Table of Contents Introduction & Key Terms................................................................................1 Chapter I. Heptatonic Modes.............................................................................3 Section 1.1: The Church Mode Set..............................................................3 Section 1.2: The Melodic Minor Mode Set...................................................10 Section 1.3: The Neapolitan Mode Set........................................................16 Section 1.4: The Harmonic Major and Minor Mode Sets...................................21 Section 1.5: The Harmonic Lydian, Harmonic Phrygian, and Double Harmonic Mode Sets..................................................................26 Chapter II. Pentatonic Modes..........................................................................29 Section 2.1: The Pentatonic Church Mode Set...............................................29 Section 2.2: The Pentatonic Melodic Minor Mode Set......................................34 Chapter III. Rhythmic Modes..........................................................................40 Section 3.1: Rhythmic Modes in a Twelve-Beat Cycle.....................................40 Section 3.2: Rhythmic Modes in a Sixteen-Beat Cycle.....................................41 Applications of the Generative Modal Framework..................................................45 Bibliography.............................................................................................46 O1 O Introduction Western musicians, composers, and music theorists have explored the concept of modal- ity since antiquity. From ancient Greek music to modern jazz, modes have played an important and diverse array of roles in a myriad of Western musical traditions and genres. Studies in the psychology of music perception have shown that the variable of mode correlates with Western listeners’ perception of emotional meaning in music, and that this correlation is significant for both musicians and nonmusicians.1 Despite the prevalence of modality in Western music, no extant music theory text offers a comprehensive and accessible delineation of the mathematical patterns found in musical modes. Although music theorists and mathematicians have explored the world of modes in the language of formal logic and set theory, musicians and composers have been largely (and regrettably) excluded from these fascinating discussions due to the advanced technical language employed therein. When creators of music are shut out from discussions about music, however abstract, entire veins of potential for artistic innovation are inevitably left unexplored. The purpose of this paper is to introduce a musically literate (but not necessarily mathematically literate) audience to a colorful world of advanced modal concepts by presenting these concepts clearly and thoroughly in the language of Western music theory. Fundamentally, the generative modal framework outlined in this paper is a procedural framework for generating new rhythmic and tonal possibilities—in the form of mode sets and supersets—through pattern recognition, recombination, and analogy. This framework does not suggest a universal underlying musical grammar similar to the universal grammar proposed in Noam Chomsky’s linguistic theory of generative grammar. The generative modal framework has little to do with musical grammar, except insofar as it assumes equal spacing of pitches and beats. It is neither a prescriptive nor descriptive account of any culture- or genre-specific use of modes. Rather, this framework is intended to shed light on a series of beautiful mathematically-based rhythmic and tonal resources for musicians and composers to add to their musical vocabularies and use at their discretion. Most importantly, a complete understanding of the framework must include knowledge of how to use its techniques to generate and explore new rhythmic and tonal resources on one’s own. The countless potential applications of the procedural framework outlined in this paper are far from exhausted in the discussions that follow, as this paper aims only to point the way towards a generative framework for understanding musical modes. The vast majority of musical material available through the framework lies outside the scope of this paper, and the reader is encouraged to explore these many avenues using the tools and techniques outlined in the follow- ing chapters. 1 D. Ramos, J. L. O. Bueno, and E. Bigand, "Manipulating Greek Musical Modes and Tempo Affects Perceived Musical Emotion in Musicians and Nonmusicians," Brazilian Journal of Medical and Biological Research 44, no. 2 (February 2011): 165-72. 2 Key Terms Validity In this paper, the term valid applies to any scale all of whose scale degrees are enhar- monically distinct. All common scales in Western music are valid. The scale spelled 1-2-3-♯4-♭5- 6-7 is an example of an invalid scale. This scale’s fourth and fifth scale degrees are enhar- monically equivalent, rendering the scale essentially a six-tone (hexatonic) scale written as a seven-tone (heptatonic) scale. The concept of validity applies only to scales in twelve-tone equal temperament (12ET). Scale, scale family, mode, and mode set The word scale denotes a set of pitches identified by their intervallic relationship to a tonic pitch. The scale family of scale s is the set of every valid scale of a particular size (e.g. heptatonic) that contains every pitch in s. For example, the heptatonic scale family for the scale spelled 1-2-3-4-5 consists of the scales 1-2-3-4-5-6-7, 1-2-3-4-5-6-♭7, 1-2-3-4-5-♭6-7, and 1-2-3- 4-5-♭6-♭7, or Ionian, Mixolydian, Harmonic major, and Melodic major, respectively. ♭6-7♭ ♭Ionian♭ ♭6-♭7♭ ♭Mixolydian♭ ♭♭6-7♭ ♭Harmonic major♭ ♭♭6-♭7♭ ♭Melodic major♭ Table 0.1. The 1-2-3-4-5 heptatonic scale family. The term mode set denotes a set of scales that all share the same interval structure yet all have a different tonic pitch. The word mode refers to an individual scale within a mode set. Other Notes About Terminology Most mode sets are named after a particular mode contained within the set. In general, the first letter of the names of specific modes (e.g. Melodic minor, Harmonic major) is capitalized, while the first letter of the names of mode sets (e.g. melodic minor mode set, harmonic major mode set) is lowercase. Because absolute pitch values are irrelevant to the generative modal framework, the cycle of fifths is represented here using scale degrees instead of note names, so that C becomes 1, D becomes 2, A♭ becomes ♭6, and so on. Interval structures in this paper are written in ascending order using numerals, where 1 indicates a half step, 2 a whole step, 3 a minor third, and so on. For example, the interval structure of the natural minor (Aeolian) scale is 2122122. 3 I. Heptatonic Modes This chapter will cover the derivation of a series of four generations of heptatonic mode sets in 12ET, starting with the most ubiquitous mode set in Western music—the diatonic church mode set—and moving on to increasingly rarer, more jagged mode sets. We can visualize this “family” of mode sets as a family tree: Figure 1.0.1. The first four generations in the 12ET heptatonic/pentatonic modal family tree. While this particular visualization applies only to heptatonic and pentatonic mode sets in 12ET, the reasoning behind it can be applied to mode sets of any size in any equal-tempered tuning system (or, in the case of rhythmic modes, any equally-subdivided beat cycle—more on this in Chapter III). Although only four generations are shown here, a theoretically infinite number of generations can be derived using the methods introduced in the following sections. 1.1. The Church Mode Set The advantage of beginning our exploration of the generative framework with the church mode set lies not only in its familiarity to most Western musicians, but also in two different ways it can be plucked directly out of the framework of 12ET: stacked fifths and maximal evenness. 4 By starting on the tonic and traversing the cycle of fifths clockwise until seven scale degrees are covered, we obtain the scale degree set 1-5-2-6-3-7-♯4. In ascending numerical order, 1-2-3-♯4- 5-6-7 spells Lydian. Figure 1.1.1. Derivation of Lydian by stacked fifths. Conversely, by starting on the tritone and traversing the cycle of fifths clockwise until seven scale degrees are covered, we obtain the scale degree set ♭5-♭2-♭6-♭3-4-♭7-1. In ascending numerical order, 1-♭2-♭3-4-♭5-♭6-♭7 spells Locrian. Figure 1.1.2. Derivation of Locrian by stacked fifths. 5 The same method can be used to derive the other five church modes simply by shifting the initial and final scale degree: Figure 1.1.3. Derivation of Ionian, Mixolydian, and Dorian by stacked fifths. 6 Figure 1.1.4. Derivation of Aeolian and Phrygian by stacked fifths. Numerous theorists have pointed out the possibility of deriving of the church mode set— particularly the Lydian mode—by stacked fifths. Most prominently, George Russell used this derivation as the foundation of his Lydian Chromatic Concept of Tonal Organization, one of the most influential jazz theory texts of all time.2 In x-tone equal temperament, where x is any positive integer, the precise frequency ratio f(n) of the nth interval in the chromatic scale is given by the following formula: f(n) = 2n/x For example, the frequency ratio of the seventh interval in 12ET (the 12ET approximation of a just perfect fifth with frequency ratio 1.5) is 27/12, or approximately 1.498. By comparing