**Hexagons: Intersections of Symbolism and Geometry in Harmonics, Pythagoreanism, and Qabalah**

#### by Dr. k. Scott Eggert

Symmetricality is a key aspect of modal collections such as the hexatonic or octatonic, and factors widely into the “modes of limited transposition” so favored by Messiaen. My system, developed and used in the majority of my compositions of the last ten years, focuses specifically on the qualities of a) heptatonicism and b) inversional symmetry across the axis of the tonic.

If we move beyond a purely linear representation of a mode or scale, we find that the tonic is represented as a central point surrounded by six outer points in a hexagonal formation. Lines originating from the center move out in opposing directions to pitches at equal intervals from the tonic, therefore having an inversional relationship to each other across the axis of the central point. It was through application of this hexagonal blueprint that the other nine collections revealed themselves, and so I have dubbed these collections *hexagonal modes*.

The *hexagonal mode, *its shape and its symmetry, is directly connected to a number of such concepts from both world culture and esoteric spiritual practice. Thus an aspect of composing with *hexagonal modes *is *hexagonal symbolism*, which I shall expound upon in my dissertation. The hexagram, as the union of two opposite-pointing triangles, represents the unification of opposites, and the point at the center is the place beyond polarities, the place where division ceases and unity holds sway. Therefore a mode contrived through *hexagonal symmetry *creates music as a prayer for balance, unity, equity, harmony, peace.

To read the prospectus for this dissertation project, click here.

To listen to the compositions exemplifying hexagonal symmetry, click here.

To read Scott’s research statement, click here.

### Pythagorean Lambdoma Harmonic Keyboard: System Analysis

#### by Scott Eggert

**Overview**: This project deploys a systems analysis of the Pythagorean Lambdoma Harmonic Keyboard invented by Barbara Hero to accord with the principles of the harmonic series in The Pythagorean Table. The Pythagorean Table is an infinitely extendable grid of ratios, starting at its apex with 1/1 and extending along one arm to x/1 (where x = any number greater than 1, up to infinity) and along the other arm to 1/x. The Table is, in musical terms, a map of all conceivable tonal relationships. The harmonic ratios and intervals represented by the seminal Pythagorean system embody a far more expansive harmonic system than the Western 12-tone equal temperament, the standard tuning paradigm. In fact, the 12-tone equal temperament system that emerged in the history of Western music is actually the musical equivalent of forcing a square peg into a round hole.

Through analysis of the Lambdoma system, and the ability to hear *all* intervals of the Lambdoma in just tuning, we are given the chance to ponder: just how much are the commonly accepted principles of Western music theory applicable to the pure interval system of the Lambdoma? Must all its concepts be abandoned at the Lambdoma’s gates, or are at least some of its truths universal in nature?

Click here for a pdf of the full prospectus.

### Musica Universalis: From the Lambdoma of Pythagoras to the Tonality Diamond of Harry Partch

#### by Scott Eggert

**Overview**: For most Western musicians, twelve-tone equal temperament is simply a fact of life, the undisputed foundation of our art. Although microtonal composers and theorists do posit their alternative approaches, rarely is the question asked: what originally compelled us toward a twelve-tone system? The answer is to be found in the intersection between music and philosophy. The number twelve is directly connected to astrology, the signs of the zodiac. Each tone was meant to represent a position of the sun in the heavens, a connection to musica mundana or musica universalis, the “music of the spheres,” an idea we trace back to the philosopher who supposedly discovered the relationship between tone and number: Pythagoras of Samos (ca. 570—ca. 490 BCE).

Pythagoras believed that number was the essence of the universe, and that understanding the numerical proportions of harmonics was the key to understanding the universe—of literally unifying our consciousness with the mind of God. To this end he discovered or devised mathematical concepts such as the mensa pythagorica, a table of ratios derived from the study of the monochord, also known as the Lambdoma. The Lambdoma is considered by modern Pythagorean researchers to be one of the cornerstones of his philosophy.

Harry Partch, pioneer of the modern microtonal movement, studied the art and philosophy of the ancient Greeks at length, tracing what he learned from Helmholtz’s *On the Sensations of Tone* and Kathleen Schlesinger’s writings on Greek music back to the original sources, and what he perceived to be the original Greek aesthetic. Though it is doubtful that Partch studied the Pythagorean Lambdoma at any length, or ever discovered the writings of neo-Pythagoreans such as Albert von Thimus or Hans Kayser, the configuration and underlying concept of Partch’s Tonality Diamond bears so much similarity to the Lambdoma that it must be regarded as a further development and distillation of the original abstract idea into practical application and corporeal form.

By examining this connection we trace a line from the modern work of a twentieth century American composer back to a mathematical construct at the heart of ancient Pythagorean (and hence Platonic) teaching and philosophy, analyzing likewise how an instrument such as the Diamond Marimba is incompletely understood without pressing how ancient philosophy manifests itself in a mathematical conception of microtonality.

To view the full article, click here.

To view the full 80-page study that situates the history of the Pythagorean table as it manifested in the work of nineteenth-century and modern German theorists as well as American composers and instrument creators, click here. The following is the table of contents for Scott’s longer study of the Lambdoma across time and culture:

### Musica Universalis

TABLE OF CONTENTS

APOLOGIA……………………………………………………………………………….6

Introducing the Lambdoma………………………………………………………….9

CHAPTER 1: PYTHAGORAS………………………………………………………….11

His Life…………………………………………………………………………………11

Number as Essence…………………………………………………………………..12

Math as Music…………………………………………………………………………13

CHAPTER 2: HARRY PARTCH……………………………………………………….15

Helmholtz………………………………………………………………………………16

CHAPTER 3: JUST INTONATION…………………………………………………….19

The Powers of 2 and 3………………………………………………………………..25

The Tonality Diamond……………………………………………………………….27

CHAPTER 4: PLATO……………………………………………………………………32

Republic and Timaeus………………………………………………………………..32

The Nicomachus-Iamblichus Connection………………………………………..36

CHAPTER 5: THE GERMANIC REVIVAL…………………………………………..38

Albert von Thimus…………………………………………………………………….38

Hans Kayser…………………………………………………………………………….40

Levy, Levarie, McClain………………………………………………………………..42

CHAPTER 6: THE NEXT GENERATION OF PYTHAGOREANS………………….47

Rudolf Haase & Werner Schulze……………………………………………………..47

CHAPTER 7: THE INSTRUMENT BUILDERS……………………………………….51

Barbara Hero & the PHLK…………………………………………………………….51

Harry Partch & the Diamond Marimba…………………………………………….54

CONCLUSION…………………………………………………………………………..60

APPENDIX I: The Theology of Numbers…………………………………………..64

APPENDIX II: Glossary of Pythagorean Musical Terminology………………..66

APPENDIX III: The Quarternaries of the Tetraktys………………………………69

APPENDIX IV: The Complete Timeline of the Lambdoma……………………..71

APPENDIX V: The 43-Tone Scale of Harry Partch…………………………………73

BIBLIOGRAPHY…………………………………………………………………………..75

PHOTOS…………………………………………………………………………………….79