THE HARMONIC MATRIX
Prime Materials of Harmony  Condensed Summary
N.B. THIS ARTICLE IS OUT OF DATE.
A NEW PAPER "The Harmonic Matrix: Exploring the geometry of pitch" HERE
]]>The Harmonic Matrix:
prime materials of harmony & intonation
Daryn Bond
edited: 2009.10.10.
source material composed: 2008.07.25
1st draft completed: 2008.09.16
discussion document
For my sister, Jolene.
One day we will swim in rooms of jello!
“With the divine ear element which is purified and surpasses the human, may I hear both kinds of sounds, the divine and the human, those that are far as well as near.”
Anguttara Nikaya 3:100 110; I 25356
Translated by Bhikkhu Bodhi. In the Buddha’s Words.
Contents
0. Imperfect Pitch................................................................................................................ 4
1. It Goes Like This.............................................................................................................. 5
2. Thus Have I Heard........................................................................................................... 7
3. Oh, the inJustice!......................................................................................................... 12
4. Ring the Bells – a crack in the universe............................................................................ 14
5. Spacing the Spheres – the protractions of scale building................................................. 19
6. The Devil’s Playground  Diablos en Musica.................................................................... 22
a. the howling wolf.......................................................................................................... 22
b. pitchforks: tunings and temperaments......................................................................... 23
c. the unbinding of heaven [fixing the piano].................................................................... 26
d. in the garden.............................................................................................................. 28
7. The Harmonic Matrix...................................................................................................... 30
8. The Reflecting Mirror: Mysterium Contrapunctus............................................................. 35
10. The Great Work of Counting......................................................................................... 40
11. As Above and Below…................................................................................................ 43
12. The Completion Stage................................................................................................. 47
References & Works Cited................................................................................................. 53
 There is no place for speculation in scientific discourse.
Sir Isaac Newton
I don’t have perfect pitch. I have trouble identifying intervals by ear. I am a terrible piano player. I’m an even worse computer programmer. That I should find myself compelled to write this paper at this time, I find quite phenomenal.
These failings notwithstanding, I will be presenting a brief discussion on the prime materials of musical harmony and intonation. I use the term ‘prime materials’ alluding perhaps to alchemical prima materia among other things but mostly as an indication of the importance that prime numbers and their interactions have in this theory of harmony.
I have tried to be as clear as possible within the limits of my own understanding. Please remember, this is not a text book. I make no claims of mathematical completeness or scientific rigour. I possess no degree of authority with the exception of that insight and knowledge that I have attained through my own studies and work on this subject. All mistakes found within this work are my own. I pledge to correct and clarify to the best of my ability as time passes.
Being thus unencumbered by academic constraints, I feel at liberty to include criticism, speculation and metaphors a good editor would likely find superfluous. I leave this in, in hopes it may provoke further thought, and perhaps entertain a bit too.
I would include here the names of those kind few who have helped me remove errors and clarify this paper, but so far there haven’t been any takers.[1] Clearly, more work needs to be done in this regard. This is far from the final version of this paper. Comments and questions most welcome [bond@bondinstitute.net].
Thus:
Here begins an exposition of the prime materials of harmony.
[1] Until recently: Thank you to Professor Orjan Sandred for reading and listening, and for his instruction in high quality recording techniques. I have failed to edit this paper based on his recommendations, choosing instead to write a fresh summary article, and to work on my sampling technique. This version, as Dr. Sandred rightly pointed out, offers little that is new until page 30, aspiring to the condition of a ‘textbook’.
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it goes like this:
the fourth, the fifth,
the minor fall,
the major lift.
Leonard Cohen
In the computer age, a study of musical harmony can now be conducted with a precision unknown to previous ears.[1]
In this discussion we will revisit concepts of pitch, intonation, consonance and dissonance, with the aim at advancing a cohesive theory of harmony.
Naturally occurring acoustic phenomena, such as the harmonic series, summation and difference tones, beats, et al, will serve as our reference point, providing the theoretical and experiential foundation for the discussion, and provide us with a motive for engaging in this exploration.
These three  Just Intonation, Pythagorean tuning and Equal Temperament [ET]  will be examined. Their histories will be analyzed insofar as the fundamental principles can be demonstrated. These principles will form the basis of a generalized system of intonation and harmony.
This effort serves to expand the musical resources of composers beyond the traditional 12 equally tempered pitches, to include the realm of harmonic and acoustically sound relationships founded upon the Pythagorean and just intonational systems, and beyond. The theory presented here will be shown to be flexible enough to accommodate alternate, irregular, irrational systems of arbitrary complexity.
Over the course of this discussion we will describe a variety of sets, each consisting of an infinite number of pitches organized with respect to their relationships. These sets have various properties of symmetry, inversion, repetition and selfsimilarity. We will begin to use the basic properties of these harmonic sets and in a preliminary fashion, generalize and expand upon them.
Along the way, aspects of the aesthetics and performance of music using this system will be investigated with reference to human perception, the limits of the ear, and instrumental practice. Whether the ear is capable of discriminating the enormous variety of relationships described here, and whether one could expect traditional performers to consistently realize these in live concert situations will be addressed. Challenges involved in the notation of these discreet relationships will be touched upon briefly.
In future writings and compositions, I plan to take up the vast topic of suggesting various ways of combining and manipulating these sets in compositionally interesting ways and the design of instruments and interfaces for the management and control of these resources. For now we keep it simple, using only a traditional electronic keyboard.
To begin, we set out on this exploration with a computer running the swissarmy knife of music gear, Max5/MSP/Jitter, a MIDI keyboard input and 4 octaves of keys. Example patches are included in a folder, but also compressedtext files will be included with this document.[2]
[1] John Cage once wished us all ‘Happy New Ears’ and lifting a glass in return, we begin.
[2] This will not be done  yet. People wishing to inquire about these patches can write to music@bondinstitute.net. I’d be happy to share them.
Acoustic Observations
The following will present a few features of acoustic theory salient to this discussion. No attempt is made to cover this topic in detail and interested readers may consult the references listed in the bibliography for further reading.
We begin with a pure tone and demonstrate consonance inherent within the ‘nonlinearity of the ear’, that is, our perceptions curve around a median range. The amplitude of a pure tone varies the perception of harmonics we hear in a sound.[1] The pitch of a tone appears to shift upwards as its loudness increases and more harmonics are observed.
For demonstration purposes a simple patch has been created to play pure sine tones. It plays these tones in relation to a “global_pitch_center” in cycles/second or Hz, that can be adjusted as required. Instead of using MIDI notes, ratios [converted to decimals] are sent to a “poly~” sound module.
Upon trying various pitch centers, harmonics can be heard prominently at the octave and fifth and perhaps even beyond [given some patience, ear training and the right combination of amplitude and frequency].
These harmonic overtones are related to the amplitude of the fundamental in their relative strength but moreover, in terms of their perceived pitch the relationship is constant i.e. if the fundamental is perceived to be slightly higher in pitch due to an increase in amplitude, the harmonics become more noticeable and proportionately shift upwards. This property demonstrates a natural consonance inherent within the perception of a pure tone itself.
Theory has it, if we play two tones together we begin to hear difference and summation tones. This table appears in Physics and Sound of Music [Rigden, 84]:
Interval  f_{1}  f_{2}  f_{2}f_{1}  2f_{2}f_{1}  3f_{1}2f_{2} 
Octave  f_{1}  2f_{1}  f_{1} 


Fifth  f_{1}  3/2f_{1}  1/2f_{1}  1/2f_{1} 

Fourth  f_{1}  4/3f_{1}  1/3f_{1}  2/3f_{1}  1/3 f_{1} 
Major third  f_{1}  5/4f_{1}  1/4f_{1}  3/4f_{1}  1/2 f_{1} 
Major sixth  f_{1}  5/3f_{1}  2/3f_{1}  1/3f_{1} 

Major third  f_{1}  6/5f_{1}  1/5f_{1}  4/5f_{1}  3/5 f_{1} 
The reader is challenged to verify these relationships using the tools provided.
I can offer only this brief summary of combination tones as they inform our discussion of consonance. The reason I cannot provide more detail is that personally I have great difficulty in hearing them. This may be because my audio equipment is not of sufficient quality, or that I cannot generate the sound at a high enough volume to create the effect.
Notice that according to the chart, if we play a perfect fourth at high enough volume and frequency, we should hear a missing fundamental, or the ‘fourth’ two octaves below [1/3]. Do you hear this? I’m not sure that I do. It may be interference with the sound of my fan in the background, or the humming refrigerator, or the neighbour’s dog barking in the background that make me sometimes believe that I do.
Honestly, I don’t really care to. I don’t find pure sine tones appealing to listen to for long as they can become somewhat tedious. Of pedagogical value no doubt, but the hearing of these difference tones and fundamentals I would equate to being taken into a desert and being told to stand there until one can see ‘the mirage’. After standing in the dust and heat long enough, you start telling your guide that ‘yes, you do see them’, when in fact you’re simply too hot and thirsty to care anymore.
In future demonstrations, we will employ examples using “groove~” and “sig~”, a simple form of granular synthesis [2] to produce complex tones. This alleviates aural fatigue, and with a bit of creativity we can toy with granular variables to introduce compositional processes operating at the micromeso level [20500 ms] for interest.[3] It is also important for another reason: complex tones are less subject to changes in pitch as a result of changes in sound level.[4]
The sine tone “micro_beep” module will always be available for use in testing of precise relationships when needed. Any of our tuning configurations will work with it too. Any sound producing module capable of accepting floats could be configured to work with this system.
Fire up the granular “etheramin” to play two pitches, holding one constant and allowing the other to slide freely. This little patch allows us to set the interval and the range that the free pitch can bend.
max patch: etheramin
Setting the bend range to roughly (0.95  1.05) tune the unison and explore the space around it. Experiment and notice at what point you begin to hear two sounds. I’m not telling my answers. This is called the ear’s ‘limit of detection’.
Now try settings of (about 1.95, 2.05) and listen to the octave and the space around it. Close your eyes and tune the octave. Did you get the right answer? I sure as heck didn’t! Close, but not quite.[5] Seems to me there’s more ‘room’ at the top of the octave, no?[6]
Next, tune the octave below (.49 .51). A phenomena called beats, the result of the phase cancellation of waves guides our ears in being able to tune these intervals.[7] I found it much easier to be in tune this time. How about you?
There is much to learn here. We see that there are perceptual tastes and preferences with regard to the intonation of various interval sizes and relationships. The amount of sliding and bending room we have to play with influences our decisions. This little experiment helps one appreciate how difficult it is to be ‘in tune’ or play with perfect intonation. It seems hopeless for acoustic instruments [as they are understood today] to be capable of producing these precise ratios.
Test this: tune an octave blindfolded and leave it untouched for a while. Now type in the number 2.0 and compare. Which can you tolerate longer? There is certainly an element of satisfaction in having access to this kind of precision.
Now for some fun: set the etheramin to (0.5, 2.0)
Do you draw the same picture that I do? [8]
I do not believe that limitations of traditional performance practice or the fluctuating tastes of the ear should be used as a criteria by which the usefulness of this method is to be evaluated. If our music doesn’t fit our instruments, then we are compelled to find a way to make the instruments fit the music. We can always retreat if it doesn’t work out. If it does, well, then there is the exciting possibility of something new, rich and strange developing that will alter the way humans perceive and interact with music.
The next example demonstrates the harmonic series, the series of integer multiples above the fundamental, or global pitch center. The MIDI keyboard has been adapted so that D# above middle C or MIDI note 63 has been designated as 1, E as 2, F 3, etc. This arrangement has been chosen for reasons of symmetry that will become obvious later, and because the pattern of integers in the series falls nicely across the keyboard. Notice that 1, 4, 8, 16 the octaves (2 being the exception), are all black notes, the odd numbers 3 5 7 9 are white notes, the pattern changes with 10 12 14 on the white notes, and 11 and 13, the prime pair C# and D# respectively.[9] ‘D’ is left undefined, producing division by 0 and other nasty nonsense. In this patch, it triggers a ‘bang’.
Obviously, one would want to develop a more logical pattern for ease of navigation if one were to build an instrument based on this series.
When playing the adapted keyboard, the series of tones we hear historically has been called the ‘harmonic series’. I will be calling it the ‘dominant harmonic series’ or ‘P series’ for reasons that I will explain as we continue the argument. It can be notated like this:
The pitches we hear in this sequence are integer multiples of their fundamental. This can be expressed as:
P = f*1, f*2, f*3, f*4 …
One can hear some absolutely beautiful elements of this series that we do not currently have in common practice music. Particularly the tones 7, 11, 13. Some elements are familiar – the perfectly intoned major triad between (3) 4 5 6, the major triad built on the dominant (3) found at (6) 9 12 15 [this time (3 4 5) * 3], yet there are also some combinations that may sound strange to our ears, perhaps out of tune! Yet I have found that if one takes time to familiarize themselves with the sound of these relationships, one comes to a new appreciation. It feels to me that no matter what combination of integer relationships I choose, the sound feels as though it ‘floats within its own ocean’.[10]
Before we leave this all too brief exposure to harmonic concord and purity, let us present a quick comparison: The relationship (3) 4 5 6 as previously mentioned creates a major triad. This can also be reduced to 1/1, 5/4, 3/2 or 1.0, 1.25, 1.5. Now listen to the triad produced by the equivalent relationship converted to equal temperament [ET]. The similarity cannot be denied, yet there is a quality of harshness involved in the ET version that is not present in the pure harmonic version.
Now if you’re ready for a shock, test the difference between 4 5 6 7 and its equal temperament version. Notice that the dominant seventh chord which we traditionally considered as being a dissonance that requires resolution, in the harmonic series has a quality of consonance and stability that does not demand any subsequent movement. Certainly not as consonant as the octave, perfect fifth or major 4 5 6 triad, but the ear accepts it as having an interior logic and harmony not found in the very turbulent ET version of the chord.
[1] Science of Music.
[2] Roads, Curtis. Microsound
[3] I use a simple bit of code based on an example posted by Andrew Benson, that I affectionately call, the ‘winds of chaos’.
[4] Rossing, p.113
[5] You get better with practice.
[6] would be interesting research project to invite musicians and nonmusicians to tune these intervals and compare.
[7] Rigden, 74.
[8] It helps if you’ve got a tablet to draw with. My sarcastic mantra invented to avert impending carpal tunnel syndrome: ‘I love clicking the mouse.’ When in the world are we going to come up with something better? No offense mouselovers, I got beefs with the standard typewriter keyboard too! Just don’t get me started…
[9] I gather most working in intonation put tape on their keyboards [as long as keyboards persist in their current form, that is ; ]. Reminds me of when I was a child and wrote the note name letters in permanent black marker on the plastic keys of my first Bontempi organ. Once I had learned the names I wanted to rub them out but they wouldn’t come off. I tried to cover them up with tape, working in reverse as to how I do today [and the way I would personally recommend others work] but much to my chagrin, the names of the notes kept showing through!
[10] try this to hear a familiar tune the way the aliens intended: 9 10 8 4 6
This thing began with truth, and truth does exist. For some hundreds of years, the truth of Just Intonation, which is defined in any good music dictionary, has been hidden. One could almost say maliciously.
 Harry Partch
At this point one may be tempted to place value judgments on the two systems, hailing Just Intonation as superior to Equal Temperament [ET], or go in the direction of the mystic and claim one as more soothing to the nervous system, having healing properties etc. [as I admit, I am inclined to do], but for now we just observe the difference.
Timbral considerations aside, there is a special color and quality present within each of the two chords. Composers and musicians should understand these as not equivalent and can make decisions regarding the quality of sound they hope to achieve using whichever intonational approach best suits their goal.
Clearly, there must be good reasons for sacrificing the intrinsic beauty of the harmonic series and its logical extension into just intonational systems. We will consider these reasons as we continue the argument and review the historical development of intonation practice in the West.
Given the preceding examples, we are approaching an understanding of consonance and dissonance that enable us to form a provisional definition of harmony. What has often been evaluated as a subjective perception we now in a position to describe in terms of fundamental acoustic phenomena and harmonic relationships appearing as properties of the harmonic series and its transpositions.
Most textbooks simply list commonly regarded intervals as consonant and others as dissonant, pointing at the harmonic relations of the tones in terms of their interval relations [and oddly, almost always from the basis of ET]. Rarely do they mention that these subjective opinions have changed over history, tending towards the acceptance of more complex relationships as consonant. For example, in the Medieval period, octave and perfect fifths alone were considered consonant and all others as dissonant, later fashion admitted thirds and sixths as secondary consonances, and in the modern era we permit the ET Maj 7^{th}, 9^{th}, 13^{th}, #11, etc. as completely acceptable. It is worth asking why this has happened, and particularly in this order: moving from 5ths to chords; from diatonic to chromatic saturation.
One may argue that the epitome of what is considered ‘in tune’ as heard in ET, would be considered as horrifyingly dissonant to a Medieval composer [or musician]. Yet somehow this sound has come to be accepted in the West as the only consonance. I recall, prior to my own personal investigations into intonation, I heard the pipe organ at the Trapiste Monastery[1] in Holland Manitoba during a short retreat there. The organ, which I now assume to be tuned to just or mean tone relationships, I found strangesounding and out of tune to my ears – yet oddly appealing, mysterious. I put it down to old age: that the instrument had simply gone off. It only occurred to me later that the instrument is intended to sound that way!
He restoreth my soul.
His mercy lasts forever.
World without end. Amen.
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Ring the bells that still can ring
Forget your perfect offering.
There is a crack in everything
That's how the light gets in.
Leonard Cohen 'Anthem'
Consider the harmonic series again. One notices that this simple arithmetic series (1, 2, 3, 4 … ) refers to a series of multiplications of a base frequency, and within this lies an even deeper identity: the series 1, 2, 4, 8, 16… , the octaves above the fundamental. We can also express this as a power of 2 series
2^0, 2^1, 2^2, etc.
This relationship based on the power of 2’s is so strong in music that it provides us with a definition of a ‘note’ that is, a name of a pitch or specific frequency irrespective of octave displacement. Any frequency in a relation of a power of 2 to another is called by the same name. 220 440 880 1760 etc. are all the note ‘A’.
f*2^n n={ …2, 1, 0, 1, 2…}
We can now construct a second series, the power of 3 series. It gives the fundamental, then 3 * f, 9 * f, which starts to get very high in frequency quickly but we can transpose downward using the octave identity, or power of 2 series.
3, 3/2, 3/4, 3/8 are all E
We have already seen that 3f, the 3^{rd} harmonic, is referred to as the musical fifth one octave above the fundamental. Thus 3/2f is the same note by the octave transposition identity
f * 3 * 2^1
I begin to build a set that looks like this:
G D A E B F# C# etc.
Perhaps we have waited too long to travel backwards in time to meet Pythagoras. This is his series after all. Certainly, world musics have understood instinctively these relationships and base their music upon them – the octaves, fifths, building 5 note, 8 note scales, etc. and these basic elements appear in most musics of the world – but credit for elucidating the numeric relationship between these fundamentally appealing and harmonious sounds is given to Pythagoras.
The story goes that Pythagoras passed by an iron workshop hammers were striking anvils. Some sounds were pleasant and in harmony with each other, others were not. After testing various hypotheses, he found that the difference was due to the relative weight of the hammers. Finding weights in proportion 6 8 9 12 Pythagoras identified the numeric relationships of the octave [6:12] the fifth [6:9] the fourth [8:12] and the second [8:9].[1]
This beautiful set just happens to be the lowest integer relation of its kind, [perhaps a finding as beautiful as the first perfect right triangle 3 4 5]. We can reduce and transpose the relationships:
1 1.33 1.5 2
1/1 4/3 3/2 2/1
I IV V I
or
.75 1 1.125 1.5
3/4 1 9/8 3/2
V I II V
or
.667 .889 1 1.333
2/3 8/9 1 4/3
IV bVII I IV
Pythagoras was duly impressed [as am I] by this and began to extrapolate upon the principle. Seeing no need to go beyond the integers 1, 2, 3, 4 – the basis of his holy Tetrakys[2] – Pythagoras devised what will be termed here, the Pythagorean series.
P3/2(n) = f*(3/2)^n n={0,1,2,3…}
Notice that this is not a scale but a series of interval leaps of a fifth. The inversion gives a descending series of musical fourths.
P2/3(n) = f * (2/3)^n n={0,1,2,3…}
To create a scale then, we would need to apply our octave identity to keep the series within a specified range. This is usually done by brute force, transposing as needed, enabling the creation of various modes. Historically the Pythagorean lyre had 8 strings tuned as:
1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1
In general, I will use here a function called “ratio_limit”, that will bring all ratios > 2 to between 1 and 2, and ratios < 1/2 back up to the 1/2 to 1 range:
RL(n) = pow(2, (log2(n)  trunc(log2(n)) [N.B. [3]]
Now here’s the rub: if we continue with the series (not the scale) up to n=12 we have traveled up 7 octaves, but
(3/2)^12 ≠ 2^7
P(0) 1.5 69. A3
P(12) 1.520465 69.234406 A3
Tantalizingly close. Here is the first obvious crack in the universe! [As always – more perfect than perfect]. This small difference, known as the Pythagorean comma has been the source of great trouble both theoretically and philosophically. In fact, it is said that Pythagoras knew of this comma but only revealed the fact to his closest initiates. Telling others of this secret, including lesser initiates, was punishable by drowning![4]
In observing the Pythagorean series limited to an octave, after 12 luscious notes, each a glorious perfect fifth away, we find a unison that wobbles, or alternately an octave that is stretched slightly. The sensation is one of almost of not being able to breathe because they’re suddenly so much more oxygen, much more room. Fill this plussized octave in with a biting Pythagorean triad however, and one begins to feel like a cat being skinned alive for gut. Certainly, no serious musician would abide such caterwauling. The natural tendency is to fix this interval to the perfect 2/1 octave. For now, enjoy the expanse and the throbbing in the room [and possibly in your head].
Resist the temptation to reorder the series for now. See how the cluster of major seconds on the keyboard produce the first notes of the scale, that the major third {K:M3} produces the socalled Pythagorean third {P:M3}
(12) 7 9 11 6 8 10 (locrian) (Lydian)
p +1 +4 major +10 dom7th\
p +1 +9 minor
p +2 +4 +1 +3
After taking time to familiarize yourself with this series, we’ll do a few more comparisons. See if these features stand out:
* the Pythagorean third is higher than the third of the harmonic series.
* the Pythagorean sixth is awkwardly high as a melodic step – [at least, this is what it says in most textbooks].
* compare again the triads in ET, the harmonic, and Pythagorean series.
Mathematically speaking, we note that while the harmonic series is multiplicative and linear, the Pythagorean series is a power series [logarithmic].
Most contemporary textbooks speaking from a modern bias describe the stretched octave [or wobbling unisons], the high thirds and ‘difficult to sing’ sixths as “problems”. Here we see them simply as features of the series. Certainly, it would be nice to construct a Pythagorean scale with a perfectly closed octave, but this cannot be done using any method. This mathematical fact has not stopped people from trying, however, and failing that, going to extreme measures to fix ‘the problem’.
Moreover, there is another complication: that of the harmonic series. People just seem to like the sound of 5/4 at times and at other times the sound of 81/64.
For fun, we use modern computer to crunch away forever and never come back to our beloved 2:1. A simple algorithm for generating the Pythagorean spiral of fifths is given. This algorithm uses a conditional statement rather than the log function used above. The result is similar.
On a technical note, this bit of code satisfies the formal definition of an algorithm, a process with a beginning and an end.[5] The beginning here is t=1 or the first step from 1 to 1.5. The end occurs when the algorithm returns to 1.0. I turned my version of this patch on in 2008/7/11 and it’s still running. Let me know if your version should ever stop!
I have sometimes been tempted to work out the pattern, but as usual I get sidetracked by the multitude of variations and complexities implied by this method and must leave that bit of math for another time to work out.
I will offer some preliminary observations: first, that our 12 notes migrate upwards by the distance of the Pythagorean comma at each turn of the wheel. Given ‘D’ as the center and keeping the note names constant, eventually after [x] number of turns, C# is closer to the ratio 1 than D now is, which is now close to Eb, etc. But this migration misses being exact by some amount I don’t have the patience to calculate. This series never repeats. If we turn up the tempo to a ridiculous number [480 BPM or so], I hear a pulsing firebrand of consonance whirling throughout spacetime. How about you?
Going further we find that there is no whole number ratio (a/b)^n which will divide the octave evenly.[6] The proof of this is related to Fermat’s Last Theorem and thus need not be discussed here. I’ll write it in the margin later.[7]
[1] Divine Harmony, Fermat’s Last Theorem.
[2] an equilateral triangle consisting 10 points arranged in 4 rows, 1+2+3+4.
[3] if we wish to keep the octaves, 1/2 and 2/1, then conditional expressions are required. See appendix.
[4] Temperament.
[5] Current Directions in Computer Music Research, Max Mathews, Ed.
[6] though 13 comes freakishly close, but has no 5ths. Didn’t god and the devil have an argument at one point?
[7] or put it in a footnote! Referring, of course, to Fermat’s famous margin note that claimed he had proof but didn’t write it out. Anything but obvious, the proof of this remained unsolved until very recently. See Fermat’s Last Theorem for the entertaining story of this problem and its eventual solution.
There is geometry in the humming of the strings, there is music in the spacing of the spheres.
Pythagoras
One begins to see how complicated the whole business of building scales out of just harmonic intervals becomes when one tries to construct scales that preserve the pure relationships of the harmonic and Pythagorean series. Beginning with the familiar pillars of harmony:
1/1 4/3 3/2 2/1
we start to choose relationships to fill in the gaps. A variety of solutions are possible. We look at the most basic one.
1/1 9/8 6/5 5/4 4/3 3/2 5/3 ? 15/8 2/1
d r ma mi f s l ta ti d
So now, what about ta [b7]? How about 16/9, the perfect fourth above the perfect fourth. 4/3 * 4/3? or the exotic but harmonic 7/4? or perhaps the perfect fifth above the minor third [6/5 * 3/2 = 9/5]?
If I choose to include 7/4 then maybe I’d also want the perfect fifth above it, 7/4 * 3/2 = 21/8, [down an octave to 21/16]. The major third between this pair would be 7/4 * 5/4 = 35/16, etc. [1] [21/8 * 5/6 = 35/16. Wow, math works!]
Clearly not all rational relationships can practically be included within a single finite scale. Already we see that the Pythagorean scale does not contain 12 pitches, but an infinite number of them. If we include more intervals than just 3/2, then for each new relationship this infinity becomes infinitely more complex, a higher order of infinity. The difficulty we experience in tackling this, as I will demonstrate later, is brought about by the desire to create ordered series of pitches called a scale, and is confounded by the complexity arising from the mixing of harmonic ratios, which follow a multiplicative series (r * a/b), with the coincident but divergent power series ((a/b)^n). For now, we simply conclude that no discreet collection of relationships [with the exception of the identity octave, and even this is debatable] will yield a finite collection of pitches.
So, inevitably theorists and musicians draw a line or put a limit upon the relationships they choose to work with, building scales and compositions out of a limited selection of materials. The number and variety of resultant scales is astonishing, and one need only look at some internet resources [Scala, et al] to find literally thousands of tuning tables.[2]
Ancient music theory has been largely concerned with the making of 8 note scales and modes. The Ptolemic sequences and modes of the Harmoniai [according to Schlesinger are found in Partch]. These present a lovely collection of historic scales. It is easy to construct a simple patch using precise ratios mapped along the keyboard octave to hear these beautiful constructions.
Now, I’d like to bring your attention to the 22tone Indian sruti scale. Its ratios are:
1/1 256/243, 16,15 10/9, 9/8, 32/27, 6/5, 5/4, 81/64, 27/20, 4/3, 45/32, 64/45, 3/2, 128/81, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 243/128, 2/1
Notice that this scale is symmetrical and invertible. The reference says “There are a great many takes on this”, but “the tonic, Sadja, and the pure fifth, Panchama are inviolate.”
A simplifer arrangement is often cited in musical texts as being the basis for the modern 12tone scale. We will return to this basic scale and use it as an example when we come to the longstanding problem of modulation in just intonation.
1/1, 16/15, 9/8, 6/5, 5/4, 4/3, [64/45 or 45/32], 3/2, 8/5, 5/3, 7/4,15/8, 2/1
Hindemith, in defending his bias towards ET claims this set of ratios as the basis of 12tone ET harmony in his book, The Craft of Musical Composition[3]. This theory falls far short, of course. Certainly better fractional relations can be devised to approximate ET if we so desired.[4]
next 6. The Devil's Playground
[1] Not even Partch goes this far in preserving these extended relationships in his socalled ‘monophony’, see below.
[2] This gives me an opportunity to share the reason this work is being written. I once posted a ravingly stupid email to the Max/MSP list and in some early morning stupor [N.B. never send emails before 2 p.m.] signed it with a signature ‘tune out, turn off, drop in.’ At the time I knew nothing about intonation, this being a phrase of social commentary, and frankly wasn’t much interested, having bought into the notion as a piano player that ET was the crowning achievement of Western Music. While composing MIDI pieces and firing them off to an editor, I noticed a long list of alternate tunings for the keyboard [Just, Meantone, various Gamelans, just to name a very few]. I thought I would check the Max/MSP list for objects that might easily offer these options in a patch. After typing ‘tuning microtuning intonation’ into the search engine I was horrified to find that my ridiculous signature had, months later, been returned at the top of the list of results – and it had nothing at all to do with tuning, or intonation! Ashamed, I decided then and there I needed to learn something about the subject and rectify the situation by posting something more useful later if I could. (That forum is terribly intimidating and I find I manage to embarrass myself, or give away the essence of my work whenever I post!)
[3] source Partch, verify story in Hindemith or elsewhere. ratios correct.
[4] Partch offers a few historical examples, particularly Ling Lun of China, 2600 B.C., p. 401.
Oh, Diamond, Diamond! Thou little knowest what misfortune thou hast done!
Sir Isaac Newton[1]
The very fundamental progression VI or 3/2  1/1 or the tendency of 3/215/89/8 to move to 1/1 5/4 3/2 and still retain the same proportions is termed a modulation. Similarly IIV retains the same proportions as VI
1/1 5/4 3/2 to 4/3 5/3 2/1
The logic implies that IV moves to bVII in the same way, bVII, bIII, etc. This would not be a problem using a precise calculation of the ratios, but becomes one when one is restricted to 12 pitches and an equal octave. The Lords and Ladies of the Renaissance were disturbed to find that the lovely just major triad, with its harmonious thirds did not sound as beautiful in all keys.
They found within their keyboards was what is called a wolf tone – a harmony that beats and howls.
[max patch: fixedratioKeys]
This led to a variety of strategies aimed at permitting more distant modulations while trying to maintain the same perfect ratios. Some added extra keys to the traditional keyboard. These models did not become popular, though some 17 and 19 keys per octave instruments were built.[2]
Stuck with the big twelve, people began to tinker with the absolute pitches, raising and lowering various relationships by small amounts, lessening the howling at the expense of the pure relationships.
Ultimately the perfect fifth and fourth were sacrificed to preserve the octave, or ‘soured’ in the language of the time. The fifth was crushed slightly, by a few cents, the fourth sharpened all within what was considered to be an ‘acceptable degree’.
A variety of temperaments [mean tone, well tempered] were invented to smooth the dissonances yet preserve at much as possible just relationships. Some composers today, notably Terry Riley, swear by mean tone temperament as superior to ET by virtue that its intervals are more harmonious than those of ET and each key has its own particular flavour or quality.
Many are surprised to learn that Bach’s 48 Preludes and Fugues are written for, as the title clearly indicates, a ‘well tempered clavier’ and not an ‘equally tempered’ one. Chopin’s Preludes likewise take advantage of the colouristic qualities and effects noticeable between various keys in a mean tone tuning system. These colours are lost in ET and as of today, most people, myself included, have not heard the music of Bach, or indeed most western art music[3] in the way that the composers themselves would have heard it, and presumably intended it to sound.
[1] Anecdote from St. Nicholas magazine, Vol. 5, No. 4, (February 1878) :
Sir Isaac Newton had on his table a pile of papers upon which were written calculations that had taken him twenty years to make. One evening, he left the room for a few minutes, and when he came back he found that his little dog "Diamond" had overturned a candle and set fire to the precious papers, of which nothing was left but a heap of ashes.
[2] Isacoff, 105.
[3]right up into the very late romantic period, for true ET was not perfected until 1917! Duffin, 112.
Oh, Diamond, Diamond! Thou little knowest what misfortune thou hast done!
Sir Isaac Newton[1]
The very fundamental progression VI or 3/2  1/1 or the tendency of 3/215/89/8 to move to 1/1 5/4 3/2 and still retain the same proportions is termed a modulation. Similarly IIV retains the same proportions as VI
1/1 5/4 3/2 to 4/3 5/3 2/1
The logic implies that IV moves to bVII in the same way, bVII, bIII, etc. This would not be a problem using a precise calculation of the ratios, but becomes one when one is restricted to 12 pitches and an equal octave. The Lords and Ladies of the Renaissance were disturbed to find that the lovely just major triad, with its harmonious thirds did not sound as beautiful in all keys.
They found within their keyboards was what is called a wolf tone – a harmony that beats and howls.
[max patch: fixedratioKeys]
This led to a variety of strategies aimed at permitting more distant modulations while trying to maintain the same perfect ratios. Some added extra keys to the traditional keyboard. These models did not become popular, though some 17 and 19 keys per octave instruments were built.[2]
Stuck with the big twelve, people began to tinker with the absolute pitches, raising and lowering various relationships by small amounts, lessening the howling at the expense of the pure relationships.
Ultimately the perfect fifth and fourth were sacrificed to preserve the octave, or ‘soured’ in the language of the time. The fifth was crushed slightly, by a few cents, the fourth sharpened all within what was considered to be an ‘acceptable degree’.
A variety of temperaments [mean tone, well tempered] were invented to smooth the dissonances yet preserve at much as possible just relationships. Some composers today, notably Terry Riley, swear by mean tone temperament as superior to ET by virtue that its intervals are more harmonious than those of ET and each key has its own particular flavour or quality.
Many are surprised to learn that Bach’s 48 Preludes and Fugues are written for, as the title clearly indicates, a ‘well tempered clavier’ and not an ‘equally tempered’ one. Chopin’s Preludes likewise take advantage of the colouristic qualities and effects noticeable between various keys in a mean tone tuning system. These colours are lost in ET and as of today, most people, myself included, have not heard the music of Bach, or indeed most western art music[3] in the way that the composers themselves would have heard it, and presumably intended it to sound.
[1] Anecdote from St. Nicholas magazine, Vol. 5, No. 4, (February 1878) :
Sir Isaac Newton had on his table a pile of papers upon which were written calculations that had taken him twenty years to make. One evening, he left the room for a few minutes, and when he came back he found that his little dog "Diamond" had overturned a candle and set fire to the precious papers, of which nothing was left but a heap of ashes.
[2] Isacoff, 105.
[3]right up into the very late romantic period, for true ET was not perfected until 1917! Duffin, 112.
I met the Dutch composer Louis Andreissen after a talk he gave at UBC ca. 1993. Inspired by the energy of his music, I was quite interested to hear about his rhythmic processes. Instead, Mr. Andreissen gave a lecture on pitch class sets.
After the lecture, I waited outside, until after all the other admirers had greeted him. Mr Andreissen graciously gave me another moment of his time to ask my question: ‘Mr. Andreissen, I come up with elegant pitch rows and class structures but I don’t know what to do with them next.’
Louis Andreissen replied, ‘Don’t forget about your transposition levels.’[1]
This whole daunting subject of temperaments and tunings, categorizing and comparing them, is one that I have no intention of going into further. Each one seeks to find an approximate resolution to a fundamental contradiction: that no rational, harmonic ratio evenly divides another into any other number of parts. No amount of fifths will divide the octave evenly; no amount of thirds will divide the octave or the fifth evenly, and so on.
A solution then, is to forego rational relationships and calculate the interval as a power of two. Thus:
2^(1/n) where n is the desired number of divisions of the octave
We see that (2^1/12) is the precise value for the ET semitone and that 12 semitones equals an octave. Remembering that to add intervals, we multiply the ratios: 12^(2^1/12) = 2
Example max patch: equal_temperaments
We can generalize this as shown. The example patch goes on demonstrate the method to calculate the equal division of any interval, not just the octave.
What can we say about the perceived consonance of these relationships? Well, first off, as all these are irrational, the natural harmonics structures of the tones will theoretically never match up. Theoretically because, in truth, we can’t produce irrational ratios exactly. They are always rounded by computers, approximated by the ear, physically impossible to actualize. Irrational numbers have no real world expression. I’m not sure what we think we’re hearing when we play a tritone, but it sure as heck isn’t the square root of two.
Not that a difference can be heard between 1.414219 and 1.41422 anyway, and the computer is more exact! We notice that with just intervals, no matter how complex they are, eventually their waveforms will sync up, even should that mean they howl wildly into the night. In geek speak, this means there always exists a lowest common whole number multiple of any two rational numbers. With irrational relationships this magic quantity does not exist.
Debates surrounding the relative merits of ET vs. Just Intonation will doubtless rage on. I caution against this until one has examined the basis for their assumptions. Why is it so important that one preserve the octave relationship? Why not a particular version of a major sixth? or a just 11^{th}? Why should one want to modulate cyclically, endlessly? Is it necessary to always start and end in the same place? Is there ever a moment when a wolf tone or an impure triad is absolutely the most beautiful thing in the world?
In my view, the main problem of ET is simply the absence of the varieties and shades of tonal colour and nuance that are possible. Considering only the limited number of examples we have looked at so far, composers unnecessarily restrict themselves from using such devices as the stretched octave, wobbling unisons, the entire range of precise thirds and sixths [as opposed to expressive thirds varied upon taste], and have allowed for the complete omission of a wide range of pure harmonic intervals [7/4, 8/7, 11/8, to name only a few]. In short, the only regrettable thing about equal temperament is its exclusive use.
One must remember that ET offers a new set of consistent standards and sounds that do in fact solve a lot of problems particularly in the performance field and provides a measure of consistency that we use as a reference point. The existence of an absolute scale to compare various relationships against is essential for work in this field.
Moreover, in rejecting one system of irrational intonation, we are forced to exclude an infinite variety of alternate irrational systems [to say nothing of the imaginary ones!] In doing so, we infinitely shrink our potential musical territory. Number theorists will tell us that there are infinitely more irrational numbers than rationals! So let’s not be too hasty in dismissing ETs. Let’s keep our options open.
Discrimination coming from proponents of ET, can be just as damaging. Many critics, likely not having heard the difference between the various pure and tempered intervals hail ET as the solution to all musical problems. One writer goes so far as to dismiss systems of just intonation offhand saying, ‘orchestras composed of instruments with just intonation would approach musical chaos.’[2]
As if this is a bad thing.
We must not let 12tone ET become a path of least resistance – a muddy and drab rut our musics cannot drag themselves from. For me, having now heard universally true and pure fifths, an assortment of luscious thirds, sevenths, and other prime beauties, and having glimpsed strange unnamed creatures floating in an undulating ocean of consonance, I ponder the distant foreign realms I have not yet seen and must press on, taking with me ET as a yardstick and not a rule.
This is about all I wish to say regarding 12tone Equal Temperament. It is certainly a significant achievement, and provides a certain, shall we say, flavour to this particular period of history. But it is my contention that in future, sophisticated aesthetes will look upon our barbaric time as one when only this one irrational sound could be heard  and will wonder how we could have ever gone so far as to call it ‘perfect’!
I cannot resist offering one final editorial comment: 12tone Equal temperament is to me like the belief in Heaven – sure everything’s all perfect and good, but given an eternity of it, damn it gets boring!
So, if it is into the realm of ‘musical chaos’ we are to go, let’s march on bravely! There is a spectacular order shining within and it will take your breath away.
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