_______ 7. The Harmonic Matrix PDF Print E-mail
Written by D. Bond   
Tuesday, 25 May 2010 21:07

7. The Harmonic Matrix


            let x=x

                        Laurie Anderson


All of the systems we have thus considered fall within a category I term ‘constellation’ building. One result of constellation building is a scale. Another is a set of relationships between the tones. Similar to stellar constellations, we don’t see the 3-dimensional relationships between the stars, and must deduce it from other properties [spectral properties, redshift, etc]. We see the constellation as flat in the sky.


Likewise, these constructions of harmonic relationships don’t necessarily reveal their inner structure when heard as scales. Here the details of the scale are lost, only to be revealed when the notes are played together. Try singing Partch’s 43 tone scale, if you dare. Some melodic differences when played on a keyboard verge on the barely noticeable, but then try playing intervals, and the relationships becomes crystal clear.


One of the confounding factors in the study of these relationships is our tendency to reduce to simplest terms wherever possible, whenever possible. This is avoided in the following definition.


Here we define a matrix that contains all possible just intonational ratios:


            H[n,n] = (cell[0]+1)/(cell[1]+1) where n = infinity


This matrix may be somewhat large to investigate right off the hop, or in the digital realm, so let’s examine the first few simple matrices.


Given H[1,1] we derive the identity matrix [unison].




Given H[2,2] the octave identities appear:


            1/1      2/1

            1/2      2/2                 


The tendency to reduce often compels us to consider this group as simply 1/1, 2/1. Maybe, we can be forgiven for ignoring 2/2 as it is equivalent to 1/1 but what could be the reason for consistently failing to recognize poor 1/2  and the relationships that fall between it and 1?  This is perhaps less surprising when we consider that strings can be divided by placing a finger or stop, but a ‘lengthening’, any of the ratios between (0 < r < 1] each requires a different string. 

In this system, you can see that string ‘lengthening’ is represented in the rows of the matrix, and string ‘division’ in the columns.


Now we introduce the first  true ‘number’, 3:


            1/1      2/1      3/1

            1/2      2/2      3/2

            1/3      2/3      3/3


Notice we have added 5 new ratios, but only 2 new notes [2/3, 3/2]. Here for the first time appear our three great pillars of harmony: IV, I, V[1]


Please notice that IV is not presented here as it normally is as 4/3, but as it first appears, under the tonic as 2/3 in a symmetrical inverse relationship to 3/2. This relationship, I call ‘complementary opposites’.


It is for this reason, I have not adopted Partch’s designation of Otonalities and Utonalities. Seeing that these are reciprocal of one another and actually form one continuous series, there is no real ‘3’ tonality and a ‘4’ tonality. 3/1 refers to the dominant while 1/3 refers to the subdominant. This I believe, is more accurate and descriptive of the symmetry of the relationship. Besides, when other relationships that do not have conventional terms, such as  [7, 11, 13], begin to emerge, complicating the issue with history laden terminology such as ‘tonality’ only confuses the issue.


Now, when we extend to H[4,4], notice that no new notes are produced, but our two favourite friends, IV and V invert themselves and join their partner in each octave 4/3, 3/4. It may be too early to tell, but by this stage, the pattern is firmly established, the fractal shape fully defined.




When we get to 5, the next prime number, a group of new notes make their first appearance: 3/5, 4/5, 5/4, 5/3. These represent the ‘limit of the 5th’ in Patch’s terminology, or those beautiful just thirds and sixths we can now learn to love and adore all over again.


Upon reaching 6, the first perfect number according to Pythagoras (1*2*3=1+2+3), we see the only new notes are inversions and octave transpositions of previously identified pitches. A pattern is now beginning to emerge.




We now have the makings of some very beautiful triads. Music for lovers! Sacred music, powerful music, deep music…


            D Major, G major, Bb Major, G minor, D minor, B minor


Combine this with a few transposition planes at Pythagorean ratios  such as 3/2, 2/3 and you’re holding the cube[2] containing the relationships for the majority of chord progressions in the known universe! Now this is something worth patenting!


How I should like to spend more time toying with the astonishing beauty of these primal relationships, but there is much territory to cover and we’ve had literally thousands of years to hear these perfect triads [and decided to ‘sour’ them just the same], so let’s skip along to H[8,8] to find the next octave (2^3).





With this step the 7 labours are accomplished! All the new pitches that appear with the arrival of 8 are by-products of astonishing arrival of 7.


Here it is: our beloved 7th, accompanied with a retinue of new faces, 7/6, 7/5, [7/4, 7/2, 7/1, we’ve already met]. But who’s that striking beauty on the other side? Such strength, such poise! 1/7, 2/7, 4/7, 8/7 [it’s poetry, really], with her attendants 6/7, 5/7.


If 7/4 is the true form of b7, C, then this must be the true form of the m6, E. 


We begin to notice that as the dominant harmonic series ascends upwards in a just dominant 7th, D F# A ~C, the subdominant series is descending in a mirrored direction, D Bb G ~E, a G minor 6th chord or extraordinarily tenuous ~Em7b5.


Scale builders out there now begin to throw their hands up:  We have gone far enough! There are enough pitches now, too many to handle in fact, especially when we start adding all the fifths and thirds in relation to the new tones!


But we are not done yet. Not by a long shot. The next octave doesn’t occur until 2^4 or 16, and the size of our matrix will quadruple from 64 to 256! Mercifully, 16*16 is the largest input matrix I intend to work with. There are a some good reasons for this. First, my keyboard has only 4 octaves and to access this size of matrix in its entirety I would need a larger MIDI keyboard or a different approach. Also, awkward fingers stretches would be problematic. But the most important reason, is that 16 is more than enough for us to see the emerging pattern.


I imagine there may be a time when 32^2 or larger, becomes desirable and manageable, but for now we can safely say we have enough information to see the bigger picture.



Easier to read:








Limited [0.5, 2.0]


next - 8. The Reflecting Mirror

[1] As the perfect 5th  is related to ‘3’ the major 3rd  is related to 5, this becomes confusing. I rarely refer to these as IV and V but retain 3/2, 2/3 for ease of understanding. In deference to history, however, we employ both designations as needed.

[2] I’ll examine harmonic cubes in more detail in another part.

Last Updated on Tuesday, 25 May 2010 22:03