(2t), with frequencies 2π and π, every peak of the lower frequency function coincides with a peak of the high-frequency function. In sonic terms, this is the highest degree of consonance possible for two pitches of different frequencies.

History of Scales

As Western music developed from the Middle Ages through the twentieth century, the central construct was the diatonic scale. This arrangement spans an octave with seven distinct pitches arranged in a combination of five whole steps and two half steps. Interestingly, the pattern of intervals (and not the absolute pitch of the starting note) was the only distinguishing feature of scales until the rise of tonal harmony in the seventeenth century. Pitch-specific examples help illustrate the interval patterns.

The diatonic scale traces its origins to the ancient Greek genus of the same name, referring to a particular tuning of the four-stringed lyre (tetrachord) consisting of two whole steps and one half step in descending succession. An example of this tuning can be constructed with the pitches A, G, F, and E. Concatenization of two diatonic tetrachords [A-G-F-{E]-D-C-B} produces the pitches of the diatonic scale (the piano white keys). In medieval European musical practice, the distinct Church Modes (such as Lydian or Phrygian) developed from the diatonic scale by the assignment of a tonal anchor or final tone. For example, the Dorian mode is characterized by the sequence of ascending half and whole steps in the diatonic scale whwwwhw; for example D-E-F-G-A-B-C-D, while the Phrygian mode is hwwwhww: E-F-G-A-B-C-D-E. The diatonic major scale wwhwwwh (C-D-E-F-G-A-B-C) came into widespread use in the seventeenth century. The diatonic natural minor scale is whwwhww (A-B-C-D-E-F-G-A).

Intervals, Ratios, and Equal Temperament

The simplest musical interval is the octave. The frequency ratio between pitches separated by an octave is 2:1. The interval of a perfect fifth has frequency ratio 3:2. Using these two ratios, pitches and corresponding intervals for the diatonic scale can be assigned according to Pythagorean tuning. Simpler diatonic scales based on ratios of small integers are known as “just tunings.” Western music in the modern era uses a symmetric assignment of intervals known as “equal temperament.” In equal temperament, the 12 half steps that comprise the frequency doubling octave each have frequency ratio 2^1/12≈1.0595. For these three tuning schemes, frequency ratios relative to the starting pitch and intervals between adjacent scale notes are illustrated and compared in Table 1. For each intonation, the first row gives the frequency ratio from the tonic C to the given note. The second row in each case gives the frequency ratio between adjacent diatonic pitches.

Modern Scales

In contrast to the idiosyncratic pattern of intervals that comprise the diatonic scales, the chromatic scale hhhhhhhhhhh is perfectly symmetric. In particular, the set of pitches that form the chromatic scale is unchanged by transposition—there is only one set of pitches with this intervallic pattern. This set of pitches is referred to as having order one. The elements of the pitch set forming a diatonic scale, which generates 12 diatonic scales by transposition, has order 12. This point of view suggests other scales of interest with respect to transposition. The set of six pitches in a whole-tone scale wwwwww (for example, C-D-E-F≈-G≈-A≈-C) are unchanged by transposition by an even number of half steps. A transposition by an odd number of half steps results in the whole tone scale containing the remaining six pitches (for example, C≈-D≈-F-G-A-B-C≈). Thus, the set of pitches in the whole-tone scale has order two. Whole-tone scales are a characteristic feature in much of the music of Claude Debussy.

The twentieth-century composer and music theorist Olivier Messiaen codified a number of eight-tone “scales of limited transposition.” Among these are the order three scales hwhwhwhw and whwhwhwh, which are called “octatonic scales” in the music of Stravinsky and sometimes referred to as “diminished scales” in jazz performance. It can be seen that transposition by one and two half steps produce new diminished scales, but transposition by three half steps leaves the original set of pitches unchanged.

Bibliography

Grout, Donald Jay. A History of Western Music. New York: Norton, 1980.

Hanson, Howard. Harmonic Materials of Modern Music: Resources of the Tempered Scale. New York: Appleton-Century-Crofts, 1960.

Johnson, Timothy. Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Lanham, MD: Scarecrow Press, 2008.

Pope, Anthony. “Messiaen’s Musical Language: An Introduction.” In The Messiaen Companion. Edited by Peter Hill. Portland, ME: Amadeus Press, 1995.

Sundberg, Johan. The Science of Musical Sounds. San Diego, CA: Academic Press, 1991.