A NEW PAPER "The Harmonic Matrix: Exploring the geometry of pitch" HERE

This old version offers the basics of harmony and ratios but incorrectly labels the lambdoma as the Harmonic Matrix. The Grand Harmonic Matrix is now called the Harmonic Matrix, and the lambdoma retains its proper name. This mislabeling has been corrected in the new paper.

Introduction

What is presented here is a condensed overview of the "Harmonic Matrix" theory.

It seems that this is best understood by those with strong music theory backgrounds and basic mathematical understanding. The longer explanation which gives a history, anecdotes, ways of testing and discovering of the materials will follow with the online release of the original paper written in the summer of 2008. Written mainly for myself, amusement, and to share with others for advice, it has been revised somewhat but retains most of its original features. It will be presented here at a later date for those who wish to hear these sounds for themselves, rather than just reading the theory [a pdf version without notation can be obtained by writing
This e-mail address is being protected from spambots. You need JavaScript enabled to view it
].

N.B. Actually hearing these relationships of harmony, changes the way this topic is understood, & is more than just "highly recommended". The ideas here can be grasped by mathematicians easily. Listening is crucial!]. The aural discrimination of these harmonic relationships is an integral part of understanding.

THE HARMONIC MATRIX

Prime Materials of Harmony - Summary Introduction

To begin, consider a matrix, that contains ratios based on its addresses (+1) such that horizontally there is a series of multiplication ratios and vertically the divisors. This correlates to the dominant harmonic series along the horizontal, naturally occuring acoustic phenomena. i.e.. the harmonic series of the textbooks [aside: this stolen web image contains superfluous intervallic nomenclature. Simply let the ratio be the ratio: 4/1, 7/1, 11/1, 13/1, etc..]

This then is 'PA' [a term from Indian solfege, related to 'so' in the West], or the dominant harmonic series. It is a multiplication series of a fundamental frequency. Structurally, one can see the octaves occurs most, followed by the perfect fifth [ratio 3], the major third [ratio 5], then the seventh [ratio 7].

Along the vertical axis the divisor, 'MA', [related to 'fa'] or the shadow side of harmony, is revealed. The subdominant harmonic series, MA gives the harmonic basis for the minor scale, and reinforces the patterning of the dominant. Here we find harmonic basis for the minor scale with its melodic and harmonic modes.

To give MA her due, one hears that she similarly divides along the power of two series. But MA's great strength is that the division along the 3rd, is the generating force of the power of 2 series. One discovers that the underlying tonality of MA is 3, 6, 8, 12, 16, 24 ...

As just intonation describes a/b ratios, the Pythagorean series is a power series [the cycle of fifths ..., 2/3, 1/1, 3/2, 9/8, 27/16 ...] the two overlap [e.g. 9/8].

The Pythagoreans limited music theory to the octave series, and the perfect fifth, or 2 and 3. This network is the filled in with the appropriate thirds [the ratio of 5] in most music and our triads, major seconds [(3^2)/(2^3)], sevenths [7, (2^4)/(3^2)], have a quality of consonance, that is appealing to the ear. Fractally, we have a groups of sets, interlocked at the simple integer level that when extended beyond the traditionally useful [thus far in music history, that is] intervals, contain ways of infinitely dividing [alternately quantizing] the universe of harmonic relationships. The complexity arising between various versions of the intervals is hard to realize on traditional instruments, and becomes complex, very quickly. To reconcile the difficulty, Western music theory has sought, and found an irrational [numerically speaking - not a value judgment] additive series of twelve pitches (twelfth root of 2, 2^(1/12)), the Equal Temperament scale.

Returning to the MA & PA relationship, the resulting matrix displays the entire set of whole number ratios, and therefore all just intervals within the real numbers. There has always been a tendency to reduce 2/2 to 1/1. Don't do this! (yet) "Let x=x", or, a/b=a/b. In doing this all relationships with regard to any fundamental assumption, here musically the ratio 1.0, can be described and viewed structurally, revealing the relationship between all intervals.

^{[right click "view image" for larger view] }

Of note is [what I will term] the 'identity horizon along the diagonal, the repetitions of the harmonic series at each transposition level, and if one calculates the occurrence of resulting tones, we again find the two harmonic series. In the following diagram, relationships greater than an octave are transposed back into the range .5 to 2.0. Here is the "Harmony Cone" or "Code of Silence"

n=16

From this we can invert or rotate the matrices [not shown] and deduce two further matrices, the PA Squared [P^{2}] and MA Squared Matrices [M^{2}].

n=8 n=8

Put the four together and we arrive at the Grand Harmonic Matrix:

n=128

The Pythagorean Matrix can similarly be constructed through its integer and rational iterations. [The Iterative Power Matrix - descending only.]

In the representation of all the above, the limit has been 1/2 and 2, or based in log2. This is similarly arbitrary, and can be extended to any power series imaginable [log(3), log(5), etc.]. The cross multiplication of the Grand Harmonic with the Grand Pythagorean Matrix* completes a description of the prime materials of harmony.

Specialized structures [for example, Equal Temperament and other irrational divisions, the geometric divisions of any interval into any number of parts, or arbitrary divisions and processes] are also defined and included within the parameters of this discussion. These will be left until later.

This concludes a concise introduction to Harmonic Matrix Theory.