Towards a Generative Framework for Understanding Musical Modes

Home , Acoustic scale, Altered scale, Half diminished scale, Harmonic Scale, Lydian augmented scale, Phrygian mode

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 modality 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 following 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.

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.

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.

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.

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.

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 the precise frequency ratios of the intervals in 7ET to those in 12ET, each 7ET interval can be mapped onto its nearest approximation in 12ET. In the following diagram, line segments connect each 7ET interval ratio to its 12ET approximation. The seven scale degrees associated with these seven 12ET interval ratios are given in parentheses.

2 George Russell, The Lydian Chromatic Concept of Tonal Organization (Brookline: Concept Publishing Co., 1953), 14.

Figure 1.1.5. Dorian is the scale in 12ET that best approximates a 7ET chromatic scale.

The Dorian mode (1-2-♭3-4-5-6-♭7) represents the best 12ET approximation of a 7ET chromatic scale. Because the other six church modes exhibit the same interval structure as Dorian, their interval distributions are also maximally even. The church mode set is therefore the maximally even heptatonic mode set in 12ET.

Many scholars, most notably John Clough and Jack Douthett in their paper on maximally even sets, have explored maximal evenness and related properties in pitch class sets.3

Its unity on the cycle of fifths and its maximally even interval distribution make the church mode set the most convenient place to begin this exploration into the realm of mode sets and their complex, beautiful interrelationships. Let’s begin by observing some interesting properties of the church mode set.

An interesting linear pattern arises when the church modes are listed from “brightest-sounding” to “darkest-sounding” (i.e. most sharps to most flats). In the table on the following page, all scale degrees lowered from Lydian are in bold, and the most recent scale degree in the sequence to have been lowered is underlined.

3 John Clough and Jack Douthett, “Maximally Even Sets,” Journal of Music Theory 35, no. 1/2 (Spring 1991): 95.

Lydian♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭ Ionian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭ ♭Mixolydian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7 Dorian♭ ♭1♭ ♭2♭ ♭3 ♭4♭ ♭5♭ ♭6♭ ♭7 Aeolian♭ ♭1♭ ♭2♭ ♭3 ♭4♭ ♭5♭ ♭6 ♭7 Phrygian♭ ♭1♭ ♭2 ♭3 ♭4♭ ♭5♭ ♭6 ♭7 Locrian♭ ♭1♭ ♭2 ♭3 ♭4♭ ♭5 ♭6 ♭7

Table 1.1.1. The pattern in the church mode set.

The first scale degree in the sequence to be lowered is 4, followed by 7, 3, 6, 2, and finally 5. This sequence raises the question: what happens when we continue the sequence and flat the 1 in Locrian?

♭Lydian ♭1 ♭2 ♭3 ♭4♭ ♭5 ♭6 ♭7

The term “♭Lydian” indicates an alternative spelling of the Lydian mode where every scale degree in the mode is lowered. So, whereas Lydian is spelled 1-2-3-♯4-5-6-7-1, ♭Lydian is spelled ♭1-♭2-♭3-4-♭5-♭6-♭7-♭1.

Unless absolute pitch is taken into account, lowering any scale’s tonic is equivalent to raising every other scale degree. When Locrian’s tonic is lowered, the sequence cycles back to Lydian and begins anew.

To visualize the relationship between Locrian and Lydian on a keyboard, play a one-octave Locrian scale on B. Now play the same scale, but lower the tonic by a half step; that is, replace B with B♭. The original B Locrian scale has modulated to a B♭ Lydian scale.

Figure 1.1.6. Modulation from B Locrian to B♭ Lydian by lowering the tonic.

The church mode set therefore constitutes a closed, repeating cycle. The seven church modes can be arranged in cyclical order on a regular heptagon, as shown on the following page.

Figure 1.1.7. The church heptagon.

On the church heptagon, Lydian and Locrian are both spelled with a natural tonic, showing that these two modes diverge from one another on every scale degree except the tonic. The “♯1/♭1” specification between them suggests the equally valid alternative interpretation: that they diverge from one another only on the tonic.

The church modes as depicted on the heptagon are removed from one another by varying degrees of separation. For example, Phrygian is five degrees removed from Lydian in the clockwise (flat) direction and two degrees removed from Lydian in the counterclockwise (sharp) direction.

The church heptagon as depicted in Figure 1.1.7 exhibits left-right intervallic symmetry. Locrian and Lydian perfectly mirror one another’s intervals: Locrian’s interval structure (1221222) amounts to Lydian’s interval structure (2221221) spelled backwards. The same is true of the Phrygian-Ionian and Aeolian-Mixolydian pairs: Phrygian (1222122) is Ionian (2212221) spelled backwards, while Aeolian (2122122) is Mixolydian (2212212) spelled backwards.

Dorian (2122212) has no symmetrical counterpart because its interval structure is already symmetrical. Due to its intervallic symmetry, Dorian is located in the central corner of the church heptagon. To visualize Dorian’s intervallic symmetry on a keyboard, simply note that the

10 layout of black and white keys is symmetrical around the D key (the natural Dorian tonic key). Note: The same is true of the A♭ key—this will become relevant in Chapter II.

With this foundation for understanding the cyclical and symmetrical nature of the church mode set, we can move on to deriving more mode sets.

1.2. The Melodic Minor Mode Set

The melodic minor mode set can be generated from the church mode set by a simple process of modal recombination.

Every church mode amounts to a combination of its two adjacent modes on the heptagon. For example, Mixolydian is a combination of Ionian and Dorian. All three modes share scale degrees 1, 2, 4, 5, and 6, diverging from one another on scale degrees 3 and 7 (Ionian contains 3 and 7 whereas Dorian contains ♭3 and ♭7). By combining Ionian’s 3 with Dorian’s ♭7 while holding the other five scale degrees (1, 2, 4, 5, 6) constant, the scale 1-2-3-4-5-6-♭7, or Mixolydian, is produced. The following modal recombination table shows this process, with the scale degrees borrowed from Ionian and Dorian in bold.

♭Ionian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭ ♭Mixolydian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7 ♭Dorian♭ ♭1♭ ♭2♭ ♭3 ♭4♭ ♭5♭ ♭6♭ ♭7

Table 1.2.1. Ionian and Dorian combine to produce Mixolydian.

Naturally, one might ask what happens when we combine the other two scale degrees—Dorian’s ♭3 and Ionian’s 7—while still holding the other five scale degrees constant. This process yields 1-2-♭3-4-5-6-7, the melodic minor or jazz minor scale.

♭Ionian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭ ♭Mel. min.♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭ ♭Dorian♭ ♭1♭ ♭2♭ ♭3 ♭4♭ ♭5♭ ♭6♭ ♭7

Table 1.2.1b. Ionian and Dorian combine to produce the Melodic minor scale.

Since Mixolydian and Melodic minor are the only scales in 12ET that can be generated by recombination of Ionian with Dorian, Melodic minor is Mixolydian’s generative counterpart.

To derive any church mode y’s generative counterpart:

1) Determine the two church modes x and z that lie adjacent to mode y on the church heptagon. 2) Determine the two scale degrees where modes x and z differ from one another. 3) Combine these two scale degrees, holding the remaining five scale degrees constant, so as not to produce mode y.

For another example, let’s look at Aeolian, whose adjacent modes on the church heptagon are Dorian and Phrygian. These modes diverge from one another on scale degrees 2 and 6: Dorian contains 2 and 6 whereas Phrygian contains ♭2 and ♭6. 2 combines with ♭6 to produce Aeolian:

♭Dorian♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭♭7♭ ♭Aeolian♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭7 ♭Phrygian♭ ♭1♭ ♭♭2♭ ♭3 ♭4♭ ♭5♭ ♭♭6♭ ♭7

Table 1.2.2a. Dorian and Phrygian combine to produce Aeolian.

…while ♭2 combines with 6 to produce 1-♭2-♭3-4-5-6-♭7, the second mode of the melodic minor scale, known as Dorian ♭2 or Phrygian ♮6.

♭Dorian♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭♭7♭ ♭Dorian ♭2♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7 ♭Phrygian♭ ♭1♭ ♭♭2♭ ♭3 ♭4♭ ♭5♭ ♭♭6♭ ♭7

Table 1.2.2b. Dorian and Phrygian combine to produce Dorian ♭2.

Ionian’s generative counterpart is the Acoustic, or Lydian Dominant, scale (1-2-3-♯4-5-6-♭7):

♭Lydian♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭ ♭Ionian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭ ♭Mixolydian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7

Table 1.2.3a. Lydian and Mixolydian combine to produce Ionian.

♭Lydian♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭ ♭Acoustic♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭♭7♭ ♭Mixolydian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7

Table 1.2.3b. Lydian and Mixolydian combine to produce the Acoustic scale.

Dorian’s counterpart is the Melodic major, or Aeolian Dominant, scale (1-2-3-4-5-♭6-♭7):

♭Mixolydian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭♭7♭ ♭Dorian♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭♭7♭ ♭Aeolian♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭7

Table 1.2.4a. Mixolydian and Aeolian combine to produce Dorian.

♭Mixolydian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭♭7♭ ♭Mel. maj.♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭ ♭Aeolian♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭7

Table 1.2.4b. Mixolydian and Aeolian combine to produce the Melodic major scale.

Phrygian’s counterpart is the Half-diminished, or Locrian ♮2, scale (1-2-♭3-4-♭5-♭6-♭7):

♭Aeolian♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭ ♭Phrygian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭ ♭Locrian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭7

Table 1.2.5a. Aeolian and Locrian combine to produce Phrygian.

♭Aeolian♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭ ♭Half-dim.♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭Locrian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 1.2.5b. Aeolian and Locrian combine to produce the Half-diminished scale.

The process of modal recombination is less intuitive for the two mode pairs that straddle the “altered tonic zone” of the church heptagon: Phrygian-Lydian and Locrian-Ionian.

Let’s start with Locrian, whose adjacent modes are Phrygian and Lydian. Following the shortest route between them on the church heptagon, Phrygian and Lydian differ from one another on scale degrees 1 and 5: Phrygian contains ♯1 and ♯5 relative to Lydian, while Lydian contains ♭1 and ♭5 relative to Phrygian.

Because of the altered tonic, we must decide whether to treat Lydian or Phrygian as the natural mode; the other will receive an accidental. Both options are demonstrated below.

First, let’s take Lydian as the natural mode and write Phrygian in its raised form (♯Phrygian). Since “♯Phrygian” means “Phrygian with every scale degree raised,” ♯Phrygian is spelled ♯1-2- 3-♯4-♯5-6-7.

From this perspective, the two divergent scale degrees are as follows: ♯Phrygian contains ♯1 and ♯5 whereas Lydian contains ♮1 and ♮5. ♯1 combines with ♮5 to produce ♯Locrian:

♭♯Phrygian♭ ♭♯1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭6♭ ♭7♭ ♭♯Locrian♭ ♭♯1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭ ♭Lydian♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭

Table 1.2.6a. ♯Phrygian and Lydian combine to produce ♯Locrian.

…while ♮1 combines with ♯5 to produce 1-2-3-♯4-♯5-6-7, Lydian ♯5 or Lydian Augmented:

♭♯Phrygian♭ ♭♯1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭6♭ ♭7♭ ♭Lyd. Aug.♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭6♭ ♭7♭ ♭Lydian♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭

Table 1.2.6b. ♯Phrygian and Lydian combine to produce the Lydian Augmented scale.

Now, taking Phrygian as the natural mode and writing Lydian in its lowered form: “♭Lydian” means “Lydian with every scale degree lowered,” so ♭Lydian is spelled ♭1-♭2-♭3-4-♭5-♭6-♭7.

From this perspective, the two divergent scale degrees are as follows: Phrygian contains ♮1 and ♮5 whereas ♭Lydian contains ♭1 and ♭5. ♮1 combines with ♭5 to produce Locrian:

♭Phrygian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭ ♭Locrian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭7 ♭♭Lydian♭ ♭♭1♭ ♭♭2♭ ♭3 ♭4♭ ♭♭5♭ ♭♭6♭ ♭7

Table 1.2.6c. Phrygian and ♭Lydian combine to produce Locrian.

…while ♭1 combines with ♮5 to produce ♭1-♭2-♭3-4-5-♭6-♭7, the lowered rendering of the Lydian Augmented scale:

♭Phrygian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭ ♭♭Lyd. Aug.♭ ♭♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭7 ♭♭Lydian♭ ♭♭1♭ ♭♭2♭ ♭3 ♭4♭ ♭♭5♭ ♭♭6♭ ♭7

Table 1.2.6d. Phrygian and ♭Lydian combine to produce the ♭Lydian Augmented scale.

Lydian’s counterpart can be generated similarly.

Lydian’s adjacent modes are Locrian and Ionian, which diverge on scale degrees 1 and 4. Locrian contains ♯1 and ♯4 relative to Ionian, while Ionian contains ♭1 and ♭4 relative to Locrian.

We must decide again which of the two modes to take as the natural one. First, taking Locrian as the natural mode and using ♭Ionian (♭1-♭2-♭3-♭4-♭5-♭6-♭7).

From this perspective, the two divergent scale degrees are as follows: Locrian contains ♮1 and ♮4 whereas ♭Ionian contains ♭1 and ♭4. ♭1 combines with ♮4 to produce ♭Lydian:

♭Locrian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭♭Lydian♭ ♭♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭♭Ionian♭ ♭♭1♭ ♭♭2♭ ♭♭3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 1.2.7a. Locrian and ♭Ionian combine to produce ♭Lydian.

…while ♮1 combines with ♭4 to produce 1-♭2-♭3-♭4-♭5-♭6-♭7, the Altered or Super Locrian scale:

♭Locrian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭Altered♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭♭Ionian♭ ♭♭1♭ ♭♭2♭ ♭♭3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 1.2.7b. Locrian and ♭Ionian combine to produce the Altered scale.

Now, taking Ionian as the natural mode and using ♯Locrian (♯1-2-3-♯4-5-6-7).

From this perspective, the two divergent scale degrees are as follows: ♯Locrian contains ♯1 and ♯4 whereas Ionian contains ♮1 and ♮4. ♮1 combines with ♯4 to produce Lydian:

♭♯Locrian♭ ♭♯1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭ ♭Lydian♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭ ♭Ionian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭

Table 1.2.7c. ♯Locrian and Ionian combine to produce Lydian.

…while ♯1 combines with ♮4 to produce ♯1-2-3-4-5-6-7, the raised rendering of the Altered scale:

♭♯Locrian♭ ♭♯1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭ ♭♯Altered♭ ♭♯1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭ ♭Ionian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭

Table 1.2.7d. ♯Locrian and Ionian combine to produce the ♯Altered scale.

We have now derived the entire melodic minor mode set from the church mode set through modal recombination.

♭Church modes♭ ♭Generative counterparts♭ ♭Lydian♭ ♭Altered♭ ♭Ionian♭ ♭Acoustic♭ ♭Mixolydian♭ ♭Melodic minor♭ ♭Dorian♭ ♭Melodic major♭ ♭Aeolian♭ ♭Dorian ♭2♭ ♭Phrygian♭ ♭Half-diminished♭ ♭Locrian♭ ♭Lydian Augmented♭

Table 1.2.8. The seven church modes and their respective generative counterparts.

Tables 1.2.1a through 1.2.7d demonstrate a rigorous derivation of the melodic minor modes from the church modes through modal recombination. A less rigorous but more intuitive method is also available: taking shortcuts on the church heptagon.

The pattern shown in Table 1.1.1 begins on Lydian and lowers each scale degree by a half step, one by one, in the following order: 4, 7, 3, 6, 2, 5, 1, 4, 7, etc. The melodic minor modes appear when one scale degree in the sequence is skipped.

For example, starting on Lydian and traversing the church heptagon clockwise, we see that lowering the fourth scale degree gives Ionian. However, if we skip the fourth and immediately

15 raise the seventh (the next scale degree in the sequence), we are left with Ionian’s generative counterpart: the Acoustic scale.

Conversely, starting on Mixolydian and traversing the heptagon counterclockwise, we see that raising the seventh scale degree gives Ionian. However, if we skip the seventh and immediately raise the fourth (the previous scale degree in the sequence), we are again left with the Acoustic scale.

In deriving the melodic minor mode set by taking shortcuts on the church heptagon, it becomes clear that every melodic minor mode can be expressed either as a church mode with a lowered scale degree, or as a church mode with a raised scale degree:

As church mode with a As church mode with a Melodic minor mode lowered scale degree raised scale degree Melodic minor Ionian ♭3 ♭Dorian ♮7♭ Dorian ♭2 Dorian ♭2 ♭Phrygian ♮6♭ Lydian Augmented Phrygian ♭1 ♭Lydian ♯5♭ Acoustic Lydian ♭7 ♭Mixolydian ♯4♭ Melodic major Mixolydian ♭6 ♭Aeolian ♮3♭ Half-diminished Aeolian ♭5 ♭Locrian ♮2♭ Altered Locrian ♭4 ♭Ionian ♯1♭

Table 1.2.9. Each melodic minor mode can be written as either of two church modes with an altered scale degree.

By following a sequence of raising or lowering scale degrees within the melodic minor mode set, the latter can be mapped onto a heptagon similar to the church heptagon. Each pair of adjacent modes on the melodic minor heptagon exhibits two divergent scale degrees instead of just one, making this sequence quite difficult to demonstrate with a table. The melodic minor heptagon is shown on the following page.

Figure 1.2.1. The melodic minor heptagon.

Like the church heptagon, the melodic minor heptagon as depicted in Figure 1.2.1 exhibits left- right intervallic symmetry. Each pair of horizontally aligned modes is an intervallic palindrome, as the two modes in each of the three pairs mirror one another’s interval structures. Dorian’s generative counterpart, Melodic major, is located in the central corner of the heptagon due to its unique intervallic symmetry within the melodic minor mode set.

1.3. The Neapolitan Mode Set

Any two adjacent modes on the melodic minor heptagon can be recombined in exactly the same way as the church modes before. For example, take Melodic minor and Dorian ♭2, which diverge on scale degrees 2 and 7:

♭Mel. min.♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭ ♭Dorian♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭♭7♭ Dorian ♭2 ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭♭7♭

Table 1.3.1a. Melodic minor and Dorian ♭2 combine to produce Dorian.

♭Mel. min.♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭ Neapolitan ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭ Dorian ♭2 ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭♭7♭

Table 1.3.1b. Melodic minor and Dorian ♭2 combine to produce the Neapolitan major scale.

In Table 1.3.1a, melodic minor and Dorian ♭2 combine to produce the church mode whose generative counterpart lies directly opposite these two modes on the melodic minor heptagon: Dorian.

In Table 1.3.1b, Melodic minor and Dorian ♭2 combine to produce the Neapolitan major scale (to avoid confusion resulting from the ♭3 in this scale, it will henceforth be referred to as simply “Neapolitan”). We have now generated a full trio of counterpart modes: one from the church mode set, one from the melodic minor mode set, and one from the Neapolitan mode set (Dorian, Melodic major, and Neapolitan, respectively). This counterpart trio is unique in that all three modes have symmetrical interval structures.

The remaining six generative counterpart trios can be derived similarly. We can derive the Half- diminished scale’s two generative counterparts by combining the two modes that share the side located directly across from Half-diminished on the melodic minor heptagon, namely Dorian ♭2 and Lydian Augmented:

♯Dorian ♭2 ♭♯1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭♯6♭ ♭7♭ ♯Phrygian ♭♯1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭6♭ ♭7♭ Lyd. Aug. ♭1♭ ♭2♭ 3 ♭♯4♭ ♭♯5♭ ♭6♭ ♭7♭

Table 1.3.2a. ♯Dorian ♭2 and Lydian Augmented combine to produce ♯Phrygian.

♭♯Dorian ♭2♭ ♭♯1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭♯6♭ ♭7♭ ♭L.W.T.♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭♯6♭ ♭7♭ Lyd. Aug. ♭1♭ ♭2♭ 3 ♭♯4♭ ♭♯5♭ ♭6♭ ♭7♭

Table 1.3.2b. ♯Dorian ♭2 and Lydian Augmented combine to produce the Leading whole-tone scale.

Lydian Augmented and Acoustic combine to produce the Altered scale’s counterparts:

Lyd. Aug. ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭6♭ ♭7♭ ♭Lydian♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭ Acoustic ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭♭7♭

Table 1.3.3a. Lydian Augmented and Acoustic combine to produce Lydian.

♭Lyd. Aug.♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭6♭ ♭7♭ ♭L.D.A.♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭♯5♭ ♭6♭ ♭♭7♭ ♭ Acoustic♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭♭7♭

Table 1.3.3b. Lydian Augmented and Acoustic combine to produce Lydian Dominant Augmented.

Acoustic and Melodic major combine to produce melodic minor’s counterparts:

Acoustic ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭♭7♭ ♭Mixolyd.♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭♭7♭ ♭ Mel. maj.♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭

Table 1.3.4a. Acoustic and Melodic major combine to produce Mixolydian.

♭Acoustic♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭♭7♭ Lyd. min. ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭♭6♭ ♭♭7♭ ♭ Mel. maj.♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭

Table 1.3.4b. Acoustic and Melodic major combine to produce Lydian minor.

Melodic major and Half-diminished combine to produce Dorian ♭2’s counterparts:

♭Mel. maj.♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭ Aeolian ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭ ♭ Half-dim.♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 1.3.5a. Melodic major and Half-diminished combine to produce Aeolian.

♭Mel. maj.♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭ Maj. Loc. ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭ Half-dim.♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 1.3.5b. Melodic major and Half-diminished combine to produce Major Locrian.

Half-diminished and Altered combine to produce Lydian Augmented’s counterparts:

♭ Half-dim.♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭Locrian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭Altered♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 1.3.6a. Half-diminished and Altered combine to produce Locrian.

♭ Half-dim.♭ ♭1♭ ♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭Alt. ♮2♭ ♭1♭ ♭2♭ ♭♭3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭Altered♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 1.3.6b. Half-diminished and Altered combine to produce Altered ♮2.

Finally, Altered and Melodic minor combine to produce Acoustic’s counterparts:

♭Altered♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭♭Ionian♭ ♭♭1♭ ♭♭2♭ ♭♭3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭Mel. min. ♭♭1♭ ♭♭2♭ �3 ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 1.3.7a. Altered and ♭Melodic minor combine to produce ♭Ionian.

♭Altered♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ Super Alt. ♭1♭ ♭♭2♭ ♭�3♭ ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭ ♭Mel. min. ♭♭1♭ ♭♭2♭ �3 ♭♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 1.3.7b. Altered and ♭Melodic minor combine to produce the Super Altered scale.

We have now derived a complete set of generative counterpart trios:

♭Church♭ ♭Melodic minor♭ ♭Neapolitan♭ ♭Lydian♭ ♭Altered♭ ♭Lyd. Dom. Aug.♭ ♭Ionian♭ ♭Acoustic♭ ♭Super Altered♭ ♭Mixolydian♭ ♭Melodic minor♭ ♭Lydian minor♭ ♭Dorian♭ ♭Melodic major♭ ♭Neapolitan♭ ♭Aeolian♭ ♭Dorian ♭2♭ ♭Major Locrian♭ ♭Phrygian♭ ♭Half-diminished♭ ♭Leading whole-tone♭ ♭Locrian♭ ♭Lydian Augmented♭ ♭ Altered ♮2♭

Table 1.3.8. The complete set of heptatonic generative counterpart trios in 12ET.

The Neapolitan mode set can be mapped onto a heptagon of its own:

Figure 1.3.2. The Neapolitan heptagon.

The Neapolitan heptagon exhibits the same left-right intervallic symmetry as the church and melodic minor heptagons, its central corner occupied by the symmetrical Neapolitan scale.

Although pairs of adjacent modes on the Neapolitan heptagon can theoretically be recombined in the same way as pairs of adjacent melodic minor modes, recombination of this sort simply generates a church mode along with an invalid heptatonic rendering of the whole-tone scale (e.g. 1-2- 3-♯4-♭5-♭6-♭7).

The three mode sets so far derived—church, melodic minor, and Neapolitan—have been dubbed the “Heptatonia Prima,” “Heptatonia Secunda,” and “Heptatonia Tertia,” respectively.4 These mode sets will henceforth be collectively referred to as the “Heptatonia.”

The interval structures of the Heptatonia consist of five whole steps and two half steps. The Heptatonia Prima set is maximally even, its half steps located as far away from one another in the interval structure as possible. Moving one of the two half steps in towards the other yields Heptatonia Secunda; doing so once more yields Hepta- tonia Tertia.

1221222 = Heptatonia Prima 1212222 = Heptatonia Secunda 1122222 = Heptatonia Tertia

The Heptatonia superset includes all heptatonic mode sets in 12ET whose interval structures contain only half steps (1) and whole steps (2); that is, the Heptatonia—and only the Heptatonia—contain exclusively major and minor (as opposed to augmented or diminished) second and seventh intervals.

The two line segments connecting these three mode sets indicate that the Neapolitan mode set is derived from the melodic minor mode set, which is in turn derived from the church mode set. The Neapolitan mode set comprises only one-third of the third generation of heptatonic mode sets in 12ET. To derive the other two third-generation mode sets, we must return to the church mode heptagon.

1.4. The Harmonic Major and Minor Mode Sets

Modal recombination of any two church modes x and z two degrees removed on the church heptagon generates two modes: 1) church mode y, and 2) the melodic minor counterpart of church mode y. Modal recombination can also be applied to any two church modes w and z three degrees removed on the church heptagon.

4 László Somfai, “Strategics of Variation in the Second Movement of Bartók’s Violin Concerto 1937-1938,” Studia Musicologica Academiae Scientarum Hungaricae 19, no. 1/4 (1977): 170.

Let w = Ionian and z = Aeolian. w and z converge on scale degrees 1, 2, 4, and 5, and diverge on 3, 6, and 7. Since there are now three divergent scale degrees instead of just two, modal recombination will generate more new modes than in the previous two sections.

Because Ionian and Aeolian converge on scale degrees 1, 2, 4, and 5, the 1-2-4-5 heptatonic scale family contains every valid heptatonic scale in 12ET that can be generated by combining Ionian with Aeolian:

♭3-6-7♭ ♭Ionian♭ ♭3-6-♭7♭ ♭Mixolydian♭ ♭3-♭6-7♭ ♭Harmonic major♭ ♭3-♭6-♭7♭ ♭Melodic major♭ ♭♭3-6-7♭ ♭Melodic minor♭ ♭♭3-6-♭7♭ ♭Dorian♭ ♭♭3-♭6-7♭ ♭Harmonic minor♭ ♭♭3-♭6-♭7♭ ♭Aeolian♭

Table 1.4.1. The 1-2-4-5 heptatonic scale family, showing all products of combination of Ionian and Aeolian.

Every scale in Table 1.4.1 belongs either to the church mode set or to the melodic minor mode set, with two exceptions: Harmonic major and Harmonic minor. Heinrich Schenker discusses the contents of Table 1.4.1 in his book Harmony, suggesting that Ionian (natural major) and Aeolian (natural minor), along with the six products of combination thereof, serve as the foundation of Western tonality.5

The following scale family tables show the derivation of the remaining six harmonic major and harmonic minor modes.

♭3-♯4-7♭ ♭Lydian♭ ♭3-♯4-♭7♭ ♭Acoustic♭ ♭3-4-7♭ ♭Ionian♭ ♭3-4-♭7♭ ♭ Mixolydian ♭ ♭♭3-♯4-7♭ ♭Harmonic major mode 4♭ ♭♭3-♯4-♭7♭ ♭Harmonic minor mode 4♭ ♭♭3-4-7♭ ♭Melodic minor♭ ♭♭3-4-♭7♭ ♭Dorian♭

Table 1.4.2. The 1-2-4-5 heptatonic scale family, showing all products of combination of Lydian and Dorian.

5 Heinrich Schenker, Harmony (Chicago: University of Chicago Press, 1954), 87.

♭2-3-6♭ ♭Mixolydian♭ ♭2-3-♭6♭ ♭Melodic major♭ ♭2-♭3-6♭ ♭Dorian♭ ♭2-♭3-♭6♭ ♭ Aeolian ♭ ♭♭2-3-6♭ ♭Harmonic major mode 5♭ ♭♭2-3-♭6♭ ♭Harmonic minor mode 5♭ ♭♭2-♭3-6♭ ♭Dorian ♭2♭ ♭♭2-♭3-♭6♭ ♭Phrygian♭

Table 1.4.3. The 1-4-5-♭7 heptatonic scale family, showing all products of combination of Mixolydian and Phrygian.

♭2-5-6♭ ♭Dorian♭ ♭2-5-♭6♭ ♭Aeolian♭ ♭2-♭5-6♭ ♭Harmonic major mode 2♭ ♭2-♭5-♭6♭ ♭ Half-diminished ♭ ♭♭2-5-6♭ ♭Dorian ♭2♭ ♭♭2-5-♭6♭ ♭Phrygian♭ ♭♭2-♭5-6♭ ♭Harmonic minor mode 2♭ ♭♭2-♭5-♭6♭ ♭Locrian♭

Table 1.4.4. The 1-♭3-4-♭7 heptatonic scale family, showing all products of combination of Dorian and Locrian.

♭♯1-♯2-♯5♭ ♭♯Aeolian♭ ♭♯1-♯2-5♭ ♭♯Half-diminished♭ ♭♯1-2-♯5♭ ♭♯Phrygian♭ ♭♯1-2-5♭ ♭♯Locrian♭ ♭1-♯2-♯5♭ ♭Harmonic major mode 6♭ ♭1-♯2-5♭ ♭Harmonic minor mode 6♭ ♭1-2-♯5♭ ♭Lydian Augmented♭ ♭1-2-5♭ ♭Lydian♭

Table 1.4.5. The 3-♯4-6-7 heptatonic scale family, showing all products of combination of ♯Aeolian and Lydian.

♭♯1-♯4-♯5♭ ♭♯Phrygian♭ ♭♯1-♯4-5♭ ♭♯Locrian♭ ♭♯1-4-♯5♭ ♭♯Harmonic major mode 3♭ ♭♯1-4-5♭ ♭♯Altered♭ ♭1-♯4-♯5♭ ♭Lydian Augmented♭ ♭1-♯4-5♭ ♭Lydian♭ ♭1-4-♯5♭ ♭Harmonic minor mode 3♭ ♭1-4-5♭ ♭Ionian♭

Table 1.4.6. The 2-3-6-7 heptatonic scale family, showing all valid products of combination of ♯Phrygian and Ionian.

♭1-4-♭7♭ ♭Locrian♭ ♭1-4-�7♭ ♭ Harmonic major mode 7♭ ♭1-♭4-♭7♭ ♭Altered♭ ♭1-♭4-�7♭ ♭ Harmonic minor mode 7♭ ♭♭1-4-♭7♭ ♭♭Lydian♭ ♭♭1-4-�7♭ ♭♭Acoustic♭ ♭♭1-♭4-♭7♭ ♭♭Ionian♭ ♭♭1-♭4-�7♭ ♭♭Mixolydian♭

Table 1.4.7. The ♭2-♭3-♭5-♭6 heptatonic scale family, showing all valid products of combination of Locrian and ♭Mixolydian.

Tables 1.4.1 through 1.4.7 provide a comprehensive account of all valid products of modal recombination across three degrees of separation on the church heptagon. However, as with the melodic minor mode set, a simple shortcut in the church heptagon sequence is the quickest and easiest method of derivation.

For example, skipping directly from Ionian to Aeolian by lowering the Ionian sixth produces Harmonic major, while skipping directly from Aeolian to Ionian by raising the Aeolian seventh produces Harmonic minor. Similarly, skipping directly from Mixolydian to Phrygian by lowering the Mixolydian second yields the fifth mode of the Harmonic major, while skipping directly from Phrygian to Mixolydian by raising the Phrygian third yields the fifth mode of the Harmonic minor.

Every harmonic major and minor mode can be written as a church mode with a single altered scale degree. The altered scale degree is lowered for each harmonic major mode and raised for each harmonic minor mode. The two tables on the following page show all fourteen modes identified by their nearest church mode.

Harmonic major As church mode with a mode (#) lowered scale degree 1 ♭Ionian ♭6♭ 2 ♭Dorian ♭5♭ 3 ♭Phrygian ♭4♭ 4 ♭Lydian ♭3♭ 5 ♭Mixolydian ♭2♭ 6 ♭Aeolian ♭1♭ 7 ♭Locrian �7♭

Table 1.4.8. The harmonic major mode set, each mode written as a church mode with a lowered scale degree.

Harmonic minor As church mode with a mode (#) raised scale degree 1 ♭Aeolian ♮7♭ 2 ♭Locrian ♮6♭ 3 ♭Ionian ♯5♭ 4 ♭Dorian ♯4♭ 5 ♭Phrygian ♮3♭ 6 ♭Lydian ♯2♭ 7 ♭Mixolydian ♯1♭

Table 1.4.9. The harmonic minor mode set, each mode written as a church mode with a raised scale degree.

In the paper “Tonal Scales and Minimal Simple Pitch Class Cycles,” David Meredith offers an impressive list of unique mathematical properties collectively exhibited by the church, melodic minor, and harmonic major and minor mode sets, a mode superset he refers to as the set of all “tonal” scales, or T.

T is shown to be the superset of the four most even heptatonic mode sets in 12ET. In fact, according to the method for calculating evenness that Meredith proposes, the most even heptatonic mode set in 12ET is the church mode set, followed by the melodic minor, harmonic major and minor, Neapolitan, and Harmonic Phrygian mode sets, respectively.6 T is also shown to be the superset of all heptatonic mode sets in 12ET whose thirds and sixths are exclusively major or minor (as opposed to augmented or diminished); that is, all pairs of adjacent numerals in the interval structures of T add up to either three or four.

Through the generative modal framework, T is shown to be the superset of the first three mode sets in 12ET generated by altering one scale degree (at a time) of any church mode according to

6 David Meredith, “Tonal Scales and Minimal Simple Pitch Class Cycles,” in Mathematics and Computation in Music, eds. Carlos Agon et al. (Springer, 2011), 178.

26 the church heptagon sequence. For example, Dorian ♮3 = Mixolydian (church), Dorian ♮7 = Melodic minor, Dorian ♯4 = fourth mode of the Harmonic minor. In the opposite direction, Dor- ian ♭6 = Aeolian (church), Dorian ♭2 = Phrygian ♮6 (melodic minor), Dorian ♭5 = second mode of the Harmonic major. This property holds true for the pentatonic reciprocal of T (i.e. the superset consisting of the pentatonic church, pentatonic melodic minor, and pentatonic harmonic major and minor mode sets).

Figure 1.4.1. The first three generations in the 12ET heptatonic modal family tree.

On the heptatonic modal family tree, line segments connect the harmonic major and minor mode sets to the church mode set to show that the former two are derived from the latter. Although neither the harmonic major nor the harmonic minor mode set exhibits intervallic symmetry per se, these two mode sets do possess mutual symmetry: every harmonic major mode forms an intervallic palindrome when paired with a certain harmonic minor mode. To maintain the overall symmetry of the family tree, these two mode sets are located directly across from one another on the third row (generation).

Although the harmonic major and minor mode sets can be mapped onto their own heptagons, each pair of would-be-adjacent modes on these heptagons contains three divergent scale degrees, inevitably making for an ugly pair of heptagons.

1.5. The Harmonic Lydian, Harmonic Phrygian, and Double Harmonic Mode Sets

Still more new mode sets can be generated by recombination of two modes v and z four degrees removed on the church heptagon. Shown on the following page are the scale family tables for two such pairs of church modes v and z.

♭3-♯4-6-7♭ ♭Lydian♭ ♭3-♯4-6-♭7♭ ♭Acoustic♭ ♭3-♯4-♭6-7♭ ♭Harmonic Lydian♭ ♭3-♯4-♭6-♭7♭ ♭Neapolitan mode 4♭ ♭3-4-6-7♭ ♭Ionian♭ ♭3-4-6-♭7♭ ♭Mixolydian♭ ♭3-4-♭6-7♭ ♭Harmonic major♭ ♭3-4-♭6-♭7♭ ♭Melodic major♭ ♭♭3-♯4-6-7♭ ♭Harmonic major mode 4♭ ♭♭3-♯4-6-♭7♭ ♭Harmonic minor mode 4♭ ♭♭3-♯4-♭6-7♭ Double harmonic mode 4 ♭♭3-♯4-♭6-♭7♭ Harmonic Phrygian mode 4 ♭♭3-4-6-7♭ ♭Melodic minor♭ ♭♭3-4-6-♭7♭ ♭Dorian♭ ♭♭3-4-♭6-7♭ ♭Harmonic minor♭ ♭♭3-4-♭6-♭7♭ ♭Aeolian♭

Table 1.5.1. The 1-2-5 heptatonic scale family, showing all products of combination of Lydian and Aeolian.

♭2-3-6-7♭ ♭Ionian♭ ♭2-3-6-♭7♭ ♭Mixolydian♭ ♭2-3-♭6-7♭ ♭Harmonic major♭ ♭2-3-♭6-♭7♭ ♭Melodic major♭ ♭2-♭3-6-7♭ ♭Melodic minor♭ ♭2-♭3-6-♭7♭ ♭Dorian♭ ♭2-♭3-♭6-7♭ ♭Harmonic minor♭ ♭2-♭3-♭6-♭7♭ ♭Aeolian♭ ♭♭2-3-6-7♭ ♭Harmonic Lydian mode 5♭ ♭♭2-3-6-♭7♭ ♭Harmonic major mode 5♭ ♭♭2-3-♭6-7♭ ♭Double harmonic♭ ♭♭2-3-♭6-♭7♭ ♭Harmonic minor mode 5♭ ♭♭2-♭3-6-7♭ ♭Neapolitan♭ ♭♭2-♭3-6-♭7♭ ♭Dorian ♭2♭ ♭♭2-♭3-♭6-7♭ ♭Harmonic Phrygian♭ ♭♭2-♭3-♭6-♭7♭ ♭Phrygian♭

Table 1.5.2. The 1-4-5 heptatonic scale family, showing all products of combination of Ionian and Phrygian.

Three new mode sets (in bold) are generated on this level of modal recombination: the modes of the Harmonic Phrygian (1-♭2-♭3-4-5-♭6-7), Harmonic Lydian (1-2-3-♯4-5-♭6-7), and Double harmonic (1-♭2-3-4-5-♭6-7) scales. Note: The Harmonic Phrygian scale is sometimes also called “Neapolitan minor.”

Like the harmonic major and minor mode sets, the harmonic Lydian and Phrygian mode sets are intervallic mirror images of one another. They can also be derived from the church mode set directly by taking a still longer shortcut on the church heptagon. For example, skipping directly from Lydian to Aeolian by lowering the Lydian sixth produces Harmonic Lydian, while skipping from Phrygian to Ionian by raising the Phrygian seventh gives Harmonic Phrygian.

Figure 1.5.1. The first four generations in the 12ET heptatonic modal family tree.

The double harmonic mode set is the only mode set shown in Figure 1.5.1 whose derivation cannot be carried out using modal recombination tables or shortcuts on the church heptagon (only scale family tables suffice), and is therefore not connected by line segments to any other mode sets. Despite its physical isolation from the rest of the mode sets on the family tree, the Double harmonic scale can be derived by lowering the 2 and raising the ♭7 in Melodic major (1- 2-3-4-5-♭6-♭7) or by raising the ♭3 and lowering the 6 in Neapolitan (1-♭2-♭3-4-5-6-7).

Although only four generations are covered here, the reader is encouraged to explore the myriad of mode sets generated on deeper levels of modal recombination.

II. Pentatonic Modes

Heptatonic and pentatonic mode sets in 12ET have a special relationship: every heptatonic mode set has its own complementary pentatonic mode set, and vice versa. The reason for this relation-ship couldn’t be simpler: 5+7=12.

The Rubin’s vase illusion is an image that can take on two different meanings depending on which part of the image is viewed as the object and which part is viewed as the background. The viewer sees either a white vase against a black background or two black faces against a white background. Exactly as in this illusion, the “object” and the “background” can be swapped with regards to the interval structure that defines any mode set.

The entire heptatonic modal family tree can therefore just as accurately be called the pentatonic modal family tree, as each heptatonic mode set contained therein has a unique reciprocal pentatonic mode set whose name can be derived from its heptatonic counterpart. This chapter focuses on the first two generations of pentatonic mode sets and leaves further exploration up to the reader.

2.1. The Pentatonic Church Mode Set

On a regular keyboard, five black keys and seven white keys interlock to form the twelve-tone chromatic scale. The white and black keys are complementary: together, they form the complete twelve-tone chromatic scale. Since the mode set outlined by the interval structure of the white keys is the heptatonic church (7c) mode set, the mode set outlined by the interval structure of the black keys is the pentatonic church (5c) mode set.

For some pentatonic mode sets (including the 5c mode set), each individual pentatonic mode in the set has a specific heptatonic counterpart. The 7c mode that corresponds to each 5c mode can be found by comparing overlapping scale degrees across mode sets.

The major pentatonic scale (the black keys on G♭) contains scale degrees 1, 2, 3, 5, 6. Three 7c modes contain these five scale degrees: Lydian, Ionian, and Mixolydian. Since Ionian lies on the midpoint of these three modes on the church heptagon, the major pentatonic scale corresponds with Ionian. We can call the major pentatonic scale Ionian pentatonic, or Ionian5 for short. Each 5c mode shares its scale degrees with exactly three 7c modes and corresponds with whichever 7c mode lies on the midpoint of these three 7c modes on the church heptagon.

♭5c mode tonic♭ ♭Scale degrees♭ ♭7c mode 1♭ ♭7c mode 2♭ ♭7c mode 3♭ G♭ ♭1-2-3-5-6♭ Lydian Ionian Mixolydian D♭ ♭1-2-4-5-6♭ Ionian Mixolydian Dorian A♭ ♭1-2-4-5-♭7♭ Mixolydian Dorian Aeolian E♭ ♭1-♭3-4-5-♭7♭ Dorian Aeolian Phrygian B♭ ♭1-♭3-4-♭6-♭7♭ Aeolian Phrygian Locrian

Table 2.1.1. Correspondences between 5c and 7c modes shown by overlapping scale degrees.

In each row of Table 2.1.1, the 7c mode that lies on the midpoint of the three consecutive 7c modes is in bold. The black keys therefore outline Ionian5 on G♭, Mixolydian5 on D♭, Dorian5 on A♭, Aeolian5 on E♭, and Phrygian5 on B♭.

We can also view this overlap from the opposite perspective to see which 5c modes fit into each 7c mode:

♭7c mode♭ ♭Scale degrees♭ ♭5c mode 1♭ ♭5c mode 2♭ ♭5c mode 3♭

♭Lydian♭ ♭1-2-3-♯4-5-6-7♭ ♭Ionian5♭

♭Ionian♭ ♭1-2-3-4-5-6-7♭ ♭Ionian5♭ ♭Mixolydian5♭

♭Mixolydian♭ ♭1-2-3-4-5-6-♭7♭ ♭Ionian5♭ Mixolydian5 ♭Dorian5♭

♭Dorian♭ ♭1-2-♭3-4-5-6-♭7♭ ♭Mixolydian5♭ ♭Dorian5♭ ♭Aeolian5♭

♭Aeolian♭ ♭1-2-♭3-4-5-♭6-♭7♭ ♭Dorian5♭ ♭Aeolian5♭ ♭Phrygian5♭

♭Phrygian♭ ♭1-♭2-♭3-4-5-♭6-♭7♭ ♭Aeolian5♭ ♭Phrygian5♭

♭Locrian♭ 1-♭2-♭3-4-♭5-♭6-♭7 ♭Phrygian5♭

Table 2.1.2. Correspondences between 7c and 5c modes shown by overlapping scale degrees.

Here is a useful trick to remember the name of any 5c mode x:

1) Locate the pitch one tritone away from the natural tonic of x. 2) Find the 7c mode whose tonic sits naturally on that pitch. 3) This 7c mode corresponds with x.

For example, if I can’t remember the name of the 5c mode whose tonic lies on B♭ (1-♭3-4-♭6-♭7), I need only determine which 7c mode’s natural tonic lies a tritone away from B♭. The pitch located a tritone away from B♭ is the natural Phrygian tonic pitch, E. The 5c mode on B♭ is therefore Phrygian5.

The natural tonic pitches of the five 7c/5c mode pairings are as follows: C/G♭ = Ionian; G/D♭ = Mixolydian; D/A♭ = Dorian; A/E♭ = Aeolian; E/B♭ = Phrygian. Because Lydian and Locrian find their natural tonic pitches on F and B—the only two keys of the same color that sit a tritone apart—these two modes lack a black key tritone pairing and thus also lack a 5c counterpart.

Derivation by complement with the 7c modes is only one way to obtain the 5c mode set. Both methods previously used to derive the 7c modes (maximal evenness and stacked fifths) can also be used to derive the 5c modes.

When the frequency ratio of each interval in 5ET is rounded to its nearest 12ET approximation, the 5c mode set emerges:

Figure 2.1.1. Dorian5 is the scale in 12ET that best approximates a 5ET chromatic scale.

Stacking four consecutive fifths yields the 5c modes:

Figure 2.1.2. Derivation of each 5c mode by stacked fifths.

When the 5c modes are arranged from brightest (fewest flats) to darkest (most flats), a simple pattern emerges that directly parallels the corresponding pattern in the 7c mode set. The table on the following page is the pentatonic version of Table 1.1.1.

Ionian5 ♭1♭ ♭2♭ ♭3♭ ♭5♭ ♭6♭

Mixolydian5 ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭6♭

Dorian5 ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭♭7♭

Aeolian5 ♭1♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭7♭

Phrygian5 ♭1♭ ♭♭3♭ ♭4♭ ♭♭6♭ ♭♭7♭

Table 2.1.3. The pattern in the 5c mode set.

Note that as the modes progress from bright to dark in mood, each scale degree is sequentially raised rather than lowered. The first scale degree in the sequence to be raised is 3, followed by 6,

2, and 5. When the tonic in Phrygian5 is raised, the following scale emerges:

♯Ionian5♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭6♭ ♭♭7♭

Here, the label “♯Ionian5” denotes a spelling of Ionian5 where each scale degree has been trans- posed upwards not by an augmented unison, but by a minor second. While the usual augmented unison spelling would be equally accurate, the unnecessary burden of unfamiliar labels like “♯3” etc. makes this option less attractive.

Just as the 7c mode set is depicted on a heptagon, the 5c mode set is represented on a pentagon.

Like its heptagonal sister, the church pentagon exhibits intervallic symmetry: Ionian5 mirrors Phrygian5, Mixolydian5 mirrors Aeolian5, and Dorian5 mirrors itself.

2.2. The Pentatonic Melodic Minor Mode Set

The image below shows a keyboard whose white keys follow the interval structure of the melodic minor mode set rather than the church mode set.

Figure 2.2.1. A keyboard laid out according to the interval structure of the melodic minor mode set.

Assuming the distance in pitch between each adjacent pair of keys is the usual semitone, the black keys on this keyboard follow the interval structure of the pentatonic melodic minor (5m) modes.

Modal recombination works just as well for pentatonic modes as for heptatonic modes; with two fewer scale degrees to consider, the process is even simpler. The following modal recombination tables show the derivation of each 5c mode’s generative counterpart in the 5m mode set.

Mixolydian5:

♭Ionian5♭ ♭1♭ ♭2♭ ♭3♭ ♭5♭ ♭6♭

♭Mixolydian5♭ ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭6♭

♭Dorian5♭ ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭♭7♭

Table 2.2.1a. Ionian5 and Dorian5 combine to produce Mixolydian5.

Mixolydian5’s 5m generative counterpart:

♭Ionian5♭ ♭1♭ ♭2♭ ♭3♭ ♭5♭ ♭6♭ ♭?♭ ♭1♭ ♭2♭ ♭3♭ ♭5♭ ♭♭7♭

♭Dorian5♭ ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭♭7♭

Table 2.2.1b. Ionian5 and Dorian5 combine to produce Mixolydian5’s 5m counterpart.

Dorian5:

♭Mixolydian5♭ ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭6♭

♭Dorian5♭ ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭♭7♭

♭Aeolian5♭ ♭1♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭7♭

Table 2.2.2a. Mixolydian5 and Aeolian5 combine to produce Dorian5.

Dorian5’s generative counterpart:

♭Mixolydian5♭ ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭6♭ ♭?♭ ♭1♭ ♭♭3♭ ♭4♭ ♭5♭ ♭6♭

♭Aeolian5♭ ♭1♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭7♭

Table 2.2.2b. Mixolydian5 and Aeolian5 combine to produce Dorian5’s 5m counterpart.

Aeolian5:

♭Dorian5♭ ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭♭7♭

♭ Aeolian5♭ ♭1♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭7♭

♭Phrygian5♭ ♭1♭ ♭♭3♭ ♭4♭ ♭♭6♭ ♭♭7♭

Table 2.2.3a. Dorian5 and Phrygian5 combine to produce Aeolian5.

Aeolian5’s generative counterpart:

♭Dorian5♭ ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭♭7♭ ♭?♭ ♭1♭ ♭2♭ ♭4♭ ♭♭6♭ ♭♭7♭

♭Phrygian5♭ ♭1♭ ♭♭3♭ ♭4♭ ♭♭6♭ ♭♭7♭

Table 2.2.3b. Dorian5 and Phrygian5 combine to produce Aeolian5’s 5m counterpart.

Phrygian5:

♭Aeolian5♭ ♭1♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭7♭

♭Phrygian5♭ ♭1♭ ♭♭3♭ ♭4♭ ♭♭6♭ ♭♭7♭

♭♯Ionian5♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭6♭ ♭♭7♭

Table 2.2.4a. Aeolian5 and ♯Ionian5 combine to produce Phrygian5.

Phrygian5’s generative counterpart:

♭♭Aeolian5♭ ♭7♭ ♭2♭ ♭3♭ ♭♯4♭ ♭6♭ ♭?♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭6♭

♭Ionian5♭ ♭1♭ ♭2♭ ♭3♭ ♭5♭ ♭6♭

Table 2.2.4b. ♭Aeolian5 and Ionian5 combine to produce Phrygian5’s 5m counterpart.

Ionian5:

♭♭Phrygian5♭ ♭7♭ ♭2♭ ♭3♭ ♭5♭ ♭6♭

♭Ionian5♭ ♭1♭ ♭2♭ ♭3♭ ♭5♭ ♭6♭

Mixolydian5♭ ♭1♭ ♭2♭ ♭4♭ ♭5♭ ♭6♭

Table 2.2.5a. ♭Phrygian5 and Mixolydian5 combine to produce Ionian5.

Ionian5’s generative counterpart:

♭Phrygian5♭ ♭1♭ ♭♭3♭ ♭4♭ ♭♭6♭ ♭♭7♭ ♭?♭ ♭1♭ ♭♭3♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

♯Mixolydian5♭ ♭♭2♭ ♭♭3♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 2.2.5b. Phrygian5 and ♯Mixolydian5 combine to produce Ionian5’s 5m counterpart.

As in the realm of heptatonic mode sets, the 5m mode set can be derived from the 5c mode set by skipping an altered scale degree in the church pentagon sequence.

The task of naming individual 5m modes is complicated by the fact that the scale degrees in each 5m mode are found in two 7m modes.

♭5c counterpart♭ ♭Scale degrees♭ ♭7m mode 1♭ ♭7m mode 2♭

♭Ionian5♭ ♭1-♭3-♭5-♭6-♭7♭ ♭Half-diminished♭ ♭Altered♭

♭Mixolydian5♭ ♭1-2-3-5-♭7♭ ♭Acoustic♭ ♭Melodic major♭

♭Dorian5♭ ♭1-♭3-4-5-6♭ ♭Melodic minor♭ ♭Dorian ♭2♭

♭Aeolian5♭ ♭1-2-4-♭6-♭7♭ ♭Melodic major♭ ♭Half-diminished♭

♭Phrygian5♭ ♭1-2-3-♯4-6♭ Lydian Augmented ♭Acoustic♭

Table 2.2.6. Overlapping scale degrees in 5m and 7m modes, with 5m modes listed according to their 5c counterparts.

Individual 5m modes cannot be named based on these overlaps because no singular 7m mode corresponds with each 5m mode. This problem becomes even more apparent when viewed from the opposite perspective, as demonstrated on the following page.

♭ 7m mode♭ ♭Scale degrees♭ ♭5c ctrprt. 1♭ ♭5c ctrprt. 2♭

♭Dorian ♭2♭ ♭1-♭2-♭3-4-5-6-♭7♭ ♭Dorian5♭

Lydian Augmented ♭1-2-3-♯4-♯5-6-7♭ ♭Phrygian5♭

♭Acoustic♭ ♭1-2-3-♯4-5-6-♭7♭ ♭Mixolydian5♭ ♭Phrygian5♭

♭Melodic major♭ ♭1-2-3-4-5-♭6-♭7♭ ♭Aeolian5♭ ♭Mixolydian5♭

♭Half-diminished♭ ♭1-2-♭3-4-♭5-♭6-♭7♭ ♭Ionian5♭ ♭Aeolian5♭

♭Altered♭ ♭1-♭2-♭3-♭4-♭5-♭6-♭7♭ ♭Ionian5♭

♭Melodic minor♭ ♭1-2-♭3-4-5-6-7♭ ♭Dorian5♭

Table 2.2.7. Overlapping scale degrees in 7m and 5m modes, with 5m modes listed according to their 5c counterparts.

Because each 5m mode contains a unique tritone interval, only one 7c mode can possibly overlap with each 5m mode. The table below therefore proves more helpful in the search for meaningful names for the 5m modes.

♭5c ctrprt.♭ ♭Scale degrees♭ ♭7c mode 1♭

♭Ionian5♭ ♭1-♭3-♭5-♭6-♭7-1♭ ♭Locrian♭

♭Mixolydian5♭ ♭1-2-3-5-♭7-1♭ ♭Mixolydian♭

♭Dorian5♭ ♭1-♭3-4-5-6-1♭ ♭Dorian♭

♭Aeolian5♭ ♭1-2-4-♭6-♭7-1♭ ♭Aeolian♭

♭Phrygian5♭ ♭1-2-3-♯4-6-1♭ ♭Lydian♭

Table 2.2.8. Correspondences between 5m and 7c modes shown by overlapping scale degrees, with 5m modes listed according to their 5c counterparts.

The 5m counterparts of Mixolydian5, Dorian5, and Aeolian5 overlap neatly with the 7c modes Mixolydian, Dorian, and Aeolian, respectively. Due to the altered tonic in the counterparts of

Ionian5 and Phrygian5, these two modes overlap with Locrian and Lydian respectively.

Each 5m mode can therefore be named according to the 7c mode that shares its scale degrees. In order to differentiate the 5m modes from the 5c modes, the letter “m” is added to each 5m mode’s subscript tag. The 5m modes are therefore as follows, from brightest to darkest: Lydian5m (1-2-3-♯4-6), Mixolydian5m (1-2-3-5-♭7), Dorian5m (1-♭3-4-5-6), Aeolian5m (1-2-4-♭6-♭7), Locri- an5m (1-♭3-♭5-♭6-♭7).

A special feature of the relationship between 5c modes and their 5m counterparts is that when added together (that is, when all of their scale degrees are compiled into one mode), they yield the corresponding 7c mode. For example, Mixolydian5 (1-2-4-5-6) + Mixolydian5m (1-2-3-5-♭7) = Mixolydian (1-2-3-4-5-6-♭7).

This phenomenon is more interesting, albeit trickier to see, with the Phrygian5-Lydian5m and Ionian5-Locrian5m counterpart pairs. The following two tables show addition of Phrygian5 to

Lydian5m, first with the former as the natural mode, then the latter. Note: For reasons similar to

those discussed on the second paragraph of page 32, ♭Phrygian5 and ♭Locrian5 are spelled here as being lowered by a minor second rather than an augmented unison.

♭Phrygian5♭ ♭1♭ ♭♭3♭ ♭4♭ ♭♭6♭ ♭♭7♭ ♭Phrygian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭6♭ ♭♭7♭

♭♯Lydian5m♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭5♭ ♭♭7♭

Table 2.2.9a. Phrygian5 and ♯Lydian5m sum to Phrygian.

♭♭Phrygian5♭ ♭♭ ♭2♭ ♭♭3♭♭ ♭♭ ♭5♭ ♭6♭ ♭7♭ ♭Lydian♭ ♭1♭ ♭2♭ ♭3♭ ♭♯4♭ ♭5♭ ♭6♭ ♭7♭

♭Lydian5m♭ ♭1♭ ♭♭2♭♭ ♭♭3♭♭ ♭♯4♭ ♭6♭

Table 2.2.9b. ♭Phrygian5 and Lydian5m sum to Lydian.

The following two tables show addition of Ionian5 to Locrian5m, first with the former as the natural mode, then the latter.

♭Ionian5♭ ♭1♭ ♭2♭ ♭3♭ ♭♭ ♭5♭ ♭6♭ ♭♭ ♭Ionian♭ ♭1♭ ♭2♭ ♭3♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭

♭♭Locrian5m♭ ♭2♭ ♭♭ ♭4♭ ♭5♭ ♭6♭ ♭7♭

Table 2.2.10a. Ionian5 and ♭Locrian5m sum to Ionian.

♭♯Ionian5♭ ♭♭ ♭♭2♭ ♭♭♭3♭♭ ♭♭4♭♭ ♭♭ ♭♭6♭ ♭♭7♭ ♭Locrian♭ ♭1♭ ♭♭2♭ ♭♭3♭ ♭4♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

♭Locrian5m♭ ♭1♭ ♭♭ ♭♭3♭ ♭♭ ♭♭5♭ ♭♭6♭ ♭♭7♭

Table 2.2.10b. ♯Ionian5 and Locrian5m sum to Locrian.

Tables 2.2.9a through 2.2.10b demonstrate that these two counterpart pairs sum to whichever 7c mode corresponds to the pentatonic mode serving as the natural mode in the addition operation.

Naturally, the 5m mode set can be mapped onto a pentagon that also exhibits the same left-right intervallic symmetry as its heptagonal sister. This heptagon is shown on the following page.

Figure 2.2.2. The melodic minor pentagon.

The two pentatonic mode sets so far derived—5c and 5m—comprise the pentatonic equivalent of the Heptatonia: whereas Heptatonia denotes the superset of all heptatonic mode sets in 12ET whose interval structures contain only two distinct intervals, namely half steps and whole steps, Pentatonia denotes the superset of all pentatonic mode sets in 12ET whose interval structures contain only two distinct intervals, namely whole steps and minor thirds.

The maximally even 5c mode set can be dubbed “Pentatonia Prima” and the 5m mode set “Pen- tatonia Secunda.” Starting on Pentatonia Prima and moving one of the two minor third intervals in towards the other yields Pentatonia Secunda:

32322 = Pentatonia Prima 33222 = Pentatonia Secunda

Although this paper discusses only heptatonic and pentatonic mode sets, the reader is encouraged to experiment with other sizes of mode sets in 12ET using the concepts of maximal evenness, stacked fifths, and modal recombination. Exploring all possible sizes of mode sets in 12ET in this way reveals a number of fascinating patterns that lie beyond the scope of this paper.

III. Rhythmic Modes

3.1. Rhythmic Modes in a Twelve-Beat Cycle

The downbeat of a repeating rhythmic pattern—the point in the pattern at which the rhythmic phrase ends and begins anew—is a rhythmic equivalent of the tonic pitch. This analogy should make intuitive sense to Westerners, as scale degree one simply becomes beat one.

Any scale in 12ET can be converted into a rhythmic pattern by imposing the scale’s interval structure onto a rhythmic cycle with 12 equal subdivisions. For example, take Ionian (2212221): by interpreting each “2” as a quarter note and each “1” as an eighth note, we generate the following rhythmic pattern in 12/8:

||: ♩♩♪♩♩♩♪ :||

Note: This rhythmic pattern, generated by analogy to the most fundamental scale in Western music, is one of the most important bell patterns in the traditional and folk musics of West and Central Africa. Psychologist Jeff Pressing refers to this parallel between disparate musical traditions as a cognitive isomorphism.7

Converting each 7c mode into a rhythmic pattern generates the complete rhythmic heptatonic church (r7c) mode set:

♭7c mode♭ ♭Interval structure♭ ♭r7c (12) mode♭ ♭Lydian♭ ♭2221221♭ ♭||: ♩♩♩♪♩♩♪ :||♭ ♭Ionian♭ ♭2212221♭ ♭||: ♩♩♪♩♩♩♪ :||♭ ♭Mixolydian♭ ♭2212212♭ ♭||: ♩♩♪♩♩♪♩ :||♭ ♭Dorian♭ ♭2122212♭ ♭||: ♩♪♩♩♩♪♩ :||♭ ♭Aeolian♭ ♭2122122♭ ♭||: ♩♪♩♩♪♩♩ :||♭ ♭Phrygian♭ ♭1222122♭ ♭||: ♪♩♩♩♪♩♩ :||♭ ♭Locrian♭ ♭1221222♭ ♭||: ♪♩♩♪♩♩♩ :||♭

Table 3.1.1. The r7c mode set in a twelve-beat cycle.

Any scale in any tuning system can be converted into a rhythmic pattern in this way. However, the conversion process is much more straightforward (and arguably much more useful) for equal- tempered tuning systems.

Note that the direction of correlation between scales and rhythmic patterns is arbitrary: Ionian’s interval structure could be written perfectly accurately from top to bottom as 1222122 (1-7-6-5-

7 Jeff Pressing, “Cognitive Isomorphisms Between Pitch and Rhythm in World Musics: West Africa, the Balkans and Western Tonality," Studies in Music 17 (1983): 38.

4-3-2), which corresponds to the rhythmic pattern labeled “Phrygian” in Table 3.1.1. This means that any scale can be just as accurately associated with the rhythmic equivalent of its intervallic mirror as with its own rhythmic equivalent according to Table 3.1.1.

Also note that a rhythmic mode set need not necessarily be paired with its pitch analogue in order to be meaningful or useful. For example, a considerable proportion of the world’s rhythmic patterns contain eight or sixteen more or less equal subdivisions at the fastest pulse, yet 8ET and 16ET are seldom-used tuning systems. Rhythmic mode sets with eight or sixteen subdivisions can easily be exploited without reference to their pitch analogues.

In order to introduce the techniques for generating rhythmic mode sets, I will derive the first few generations of heptatonic and pentatonic mode sets in a sixteen-beat cycle (i.e. 16ET) and leave further exploration into the rhythmic modal realm up to the reader.

3.2. Rhythmic Modes in a Sixteen-Beat Cycle

Comparison of the interval ratios in 7ET and 16ET shows that the maximally even seven-beat temporal interval structure in a sixteen-beat cycle is 2322232. This interval structure and its modes constitute the r7c mode set in a sixteen-beat cycle. The r7c mode set can also be derived from stacked fifths in 16ET using the 16ET approximation of a perfect fifth: 29/16. The reader is encouraged to prove these derivations for herself if interested.

♭Interval structure♭ ♭Notation♭ ♭2223223♭ ♭||: ♪♪♪♪. ♪♪♪. :||♭ ♭2232223♭ ♭||: ♪♪♪. ♪♪♪♪. :||♭ ♭2232232♭ ♭||: ♪♪♪. ♪♪♪. ♪ :||♭ ♭2322232♭ ♭||: ♪♪. ♪♪♪♪. ♪ :||♭ ♭2322322♭ ♭||: ♪♪. ♪♪♪. ♪♪ :||♭ ♭3222322♭ ♭||: ♪. ♪♪♪♪. ♪♪ :||♭ ♭3223222♭ ♭||: ♪. ♪♪♪. ♪♪♪ :||♭

Table 3.2.1. The r7c mode set in a sixteen-beat cycle.

If we imagine that Table 3.1.2 wraps around so that the top and bottom mode are adjacent, each mode in the table amounts to a combination of its two adjacent modes. Any mode’s generative counterpart can be derived simply by combining its two neighbors so as not to produce itself. For example, the following table shows the derivation of the second mode (2232223)’s counterpart.

♭2223223♭ ♭2♭ ♭2♭ ♭2♭ ♭3♭ ♭2♭ ♭2♭ ♭3♭ ♭2223232♭ ♭2♭ ♭2♭ ♭2♭ ♭3♭ ♭2♭ ♭3♭ ♭2♭ ♭2232232♭ ♭2♭ ♭2♭ ♭3♭ ♭2♭ ♭2♭ ♭3♭ ♭2♭

Table 3.2.2. 2223223 and 2232232 combine to produce 2223232.

Although the use of numerical interval structures in lieu of scale names and scale degrees makes for a much uglier modal recombination table, the concept is equally simple. The rhythmic pattern generated in Table 3.1.3 (2223232) and its modes constitute the rhythmic heptatonic melodic minor (r7m) mode set in a sixteen-beat cycle.

♭Interval structure♭ ♭Notation♭ ♭3222232♭ ♭||: ♪. ♪♪♪♪♪. ♪ :||♭ ♭2222323♭ ♭||: ♪♪♪♪♪. ♪♪. :||♭ ♭2223232♭ ♭||: ♪♪♪♪. ♪♪. ♪ :||♭ ♭2232322♭ ♭||: ♪♪♪. ♪♪. ♪♪ :||♭ ♭2323222♭ ♭||: ♪♪. ♪♪. ♪♪♪ :||♭ ♭3232222♭ ♭||: ♪. ♪♪. ♪♪♪♪ :||♭ ♭2322223♭ ♭||: ♪♪. ♪♪♪♪♪. :||♭

Table 3.2.3. The r7m mode set in a sixteen-beat cycle.

Deriving the rhythmic pentatonic church (r5c) mode set is problematic because for five beats in a sixteen-beat cycle, maximal evenness and stacked fifths yield two separate interval structures. The decision as to which of the two interval structures—33433 (maximal evenness) or 25252 (stacked fifths)—to use as the basis for a modal family tree is ultimately up to taste.

Both approaches generate a profusion of syncopated rhythms that can be layered on top of one another to create an infinite (for all practical purposes) variety of complex rhythmic textures. The maximal evenness approach is demonstrated below because it happens to generate a modal family tree that includes a number of rhythms commonly heard in British and American popular music as well as Afro-Cuban music.

♭Interval structure♭ ♭Notation♭ ♭33334♭ ♭||: ♪. ♪. ♪. ♪. ♩ :||♭ ♭33343♭ ♭||: ♪. ♪. ♪. ♩ ♪. :||♭ ♭33433♭ ♭||: ♪. ♪. ♩ ♪. ♪. :||♭ ♭34333♭ ♭||: ♪. ♩ ♪. ♪. ♪. :||♭ ♭43333♭ ♭||: ♩ ♪. ♪. ♪. ♪. :||♭

Table 3.2.4. The r5c mode set based on maximal evenness in a sixteen-beat cycle.

33433 is the fundamental rhythm in Brazilian bossa nova music. 33334 is heard in a wide variety of popular music (e.g. “Here Comes the Sun” by the Beatles).

Like the r7c mode set, each r5c mode amounts to a combination of its two adjacent modes. Combining these modes so as not to produce a r5c mode yields a rhythmic pentatonic melodic minor (r5m) mode. The following two tables show recombination of 33334 and 33433.

♭33334♭ ♭3♭ ♭3♭ ♭3♭ ♭3♭ ♭4♭ ♭33343♭ ♭3♭ ♭3♭ ♭3♭ ♭4♭ ♭3♭ ♭33433♭ ♭3♭ ♭3♭ ♭4♭ ♭3♭ ♭3♭

Table 3.2.5a. 33334 and 33433 combine to produce 33343. Note: To correct for the fact that borrowing the 3 from both outer modes shortens the cycle by one beat, the intermediary 3 is raised to a 4 (underlined).

♭33334♭ ♭3♭ ♭3♭ ♭3♭ ♭3♭ ♭4♭ ♭33424♭ ♭3♭ ♭3♭ ♭4♭ ♭2♭ ♭4♭ ♭33433♭ ♭3♭ ♭3♭ ♭4♭ ♭3♭ ♭3♭

Table 3.2.5b. 33334 and 33433 combine to produce 33424. Note: To correct for the fact that borrowing the 4 from both outer modes lengthens the cycle by one beat, the intermediary 3 is lowered to a 2 (underlined).

The rhythmic pattern generated in Table 3.2.5b is equivalent to the son clave pattern, one of the more ubiquitous bell patterns in Afro-Cuban music. The son clave pattern and its modes constitute the r5m mode set in a sixteen-beat cycle.

♭Interval structure♭ ♭Notation♭ ♭43342♭ ♭||: ♩ ♪. ♪. ♩ ♪ :||♭ ♭33424♭ ♭||: ♪. ♪. ♩ ♪ ♩ :||♭ ♭34243♭ ♭||: ♪. ♩ ♪ ♩ ♪. :||♭ ♭42433♭ ♭||: ♩ ♪ ♩ ♪. ♪. :||♭ ♭24334♭ ♭||: ♪ ♩ ♪. ♪. ♩ :||♭

Table 3.2.6. The r5m mode set based on maximal evenness in a sixteen-beat cycle, or the son clave mode set.

Tables 3.2.5a and 3.2.5b show that recombination of two r5c modes two degrees removed leads to an extra or missing beat in the cycle. This problem is easily remedied by a simple alteration of the intermediary interval in each of these two tables. Recombination of two r5c modes three degrees removed creates essentially the same problem, but with a more interesting solution.

The following page shows recombination of 33334 and 34333, borrowing the 4 from both modes in order to avoid simply generating another r5c mode. In the first table, the first intermediary 3 is shortened to a 2 to account for the extra beat. In the second table, the second intermediary 3 is shortened instead. The altered interval is again underlined in each table.

♭33334♭ ♭3♭ ♭3♭ ♭3♭ ♭3♭ ♭4♭ ♭34234♭ ♭3♭ ♭4♭ ♭2♭ ♭3♭ ♭4♭ ♭34333♭ ♭3♭ ♭4♭ ♭3♭ ♭3♭ ♭3♭

Table 3.2.7a. 33334 and 34333 combine to produce 34234.

♭33334♭ ♭3♭ ♭3♭ ♭3♭ ♭3♭ ♭4♭ ♭34324♭ ♭3♭ ♭4♭ ♭3♭ ♭2♭ ♭4♭ ♭34333♭ ♭3♭ ♭4♭ ♭3♭ ♭3♭ ♭3♭

Table 3.2.7b. 33334 and 34333 combine to produce 34324.

The pattern generated in Table 3.2.7b is equivalent to the rumba clave, another common bell pattern in Afro-Cuban music. The mode generated in Table 3.2.7a belongs to the mode set whose interval structure mirrors that of the rumba clave mode set.

The rumba clave mode set and its intervallic mirror are structurally analogous to the pentatonic harmonic major and minor mode sets (the pentatonic mode sets in 12ET whose interval structures reciprocate the harmonic major and minor mode sets). Of course, associating rhythmic mode sets with pitch mode sets becomes increasingly abstract, difficult, and impractical as we stray further and further from our starting point, i.e. the heptatonic church mode set in 12ET. However, impractical though they may be, these associations do highlight the general structural similarity of different modal family trees in various beat cycles.

♭Interval structure♭ ♭Notation♭ ♭34324♭ ♭||: ♪. ♩ ♪. ♪ ♩ :||♭ ♭43243♭ ♭||: ♩ ♪. ♪ ♩ ♪. :||♭ ♭32434♭ ♭||: ♪. ♪ ♩ ♪. ♩ :||♭ ♭24343♭ ♭||: ♪ ♩ ♪. ♩ ♪. :||♭ ♭43432♭ ♭||: ♩ ♪. ♩ ♪. ♪ :||♭

Table 3.2.8. The rumba clave mode set.

♭Interval structure♭ ♭Notation♭ ♭34234♭ ♭||: ♪. ♩ ♪ ♪. ♩ :||♭ ♭42343♭ ♭||: ♩ ♪ ♪. ♩ ♪. :||♭ ♭23434♭ ♭||: ♪ ♪. ♩ ♪. ♩ :||♭ ♭34342♭ ♭||: ♪. ♩ ♪. ♩ ♪ :||♭ ♭43423♭ ♭||: ♩ ♪. ♩ ♪ ♪. :||♭

Table 3.2.9. The intervallic mirror of the rumba clave mode set.

Applications of the Generative Modal Framework

Within 12ET, the generative modal framework provides musicians and composers with a solid foundation for understanding the relationships between various modes and the sets to which they belong. In this framework, composers might find new, previously unexplored ways of exploiting polymodality, while instrumentalists—especially jazz players—may wish to incorpor- ate the heptatonic/pentatonic modal family tree into their practice regimen. Such a regimen might look something like this: play two octaves, ascending and descending, of every church mode (heptatonic or pentatonic), followed by every melodic minor mode, every Neapolitan mode, every harmonic major and minor mode, and so on. More avant-garde artists may find themselves attracted to the potential of the generative modal framework for deriving strange, unfamiliar scales and modes in other tempered (and perhaps even untempered) tuning systems. Composers interested in minimalism might be drawn more towards the rhythmic side of the framework and its potential to generate rhythmic patterns of varying lengths and groupings that can be layered on top of one another to create repetitive yet intricate textures, a la Steve Reich’s “Clapping Music.” Musicians interested in traditional and folk genres from around the world might notice structural similarities between modes derived from the generative framework and scales or rhythms heard in various Western and non-Western traditions, and wish to use the framework to add to their improvisational vocabularies. Music educators may view the generative framework as an opportunity to invite students to stretch their brains by thinking about music more mathematically, while musicologists and music historians may see the generative framework as an interesting lens through which to analyze the evolution of European classical harmony and the “emancipation of the dissonance.” The sheer number of scales and rhythmic patterns encompassed within the generative modal framework—to say nothing of the innumerable potential applications of the framework to real-world music-making—can be quite overwhelming, especially when the concepts are new and unfamiliar. It is important to remember that the entire framework stems from three basic principles: maximal evenness, stacked fifths, and modal recombination. The former two have been written about fairly extensively in the last century, while modal recombination (the real cornerstone of the generative framework) has previously remained largely unexplored, usually mentioned only as a passing addendum in the service of some larger point about functional harmony. Through recursive application of these three basic principles to distinct (yet similar) musical structures, high-level complexity arises from a remarkably simple set of procedures. It is my hope that the reader, having grasped these procedures, will continue to experiment with the generative modal framework in her own musical style in as many different ways as she can devise.

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