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"A Just Cause"

© Paul Serotsky 1998, 2004

1 Introduction

"What rubbish!" Two little words of deprecation, hardly an auspicious start to a train of events which was to transform my attitude to music. I heard them from a lady sitting just behind me at a presentation given, in 1976, by a representative of a record distributor. We had just heard an extract from one of the CRI (Composers' Recordings Incorporated) records his company was then marketing in this country. To be fair, the lady was merely giving voice to an opinion held by many, if not most, of the audience. However, I have to admit that I find it intensely aggravating when an extended musical work is summarily dismissed on the strength of a short extract, and anyway the bit of music that we had heard didn't sound all that bad to me. Nevertheless, it was quite unlike anything I had hitherto experienced, which I suppose makes it fairly "way out", and spurred by the general scorn, my inherent bloody-mindedness dictated that I investigate further.

As luck would have it, a combination of relative impecunity - a perennial problem for a "working class" record collector - and abject failure of local record shops to connect with the source of supply prevented me from getting hold of a copy of the record. So, end of story? Well, almost but, as you might guess, not quite. Fully four years went by, before two unrelated events rekindled my curiosity. The first was an item in the Bulletin of the (then) National Federation of Gramophone Societies, wherein the editor, the late Bill Bryant, to whom I shall be eternally grateful, let it slip that he had actually got hold of several of those CRI records, including that particular one.

A year further down the line, researching for a projected Huddersfield Gramophone Society programme on "Music from the New World", I came across a reference to a book called Genesis of a Music. The authorís name was the same as the composer of that scorned musical extract. Coincidence? Not likely: letís face it, Harry Partch is not the sort of name that overburdens the telephone directory. I checked the index in the Bradford Library, and to my utter amazement found that the book was actually on the shelf! Borrowing it, I found it a substantial tome of well over 500 fairly large pages, which I read from cover to cover - twice! What I found there, though understood only in part and then with some difficulty, convinced me that I should make Harry Partch the centrepiece of my programme, with the help of the loan of one of Bill's records.

By now you have every right to ask, especially if on one particular occasion you had dismissed a fragment of very strange-sounding music with a summary "What rubbish", just what it was that "transformed my attitude to music"? Let me try my best to explain . . .

2 Partch's Achievement

That lady was in good company: no less a personage than Norman Lebrecht has also dismissed Partch, calling him a "crackpot inventor". On this occasion at least, good old Norman is talking right out of the back of his head. I discovered that the California-born Harry Partch, far from being simply a "crackpot inventor" and composer of "rubbish", is in fact an archetypal "pioneer", riding his covered wagon westward into musical territories to all modern intents and purposes utterly uncharted. He turns out to be one of the most original musicians of this century, if not in the entire history of the music of Western Civilisation.

As a young man, Partch (1901-74) became increasingly dissatisfied with the entire "business" of music, with its widespread emphasis, both by teachers and authors, on high levels of instrumental skills and on playing and composing "technique". That is, if a work was "polished" in performance or compositional technique then it was automatically "good", if a player was extremely dexterous then he was ipso facto an "artist", and if he could play in such a way as to please the "experts" - whoever they might be - then he was, equally ipso facto, a "great artist" [Genesis of a Music, p4].

What seemed to irk Partch so much was that everyone was busily fêting the virtuosi and indulging in strictly ritualised concerts, but no-one seemed to be giving any real thought to the intrinsic value of the materials with which they were working - and here he meant not just the basic philosophical attitudes, but the very organisation of sounds which distinguishes "music" from "noise".

He wasn't over the moon about music schools and teachers, either. Music, of all the Arts, seems to be the only one whose proponents steadfastly refuse to "demean" themselves by acquaintance with the science in their art. Thinking about it, such is the mystique surrounding Music as an art that one of the surest ways of precipitating a fit of apoplexy in a musician is to suggest that Music is a branch of mathematics. Worse, Partch found that the little "science" that was taught was at best superficial, often sketchy, and at worst completely spurious.

Apparently, what he found so utterly exasperating was that Music was presented as something immutable, even sacrosanct: the student was fed rules and regulations by rote, and if he dutifully learned and practised them he would become a "good musician". But, it seemed to be forbidden to question the validity of the quasi-biblical "rules", or to enquire into the whys and wherefores of the structure of the art. The reason? TRADITION, or, more bluntly, "fossilisation". We had better and better pianists, conductors, composers, we increased the range of our "good" music, but the tradition itself remained undisturbed because the ideas and objects at the heart of the art were not challenged and investigated, not even by the supposed "avant-garde".

So, even in his teens, he had more or less given up on the musical system and establishment of modern Europe. He turned to the public libraries in a search for material from which to learn. At the ripe old age of 21 he came across the key to his dilemma, a work not of musicology but of physics: Ellis' translation of On the Sensations of Tone, by Helmholtz. This so convinced him that the traditional system was rotten in its very foundation - in its implementation of the concept of "tonality" - that he made an almost unimaginably bold decision, one that would freeze the blood in the veins of most mortals. Consigning his entire musical output to date to the tender mercies of a pot-bellied stove was a gesture both practical and symbolic: he turned his back on the current musical tradition and, working from first principles, set out to create an entire Music of his own from scratch.

3 The Tonal System

The first thing Partch needed to determine was the tonal system in which his music would operate. To be true to his principles, he could not just throw out the "traditional" one and concoct another one willy-nilly. To succeed, he had to be committed to justifying, scientifically and practically, whatever he devised. This is such a fundamental step that it bears reviewing in some detail. Fortunately, it is also a fascinating and revealing - though (be warned!) difficult - subject. It might help to be aware that what follows is ultimately derivable, believe it or not, from the simple physics of a vibrating string such as we are all taught at school. True, this same basis is often used in conventional music theory, but as we shall see its application is fundamentally flawed.

Ring Out the New . . .

As we all know, the scale now used universally in all music of the Western European tradition - be it "classical", "popular", "jazz", or whatever - is that playable on a piano keyboard. The "octave" is divided into 12 equal intervals, or "semitones", appropriate selections of eight of which provide all the "keys": 12 "major" and 12 "minor". And are we not taught in school of the birth of this intonational system as if it were the Messiah of some religion (with J S Bach in the role of John the Baptist)? If not, we may as well be, for all that we learn of what it really is and how it really works.

Most "traditional" musicians would probably recoil in horror if told that each degree (note) of the scale is determined exactly by multiplying the frequency of the previous degree by a factor of 21/12 - that the Holy Musical Scale should be derivable by mere common mathematics! But worse, much worse, is to come. The ONLY truly consonant interval of this hallowed system, 12-tone equal temperament (ET), is the octave, that is the interval between two degrees such as C and C'. ALL, bar none, of the so-called major intervals and chords (such as C-E-G-C), are dissonant! ALL, bar none, of the other chords and intervals are more dissonant!

We will come to the meanings of "consonant" and "dissonant" in a moment. I recall a TV documentary in which no less a figure than Olivier Messiaen explained his characteristic chord structures. He played a sequence of notes on a piano and proclaimed, "There! I have played all the harmonics." With due respect, he was wrong on not one but two counts. Firstly and perhaps a little pedantically, he had played only a dozen or so notes, which is rather too few to constitute "all the harmonics". Secondly and rather more crucially, he had played none of the harmonics, other than the octaves of his chosen fundamental.

Galileo penned a perceptive and penetrating observation:

"Agreeable consonances are pairs of tones which strike the ear with a certain regularity; this regularity consists in the fact that the pulses delivered by the two tones, in the same interval of time, shall be commensurable in number, so as not to keep the eardrum in perpetual torment, bending in two different directions in order to yield to the ever-discordant impulses."

This holds the key to "consonance" and "dissonance". What it means is that, at regularly occurring instants, the two vibrations will be in step, and, as any mathematician or physicist will tell you, this is possible only if the frequencies of the two tones are in whole-number proportions. Moreover it follows, partly from Galileo's statement, that the degree of consonance of an interval decreases from 1/1 (unison, where the frequencies are equal) as the size of the whole numbers involved increases. This is because the tones are in step less often. With big enough numbers the interval eventually approaches "complete" dissonance, as the tones are in step so rarely that the ear can no longer perceive the regularity of the pattern.

The limit of this progression from absolute consonance to absolute dissonance comes when the numbers become infinitely large, in that remote realm where the rational number locks horns with its irrational counterpart. The "irrational number" is a thoroughly nasty beast: it cannot be exactly represented by the ratio of two whole numbers or, equivalently, requires an infinite sequence of non-recurring decimal digits (B is the classic example).

Since the adjacent degrees of ET, the piano scale, are by definition related by factors of 21/12, and 21/12 is an irrational number, we cannot escape the conclusion that no two ET tones (other than the unison and octaves) can form a consonant interval. The intervals in an octave are 1 (i.e. 20), 21/12, 21/6, 21/4, 21/3, 25/12, 21/2, 27/12, 22/3, 23/4, 25/6, 211/12, 2 (i.e. 21), all of which, except 1 and 2, are irrational.

I sometimes wonder just how such a temperament ever gained its monopoly in our music. Up until around the time of Handel, all music was based on Just Intonation (JI), which we shall consider in a moment. It seemed that there was some difficulty in developing keyboards (lately gaining in popularity) which could cope effectively with JI. So, with a zeal more characteristic of late twentieth century business corporations, Keyboards Inc. set about "effective management" of its "customer perceptions".

Handel, bless his cotton socks, resisted manfully, maintaining for some time a keyboard with additional keys which did support JI intervals. Ironically, the keyboard lobbyís victory actually made life harder for players of other instruments. In particular, brass players (trombonists apart!) now had to "force" their intonation to fit, and it seems nobody has ever made any serious effort to remedy this: to this day, brass pitching is perforce "approximate". The key point, though, and the one that impressed Harry Partch, is that the crucial purity of musical intonation was sacrificed on the altar of business convenience - a close parallel to his own experiences. Exactly why this is crucial is something that we will tackle shortly.

. . . and Ring In the Old

All musical intervals are based on ratios: two tones are related by the ratio of their frequencies. There are natural laws relating to vibrating strings. A string, stretched between two fixed points, will resonate in only certain modes. These correspond to divisions of its length into integral numbers of half-wavelengths. The same is true, though less visibly, of the vibrating columns of air in wind instruments, and in a more complex way in two dimensional bodies like drumskins or three-dimensional ones such as our old friends, the "coconut shells"!

The "harmonic series" of a vibrating string is often used, all too briefly, as a starting point for traditional teaching of music theory, just before turning to the piano and demonstrating its application. Teachers seem to be either ignorant of the fact that the ET scale of the piano, based as it is on paradoxical-sounding irrational ratios, does not actually correspond at all, or at best dismiss the differences as "insignificant" - regardless of the fact that similar "insignificant" differences in the tuning of instruments in an ensemble would sound ghastly.

The "natural" JI intervals are based on ratios of small, whole numbers, corresponding exactly to the harmonics of that vibrating string. Unison is 1/1, an "octave" is 2/1, and other common ones are 3/2 and 5/4, so that for example a 3/2 implies

higher tone = 3 x (lower tone)/2

These conform to Galileo's definition of "agreeable consonances": for a 5/4 the vibrations are exactly in step once every 5 cycles of the higher tone, corresponding exactly to once every 4 cycles of the lower. Any such ratio can be "reduced" into the range 1 to 2 by a process of doubling or halving either of the numbers. For example 7/3, which is bigger than 2, becomes 7/6, which is between 1 and 2, by doubling the lower number. This is nothing more than our common practise of "octave transposition".

It also highlights the importance of the 2/1 interval. Long history and much experimentation have shown that the human ear recognises some tones as distinct, and others as distinctly not. The ear will recognise the difference in pitch of the two tones of a 2/1 interval, but will not afford them a different identity, broadly because every time the lower tone's vibration is at a peak or trough, so is the higher tone's. However, the reason for this is not too important here: it is in any case a physiological axiom, true of any scale - JI, ET, Mean Tone, or whatever. The intervals 5/2 and 5/4 have the same identity, that is, both represent the same degree of a JI scale, just as A and A' both have the identity A in ET. In JI, reducing a ratio to between 1 and 2 is equivalent to (say) referring to "G" without saying which octave it's in.

The preference shown by the human ear - and that of a dog, a cat, or a bat, for all we can tell - for JI intervals over ET ones is also a physiological axiom, notwithstanding both biophysical considerations and that this is exactly what Galileo was driving at. Partch used to stage experiments during lectures, playing corresponding chords in both systems to the "innocent ears" of his audience, who would then be invited to vote for which they preferred. Always, and overwhelmingly, the JI chords were preferred to the ET equivalents.

The reason for this is illustrated in Fig. 1, a comparison of the waveforms for an ET major triad (like C-E-G) and the equivalent JI triad (1/1-4/3-3/2). The strongest peaks correspond to the frequency of the tonic. You can clearly see that the JI curve replicates exactly in each cycle between pairs of these peaks - in accord with Galileo's dictum. The ET curve sets out in step with the JI, but soon diverges and, as mentioned earlier, is never repeated exactly, but varies progressively from cycle to cycle, i.e. it is not consonant!

Evidence abounds for the ear's inherent preference for JI. Ensembles comprising only instruments of continuously variable pitch (such as violin family, voices, trombones) in performance invariably revert to JI, offering one tempting reason why an a cappella choir, for example, sounds so extraordinarily beautiful. Then again, have you ever wondered why the ear should enjoy a smidgen of judicious vibrato, if not to blur the dissonance inherent in the ET scale? In a justly-intoned performance, vibrato would actually be detrimental to the sound, unless, that is, it was being used to mask poor intonation! More mundanely, I have heard string players grumble about how they have to play their notes "a bit wrong" to fit in with a piano, yet very rarely seem inclined to wonder why!

Musical Scales

A scale is a sequence of degrees, identities connecting 1/1 to 2/1. But what degrees, and related by what intervals, actually constitute a scale? Without any rules to define the relationships, literally "anything goes" - and possibly, right now, a "glimpse of stocking" might seem a little short of "something shocking"? Why, for example, is our ET fixed at only 12 identities? There are 12 simply because the interval is 21/12. There is no reason why there should not be 24, with a fixed interval between successive identities of 21/24. Of course, one or two composers in the Twentieth Century have dabbled with this "quarter-tone" scale. But why not 19, or 3759, or even 42 (which might appeal to aficionados of The Hitch-hikerís Guide to the Galaxy). Observing that the intervals are called "semitones" leads nowhere either: a "whole tone" is an equally arbitrary 21/6! The only reason for 12 would seem to be that this most closely approximated the JI scale current at the time that the ET scale was invented.

JI intervals are ratios of "small, whole numbers". The question is: which numbers? Mathematically, there is a sequence of special numbers: 1, 2, 3, 5, 7, 11, 13, ... These are the prime numbers, numbers which are exactly divisible only by themselves or 1. Prime numbers are crucial in JI largely because of this property of irreducibility. They prescribe sets of whole-number ratios. The "ratios of N", where N is a prime number, define potential JI scales. Take 3, for example. The ratios of 3 are

1/1 (unity), 4/3, 3/2, 2/1

These are all the ratios which are possible, using prime numbers no greater than 3. They have been transposed into the range 1 to 2: the ratio 4/3 is musically identical to 2/3 (as 4 is a doubling of 2). Seeing as ratios of 1 (1/1 alone, poor Johnny One-Note) and 2 (1/1, 2/1 only, Johnny and his boring sister Joan) provided only a single identity, and consequently virtually zero musical potential, it comes as no surprise that the ratios of 3 formed the basis of the most primitive scales. After the 1/1 (unison or unity) and 2/1 ("octave"), 4/3 and 3/2 are the most powerfully consonant intervals, both alive - though not entirely well - in ET's "perfect fourth" and "perfect fifth". Without recourse to matters celestial or divine, the meaning of "perfect" is suddenly crystal clear!

The ratios of 5 are

1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2/1

again, transposed into the range 1 to 2. Notice that the set of ratios of 3 (shown in bold) necessarily appears as a subset. As time progressed, more of these intervals became recognised as "consonant", in the sense that they became "acceptable" in musical terms. The sequence is symmetrical, the intervals between the degrees widening from the centre outwards. Such gaps were closed, to provide a more melodious spacing, by the entirely legitimate use of multiple number ratios, that is, ratios formed by combining simple ratios of 5, e.g. 16/15, 9/8. These are nothing more than "compound intervals": a new interval results from the adding of two intervals, and to add two intervals you just multiply their ratios. Thus 4/3 x 8/5 = 32/15, which is identical with 16/15, and 3/2 x 3/2 = 9/4, identical with 9/8. The ratios of 5, with associated compound intervals, form the basis of the JI scale which was "converted" to produce 12-tone ET.

Table 1 shows how the "conversion" might have been made (I say "might", because I have derived this representation myself, from first principles). The ratios of 5 identities are shown in bold, and all the multiple number ratios are also shown. Two (10/9 and 9/5) are shaded because these are not represented in ET (you win some, you lose some - these are where the pairs of JI alternatives are relatively close - another example of operational convenience?). The places where pairs of JI degrees were reduced onto a single, approximate ET degree can be seen clearly, as can the familiar resulting pattern of "white" and "black" notes.

Note how each "black" note has two possible corresponding JI identities, one a bit lower and the other a bit higher in frequency: Handel's "JI keyboard" would have retained these as distinct keys. That the scale starts on D is due (probably, though I am not going to go into that here, as it isn't really relevant) to the shift in concert pitch - the JI scale would originally have been transposed onto the key of C. Some of the approximations, as shown by the differences, are disturbingly high.

In JI it is obviously possible to use ratios of 7, 11 and so on, and derive musical scales with ever more identities to the "octave", limited in practical terms only by the ear's ability to distinguish them. Of course, the higher the numbers employed, the greater become the degrees of comparative dissonance, with consequent widening of the scale's expressive potential.

There was at one time a controversy over the "legality" of the chord of the dominant seventh, which related to an implied extension beyond the 5 limit enshrined in ET. Significantly, the introduction of ET effectively closed the door on the development of the expressive potential of musical scales. Over several hundred years, we had moved from 3-limit ratios to full use of 5-limit ratios. There is no reason to suppose that this development would have stopped there, had not the imposition of ET "crystallised" the 5-limit.

Part 2 Part 3


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