A COMPOSITION PROCEDURE FOR DIGITALLY SYNTHESIZED MUSIC ON LOGARITHMIC SCALES OF THE HARMONIC SERIES Peter Lucas Hulen Wabash College Department of Music Crawfordsville, Indiana USA ABSTRACT scales, intervals of which are in the actual simple ratios of the harmonic series; 2) Create means to realize these Discrete spectral frequencies within pitched sounds occur scales as a material component of musical composition by in a pattern known to music theorists as the harmonic mathematically translating the simple-ratio frequency series. Their simple-ratio intervals do not conform to the intervals of the scales into adjusted musical intervals of musical semitones of twelve-tone equal temperament standard 12Tet so that: a) as necessary, commercial (12Tet). New procedures for composing digitally synthesis applications could be adjusted according to synthesized music have been developed applying these musically standardized user interfaces; and, b) pitches intervals to fundamental frequencies. Digital synthesis could be graphically represented for vocal or acoustic provides for exact frequency specification. In 12Tet, each performers according to standard music notation; 3) semitone is divided into 100 cents for tuning, dividing the Develop a series of scale transpositions sharing common octave into 1,200 fine increments. Some user interfaces for tones by which modulation from one to another may be synthesis correlate to the 1,200 cents per octave of 12Tet, effected; and, 4) Compose, perform and record new music as opposed to frequencies in cycles per second. Octave according to technical and aesthetic implications of frequencies being a ratio of two, with 12Tet dividing the applying the new scale and tonal system. octave into twelve equal intervals, the value of any one thus being the twelfth root of two, a logarithmic equation 2.1. No True Overtone Scale can be applied to calculate values in cents for the simple- Certain music of the 20th century was composed applying ratio intervals of the harmonic series. Synthesis a Lydian-Mixolydian scale [1], as shown in Figure 1. applications that assume 12Tet can be adjusted accordingly. New music applying these intervals has been composed and performed. 1. INTRODUCTION This concerns application of what are essentially musical Figure 1. Lydian-Mixolydian scale. scales to the composition of computer music. Intervals within the scales match the harmonic series. Their exact When this scale is applied in jazz improvisation, it is pitches are virtually impossible for ordinary acoustic sometimes called an “overtone” scale, or an “acoustic” instruments, but are easily realized through digital scale, even though its standard 12Tet pitches only synthesis. As such, works composed applying these scales approximate frequency intervals in the fourth octave of the represent a style of computer music composition. harmonic series [2]. It is still not a true overtone scale To adjust tuning through interfaces of digital synthesis because its intervals do not actually conform to those of applications that assume 12Tet, and to graphically the observable harmonic series. represent music applying these adjustments for performers Designing and building nonstandard acoustic of acoustic parts, the simple-ratio intervals of the scale instruments that would play a true overtone scale must be converted into the 1,200 cent-per-octave standard according to actual intervals of the harmonic series and of twelve-tone equal temperament (12Tet) using a then training musicians to play them would be impractical logarithmic equation. at best. Writing code or creating object-based digital synthesis patches for such intervals could be distracting 2. CHALLENGES and time consuming for those approaching the computer music studio as classically trained composers or The technical and artistic challenges behind creation of performers. Commercial synthesis applications bring their these new scales were fourfold: 1) Create true “overtone” own problems, as detailed below. 2.2. Musical Standards vs. Acoustic Reality 3. REALIZATION OF SCALE AND MUSIC The 12Tet standard is assumed for many commercial Developing frequency relationships of the harmonic series synthesis applications, and for standard musical notation. into musical scales begins with regarding the fundamental The actual intervals of the harmonic series conform neither frequency as tonic, against which other pitches of the to pitch relationships of 12Tet nor to standard musical series are arrayed intervallically. To represent intervals notation or notation applications, inasmuch as they assume musically, they are translated to values in cents using a and refer to 12Tet. logarithmic formula. Once represented musically, they can be applied in composition using digital synthesis resources 2.2.1. Synthesis interface standards and representative musical notation. Many digital synthesis applications can produce finely specified pitch relations; however, where user interfaces 3.1. A True Overtone Scale translate into a standard cents division of the 12Tet Taking the frequency of C (32.703 Hz) as a fundamental, semitone, making correct pitch adjustments so intervals 1 the frequencies of tones having the same relationships as conform to simple ratios of the harmonic series requires a harmonics in the harmonic series over that fundamental logarithmic formula. can be calculated accordingly. So, a true overtone scale based on the harmonic series 2.2.2. Music notation standards can be expressed as a list of frequencies, or as a series of Musicians can match, play or sing in concert with pitch- ratios by which those frequencies are related: adjusted musical tones that do not conform to 12Tet; however, the standard music notation they read and the HARMONIC FREQUENCY HZ RATIO ways they read it assume 12Tet. There is no standard for 1 (C1) ............................... 32.703 ..................................... 1/1 indicating the exact measurements in cents of pitch 2 (C2) ............................... 65.406 ..................................... 2/1 adjustments that deviate from 12Tet, yet some means of 3 ....................................... 98.109 ..................................... 3/2 indication must be applied for the sake of musicians who 4 (C3) ............................... 130.81 ..................................... 4/3 would perform pitch-adjusted musical materials according 5 ....................................... 163.52 ..................................... 5/4 to a notated musical score. 6 ....................................... 196.22 ..................................... 6/5 7 ....................................... 228.92 ..................................... 7/6 2.3. Means of Modulation 8 (C4) ............................... 261.62 ..................................... 8/7 9 ....................................... 294.33 ..................................... 9/8 The formal dynamics of many kinds of music depend on 10 ..................................... 327.03 ................................... 10/9 some kind of exposition, development and recapitulation 11 ..................................... 359.73 ................................. 11/10 of sonic materials. This can involve modulation from one 12 ..................................... 392.44 ................................. 12/11 series of intra-related pitch classes (scale) to another, and 13 ..................................... 425.14 ................................. 13/12 some kind of return to the initial series. In many cases such 14 ..................................... 457.84 ................................. 14/13 modulation depends on what pitch classes and motifs are 15 ..................................... 490.55 ................................. 15/14 shared between a given series and its successor. 16 (C ) ............................. 523.25 ................................. 16/15 As analogous to such types of music, it was found 5 desirable to develop transpositions of any new scales that Table 1. Overtone scale as list of frequencies and ratios would produce a network of pitches shared among all transpositions. This would depend on intervals between the 3.2. Translation into Musical Terms tonics of each scale transposition. This would facilitate modulation as a compositional procedure. There is a way to mathematically translate simple-ratio intervals of the harmonic series into musical intervals of 2.4. Application to Composition the 1,200 cents-per-octave 12Tet standard. Remembering that the logarithm of the product of two Provided that simple-ratio intervals of the harmonic series numbers is equal to the sum of the logarithms of the are mathematically translated into 12Tet pitches adjusted numbers: by certain numbers of cents, that the means of synthesis and graphic notation are adjusted accordingly, and that log xy = log x + log y, (1) theoretical pitch relationships are developed to provide for musical modulation, it would remain to compose new it follows that music applying those materials, and to hear it realized in performance. log xn = n log x. (2) Setting this equation aside for the moment, it is and consequently, known—given any octave of exactly 2:1—that when the value of an equal semitone in 12Tet is denoted by a, it n = 3986 ✕ 0.176 provides that = 702 cents. 12 a = 2, (3) So, while an equally tempered fifth, at seven semitones, has a value of 700 cents, a just fifth with a frequency ratio 3 thus, a is the twelfth root of two, or of /2 has a value of 702 cents (rounded to the