_______________ b. pitchforks: tunings and temperaments PDF Print E-mail
Written by D. Bond   
Saturday, 31 October 2009 10:57

b. pitchforks: tunings and temperaments


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 semi-tone 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 11th? 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 12-tone 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 12-tone 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: 12-tone 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.


next 6.c. the unbinding of heaven

[1] True story! Apparently, he almost didn’t do the lecture because it was during his daily afternoon ‘nap-time’.

[2] Rossing, 174.

Last Updated on Tuesday, 25 May 2010 22:20