These three books are sources of inspiration to me, highlighting how the Pythagorean sense of number differs from our typical modern way of seeing mathematics.

Fideler, David. Jesus Christ, Sun of God: Ancient Cosmology and Early Christian Symbolism. Wheaton: Quest Books, 1993. A quote:

We are led to the primary realization that the laws of harmony are truly universal principles, of the same order of magnitude as other universal phenomena, such as light, energy, or gravitation. When someone decides to create a tuning system, certain elements of that system may be based on particular motive, but the underlying framework of every tuning system, and the key intervals encompassed therein, are dictated by the inescapable universal laws of harmony: the perfect consonances. This is confirmed by musicologists, who have discovered that music from around the world is based on these same perfect consonances, which provide the framework for all harmonic expression (p. 189).

Schure, Edouard. Pythagoras and the Delphic Mysteries. Whitefish, MT: Kessinger Publishing, 1995. A quote:

Pythagoras called his disciples mathematicians because his higher teaching began by the doctrine of numbers. These sacred mathematics, however, or science of principles, were both more transcendent and more living than profane mathematics, which alone are known to our savants and philosophers. In them Number was not regarded as an abstract but as the intrinsic and active virtue of the Supreme One, of God the source of universal harmony. The science of numbers was that of the living forces, of the divine faculties in action and in the universe and in man, in the macrocosm and in the microcosm. In examining them, Pythagoras was evolving nothing less than a rational theogony or theology (pp. 84–85).

Strohmeier, John, and Peter Westbrook. Divine Harmony: The Life and Teachings of Pythagoras. Berkeley: Berkeley Hills Books, 1999. A quote:

For Pythagoras, mathematics was a bridge between the visible and invisible worlds. He pursued the study of mathematics not only as a way of understanding and manipulating nature, but also as a means of turning the mind away from the physical world, which he held to be transitory and unreal, and leading it to the contemplation of eternal and truly existing things that never vary. He taught his students that by focusing on the elements of mathematics, they could calm and purify the mind, and ultimately, through disciplined effort, experience true happiness.

Resources on Music

Danielou, Alain. Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness. Rochester, VT: Inner Traditions, 1995. A quote:

All music is based on the relations between sounds, and a careful study of the numbers by which these relations are ruled brings us immediately into the almost forgotten science of numerical symbolism. Numbers correspond to abstract principles, and their application to physical reality follows absolute and inescapable laws. In musical experience we are brought into direct contact with these principles; the connection between physical reality and metaphysical principles can be felt in music as nowhere else. Music was therefore justly considered by the ancients as the key to all sciences and arts — the link between metaphysics and physics through which the universal laws and the multiple applications could be understood (p. 1).

Fideler, David. “A Note on Ptolemy’s Polychord and the Contemporary Relevance of Harmonic Science.” In Alexandria 2: The Journal of the Western Cosmological Traditions, edited by David Fideler, pp. 167–182. Grand Rapids: Phanes Press, 1993. A quote:

In terms of music, our culture presents it as a commodity rather than a path of spiritual development. Most musicians learn technique, or how to play a particular piece, and don’t even know what a perfect fifth really is in an ultimate sense . . . Let us restore the pursuit of music to its rightful place in our lives and society, and let us recall those lines of Congreve on the magical and therapeutic power of harmony, the magical music of Orpheus:

Music has charms to soothe the savage breast,
To soften rocks, or bend a knotted oak,
By magic numbers and persuasive sound (p. 171).

Fux, Johann Joseph. The Study of Counterpoint from Johann Joseph Fux’s Gradus ad Parnassum. Translated and edited by Alfred Mann. New York: W. W. Norton, 1971. (This book has offered one of the best ways for me to become more sensitive to intonation and the subtle beauty of intervals when using and exploring tuning systems other than equal temperament.) A quote:

My object is to help young persons who want to learn. I knew and still know many who have fine talents and are most anxious to study; however, lacking means and a teacher, they cannot realize their ambition, but remain, as it were, forever athirst.

Seeking a solution to this problem, I began, therefore, many years ago to work out a method similar to that by which children learn first letters, then syllables, the a combination of syllables and finally how to read and write. And it has not been in vain. When I used this method in teaching I observed that the pupils made amazing progress within a short time. So I thought I might render a service to the art if I published it for the benefit of young students, and shared with the musical world the experience of nearly thirty years, during which I served three emperors (in which I may in all modesty take pride). Besides, as Cicero quotes from Plato: “We do not live for ourselves alone: our lives belong also to our country, to our parents, and to our friends” (pp. 17–18).

Gadalla, Moustafa. Egyptian Rhythm: The Heavenly Melodies. Greensboro, North Carolina: Tehuti Research Foundation, 2002. (I love his assignment of planets and notes to each day of the week and each hour of the day.) A quote:

In the 4th century BCE, Plato recommended that the Ideal State be erected upon the foundation of music — a theory of the psycho-physiological effects of music on the State and on man. Plato’s recommendation was the adoption of Ancient Egypt’s system and practices, as stated in Plato’s Collected Dialogues, in Laws II [656c–657c] (p. 77–78).

Lloyd, Llewllyn S., and Hugh Boyle. Intervals, Scales, Temperaments: An Introduction to the Study of Musical Intonation. New York: St. Martin’s Press, 1979. (This work provides wonderful material on the mathematics of dealing with ratios, intervals, hertz, cents, scales, and temperaments.) A quote:

The flexibility of the scale is demanded first, in order that consonant intervals may be maintained at their correct size whatever their position in the scale, and secondly, so that melodic intervals, and those included in enharmonic change . . . may be allowed to vary in size in accordance with the demands of the musical consonances.

Mathieu, W. A. Harmonic Experience: Tonal Harmony from its Natural Origins to its Modern Expression. Rochester, VT: Inner Traditions, 1997. (This book was the first to open my mind to the fact that the note A that comes from a successive cycle of perfect fifths as F-C-G-D-A is not the same A as the A that is the overtonal major third from F. This is shown on my harmonic lattice diagram. I still wonder at the fact that I found this book at the famous Bodhi Tree Bookstore in Los Angeles, as it was responsible for the beginning of my “tonal enlightenment.” This book is a treasure.)

Opsopaus, John. Greek Esoteric Music Theory Charts. ©1999. Website: www.cs.utk.edu/~mclennan/BA/GEM/ (This was my introduction to the Pythagorean Greater and Lesser Perfect Systems, and their correspondences.) A quote:

The study of Greek Esoteric Music is a lifelong pursuit . . . The present work is primarily a set of annotated and cross-linked charts to serve as an introduction to the theory and practice of Greek Esoteric Music. The focus is on the esoteric aspects, including practical exercises, as opposed to the theory and practice of mundane music (interesting and worthwhile though that may be).

Sassmannshaus, Kurt. Violin Master Class: The Sassmannshaus Tradition for Violin Playing. Website: www.violinmasterclass.com. (In the intonation section, there is a clear explanation of Pythagorean and just intonation with examples of differences and when each is appropriate.)

Resources on Number

Andrews, W. S. Magic Squares and Cubes. New York: Dover, 1960. A quote from the introduction:

Pythagoras say that number is the origin of all things, and certainly the law of number is the key that unlocks the secrets of the universe. But the law of number possess an immanent order, which is at first sight mystifying, but on a more intimate acquaintance we easily understand it to be intrinsically necessary; and this law of number explains the wondrous consistency of the laws of nature. Magic squares are conspicuous instances of the intrinsic harmony of number, and so they will serve as an interpreter of the cosmic order that dominates all existence (p. vii).

de Purucker, G. Studies in Occult Philosophy. Pasadena, CA: Theosophical University Press, 1973. (The section of this book, “Esoteric Hints on Cycles,” pp. 3–15, is where I found the beautiful concept of a number with it’s “dawn and twilight.” From my own research, I discovered that musically this “dawn and twilight” is the 6/5 ratio of a minor third.)

Michell, John. The Dimensions of Paradise: The Proportions and Symbolic Numbers of Ancient Cosmology. San Francisco: Harper & Row, 1988. (A wonderful resource for the significance and deeper meanings of numbers in the cosmological canon.) A quote:

The special regard paid to mathematical studies in the ancient world arose from the understanding that number is the mean term in the progression from divine reason to its imperfect reflection in humanity. At some very early period, by a process quite beyond explanation, certain groups of numbers were brought together and codified. Thus was created that numerical standard, or canon of proportion, which was at the root of all ancient cultures and was everywhere attributed to some form of miraculous revelation. It was taken to be the nucleus and activating principle of number generally, a summary of all the types of progression and relationships which occur within the field of number and thus a faithful image of the numerically created universe (p. 7).

Plummer, L. Gordon. The Mathematics of the Cosmic Mind: A Study in Mathematical Symbolism. Wheaton, IL: Theosophical Publishing House, 1982. A quote:

This number 72 will serve to introduce to us a large family of occult numbers, and in time we shall come to recognize them as friends as they appear time and again. Some of them that we should recognize right away are the all-important ones which are related to the cycles of the zodiac, as for instance 2,160, as the number of years connected with 1 sign of the zodiac, and 25,920 as the number associated with the complete Precessional cycle (p. 32). (See gematria section of this website for correspondences.)

Taylor, Thomas. The Theoretic Arithmetic of the Pythagoreans. London, 1816. Reprint. York Beach, ME: Samuel Weiser, 1978. A quote:

Iamblichus in his treatise On the Arithmetic of Nicomachus observes “that certain numbers were called amicable by those who assimilated the virtues and elegant habits to numbers.” He adds, “that 284 and 220 are numbers of this kind; for the parts of each are generative of each other according to the nature of friendship, as was shown by Pythagoras” (p. 119–120).

Resources on Geometry

Lawlor, Robert. Sacred Geometry: Philosophy and Practice. London: Thames & Hudson, 1982. (This book offers wonderful reading as well as challenging exercises for drawing with a compass and straightedge.) Two quotes from the chapter “Mediation: Geometry Becomes Music”:

The most important and mysterious character of the harmonic proportion is the fact that the inverse of every harmonic progression is an arithmetic progression. Thus 2, 3, 4, 5, is an ascending arithmetic progression while the inverse series, 1/2, 1/3, 1/4, 1/5 is a descending harmonic progression (p. 81).

The law of musical harmony, when viewed from the idea of mediating proportion, becomes a symbol for the law of natural order, the Tao of the created worlds, where oppositional yet simultaneous movements interact to create both sound and form (p. 82).

Macaulay, Anne. “Apollo: The Pythagorean Definition of God.” In Homage to Pythagoras: Rediscovering Sacred Science, edited by Christopher Bamford, pp. 245–270. Hudson, NY: Lindisfare Press, 1994. (This incredibly beautiful article links the gematria of the name Apollo with ancient tunings of the kithara and presents them geometrically. The geometrical figure on the homepage of this website is the Apollo geometry figure based on this article.)

Resources on Color

Achilles, Constance. “Color and the Trigrams.” Fulcrum: The Science Journal of the University of Science and Philosopy 2:3 (February 1994), pp. 13–29.

Gerritsen, Frans. Evolution in Color. West Chester, PA: Schiffer, 1982. (I studied color theory for a number of years, and this book was of tremendous value in developing my understanding. I use color as information in all my work. Color coding of notes has been invaluable when trying to separate four or five different ratio versions of the same note name.)

Resources on Gematria

Barry, Kieren. The Greek Qabalah: Alphabetic Mysticism and Numerology in the Ancient World. York Beach, ME: Samuel Weiser, 1999. (The appendix of Greek gematria values is a gold mine. I use it every day in trying to establish the links between number, pitch, interval, and word [gematria]). A quote from the preface:

The main thesis of this book is the Qabalah is, in fact, a late Jewish term for a gnosis that was already ancient when it emerged in Jewish mysticism. It was, in fact, the Greeks who, as early as the eighth century B.C.E., invented alphabetic numerals, the very essence of Qabalistic numerology (p. xiii).

Bond, Frederick Bligh, and Thomas Simcox Lea. Gematria: A Preliminary Investigation of the Cabala. Oxford, 1917. Reprint. London: Research into Lost Knowledge Organization, 1977. A quote from the chapter entitled “The Method of Gematria”:

About the fifth century B.C. there begin to appear in the Syro-Phoenician centre east of the Mediterranean, traces of a mode of writing in which the letters of the alphabet serve also the purpose of numerals. . . . From this parent influence two systems derived. These are the Greek and the Hebrew. . . . Their alphabets, which are also numerals, exhibit unexplained features, some of which may be described as mysterious. It is scarcely reasonable to suppose that the element of chance has in any appreciable degree entered into their framing. And this is the more unlikely in that there is evidence of a contrary belief among these peoples, who shewed a peculiar reverence for their alphabets, ascribing to each letter its own mystical value, and, to the whole, a body of symbolic teaching in which the principles of Number, Sound, and also Form as connected with each letter, all played their part (pp. 5–6).

Websites

Bernhard Deutz. Builder of monochords and other musical instruments.