"A Just Cause"

© Paul Serotsky 1998, 2004

Continued .... Part 2
Part 1 Part 3

Tonality

A musical scale does not of itself a "tonality" make. Tonality is something which makes a certain subset of identities "belong" together, which implies some underlying rule over and above that which defines the degrees of the scale. In ET, a "key" is obtained by dividing up the intervals of the "octave":

Scale

D

D#

E

F

F#

G

G#

A

A#

B

C

C#

D

Selection

D

E

F#

G

A

B

C#

D

Intervals

1

1

½

1

1

1

½

You can see the well-known pattern of tones and semitones: two groups of tone-tone-semitone, separated by a single tone. This whole template can simply be shifted up the scale, a semitone at a time, to generate all the 12 major keys. This exact translational symmetry is because all the intervals are exactly the same size. But, is there any intrinsic reason WHY? Is this pattern God-given, or what? Is there any good reason why not a sequence [1, ½, 1, ½, 1, ½, 1, ½], or anything else? And, note, we haven't yet given any thought to the "minor" keys, which in ET are a veritable can of worms.

Let's consider tonality in a JI system. A ratio involves two numbers. Each individual number represents a potential identity, because once any permitted second number is given, an identity becomes defined. If one number (say the lower) is fixed, it becomes a numery nexus (i.e. "anchor point"). Ratios with a given numery nexus are related by that nexus - they have something in common. The ratios 1/1, 5/4, 3/2 (where 1, 2, and 4 are equivalent - merely different "octaves", and hence not different identities) from the 5-limit sequence are the identities with an under-numery nexus of 1. The identities 4/3, 5/3, 1/1 (i.e. 3/3) are those with an under-numery nexus of 3. Similar groups can be formed having an over-numery nexus. All these are nothing more exotic than triads - basic chords.

But chords are the stuff of which keys are made. You can see that this is borne out by D-F#-A from Table 1. In each of the examples given above, the numbers associated with a given numery nexus are always in the sequence 1, 5, 3 (again allowing for "octave" transpositions), which is itself a fixed pattern. Now, this is starting to get interesting, so take a deep breath and brace yourselves! If the numery nexus is the lower number of the ratio, the set of ratios in the triad are the 1, 5, 3 identities of an over-number tonality ("otonality"), in ascending pitch order. Similarly, if the numery nexus is the upper number, the ratios are the 1, 5, 3 identities of an under-number tonality ("utonality"), in descending pitch order.

That in itself is an impressive mathematical and aesthetic symmetry, but there's yet more. While the members of each triad are in the same fixed relationship to one another - the middle identity is always a 5/4, and the top identity a 3/2, from the unity of the particular chord - any particular triad cannot be transformed into another of the set by the application of a common interval. Thus, while they have identical internal structures, they are truly distinct from one another: they are the chords of different keys.

It is this distinctness that obviates the possibility of "transposition" in JI, one reason often cited for its "inferiority" to ET which readily transposes. But the real reason (again) is operational convenience, the price being that ET keys are not truly distinct. JI chords of different keys may be distinct, but they are, surprising as it might seem, nevertheless related. Partch demonstrated this with a striking diagram - what he called the Incipient Tonality Diamond (fig. 2), the implications of which are truly startling. All the true triads of the 5-limit are shown. Triads formed from compound intervals do not generate tonality, and so are excluded.

The sequences ascending from lower left to upper right between solid lines are the otonalities, and those descending from upper left to lower right between dotted lines are the utonalities. Once we have tuned all the intervals of the 5-limit to a unity (1/1), we have not just one true triad based on that unity, but also five other true triads. 1 (or, equivalently, 2, 4, or 8) is always part of a ratio which is a unity ("tonic") of a triad, 5 is always part of a ratio which is the middle identity ("mediant") of a triad, and 3 (or, equivalently, 6) is always part of a ratio which is the top identity ("dominant") of a triad - and thus we have the origin of the terms "mediant" and "dominant".

From this point, things start (!) to get a bit difficult to explain simply, but it should be fairly clear by now that the 1, 5, 3 "odentities" of an otonality are a triad of tones in order of ascending pitch, equivalent to an ET "major" tonality triad such as C-E-G (or D-F#-A). The breathtaking corollary is that the 1, 5, 3 "udentities" of a utonality are a triad of tones in order of descending pitch, which is a mirror image of an otonality. This is the origin of the ET "minor" tonality triad. So we find that in JI there is no mystery about minor tonality: the major and minor keys coexist of necessity - like love and marriage, "you can't have one without the other"! Any comparison between ET and modern morality is entirely at the indiscretion of the reader.

Minor tonality in ET is difficult to explain, hence the "mystery" surrounding it. Explanations are rarely ventured, and I have yet to find one that is even remotely convincing. Even Paul Hindemith, for example, could manage nothing better than, "the minor is the 'clouding' of the major", which is, to put it mildly, somewhat woolly. In the mathematics and science of JI, as reviewed above, we find a far clearer and utterly convincing explanation of the origins of tonality, one that presents tonality as a part of the natural order. Borrowing Hindemith's phrase, we might be led to conclude that ET "is the 'clouding' of" JI. ET now really begins to look more like an artificial simulation of the fabric of tonality (shades of The Song of the Nightingale?), something that really was manufactured to suit the convenience of keyboard design and musical annotation.

However, it was also promoted with a religious zeal, quite literally so in some cases, notably by those who were in pursuit of the "Holy Grail" of a "circle of fifths", to the altar of which JI had to be sacrificed to propitiate the false gods of "Perfection". In JI, if you ascend the scale a fifth at a time, you will never drop onto an identity which is an exact number of octaves above your starting point. In ET's logarithmic world, which looks the same wherever you view it from, you can not only have a circle of fifths, you can also have a circle of thirds, fourths, or whatever you fancy.

This whole business arose from a general superstitious concern over the "perfection" of the circle, it being argued that there must be a "circle" in something as holy and mystical as music. Big Deal. With his extreme disillusionment with the apparently corrupt state of Music, it is easy to see what it was about JI that so excited and inspired Harry Partch.

But, what happens if we apply the Incipient Tonality Diamond concept to ET? The short answer is: we get a reasonable, and long overdue, explanation of the origin of ET Minor Tonalities. Fig. 3 illustrates the pattern, with the equivalents to the original JI 5-Limit tonality generating identities shown in BOLD. The full set of 12 identities is shown. In ET, all contribute equally to tonality generation, as all are smoothed into equality by the logarithmic formula. Bearing in mind that ET is a formulaic mathematical approximation of JI, let's see what happens if we follow the same procedure.

First, we lay out the C Major scale (shaded), ascending from the bottom towards the right. The "tone-tone-semitone" groupings are mapped and bracketed for clarity. On the lower left, the 12 identities are laid out in order of descending pitch, from the bottom to the left. The Major tonalities can readily be seen starting on the lower left edge, each running between a pair of solid lines towards the upper right, and having exactly the same pattern offset by a semitone at a time. So far, so good.

Minor tonalities should be revealed, in descending pitch order, by reading off the scales starting at the bottom right edge and going towards the top left (between the dotted lines, as before), following the descending tone-tone-semitone groupings. Try it. It doesn't quite work, does it? The sequence you find, compared with the ET scales of C minor, is

The identities which do not correspond in the ET scales are shown shaded. The lack of exact correspondence, and the reason there are three variants, is because the ET minor scale has been "doctored" (a minor sin to someone who has just vandalised the entire world of tonality!). The Descending Melodic Minor is closest, and interestingly if you read off the sequence starting on the C in the G major row, equivalent to the top corner of the "real" ET Incipient Tonality Diamond, you will find an exact correspondence - though why I'm not sure! The A# is replaced by a B in the Ascending Melodic Minor to re-establish a semitonal leading note, with the G# lifted to A to avoid the "discomfort" of the augmented second opened up in the process. There are a couple of other "fudges", presumably for similar reasons but nevertheless there it is, in all essential aspects: the origin of the Minor.

4 Monophony

A JI system is built on a single, generating "prime unity" (1/1 ratio). Any such system is termed a monophony. The prime unity is the root of the entire scale. You could say that JI is like a universe with a centre. In ET, the choice of a reference pitch (A=440 Hz, for example) is purely a matter of convenience for tuning. If we chose a frequency for A corresponding to a displacement of a whole number of semitones, for instance 493.883 Hz (the frequency corresponding to B), the ET scale would be identical: only the names of the identities would be different.

The only observable effect would be that instruments would have to transpose a tone. Even with a pitch shift which is not a whole number of semitones, the impact on ET is no more than the adjustment of an instrument's basic "length". However, if the JI prime unity is changed in any way at all, a completely different scale would be created, and all your instruments would have to be retuned - or even reconstructed. In JI, it seems, you cannot easily tinker with "concert pitch"!

Partch's first job was to construct his monophony. He had two tasks, the first of which was comparatively, but not completely, trivial: to decide his prime unity. He chose the pitch of the G below middle C of ET, a decision, I suspect, not unrelated to the fact that many of his early experiments were carried out using a viola!

The second task was nothing like as simple: to specify the identities of his monophony. Of course, he could simply have reversed the process which created the ET scale (as per the table), but this brought him up against one of his objections to the musical establishment. He was only too well aware that the introduction of ET had stopped further evolution of tonality dead in its tracks: this was, after all, one of the aspects of ET that he found so unacceptable.

Up to that time, there had been a slow but inexorable evolution of tonality, as JI intervals of greater relative dissonance (ratios using higher numbers) were called on to expand, by inserting further intervals and without modifying existing ones, the range and resolution of the intonational system. This kind of development is much more difficult in ET - evidently so, because the scale has changed not one jot in the 300-odd years of its existence. The musical scale was frozen solid on the equivalent of the 5-limit. This implies a limitation of expressive potential of just the same sort which the JI system simply stepped over by extending into the 5-limit in preference to the 3-limit.

The so-called "breakdown of tonality" precipitated by late Nineteenth Century chromaticism and culminating in Twentieth Century serialism was the direct consequence. It can be recognised that composers were actually battering on the prison walls of the implicit 5-limit of ET. We can thus see that Wagner, Strauss, Schoenberg et al were not trying to do away with tonality, rather they were desperate to expand it. In the event, unable to escape from the sonic strait-jacket of ET, the only recourse was to move even further from the natural landscape of JI, and one is forced to ask, "Is this is why most people find dodecaphonic music so hard to swallow?"

In this context, Partch saw that the implicit freezing of the 5-limit in ET suggested a history of under-use of the ear's abilities. Using physiological data, and experiments of his own, he discovered that the average human ear was capable of resolving and recognising far narrower intervals and differences of pitch than ET would permit.

Partch concluded that there was every reason why his monophony ought to be based on the abilities of the very organ which it was designed to serve - the human ear. After further research and experimentation, he found that the appropriate limit was not merely the 7, but the 11! This resulted in a scale comprising some 28 true identities, which were augmented by 15 multiple number ratios, giving 43 tones to the "octave" (almost entirely gratifying aficionados of The Hitch-hiker’s Guide to the Galaxy!).

These are Partch's chosen identities:

1/1	81/80	33/32	21/20	16/15	12/11	11/10	10/9	9/8	8/7	7/6
32/27	6/5	11/9	5/4	14/11	9/7	21/16	4/3	27/20	11/8	7/5
10/7	16/11	40/27	3/2	32/21	14/9	11/7	8/5	18/11	5/3	27/16
12/7	7/4	16/9	9/5	20/11	11/6	15/8	40/21	64/33	160/81	(2/1)

The true identities (the pure 11-limit ratios) are shown in bold. Some of the multiple number ratios in my earlier table do not feature here, partly because my derivation is somewhat conjectural, but largely because Partch was carefully selective from the huge range of choices. In some cases there were simpler ratios of 7 or 11, for example he preferred 14/11 (= 1.273) to 32/25 (= 1.280). However, he meticulously ensured that his monophony embraced intact all the ancient modes, including Greek and Church. Practically all of the scales prevalent in the rest of the world's "classical" and "folk" music - which can only be approximated on the piano, as Bartok was well aware - are also present and correct.

Partch's choice of a 43-tone scale was thus not, as is commonly supposed, a product of the arbitrary ravings of a crank and a crackpot, but the logical and reasonable conclusion of much careful scientific study. Music using this scale to its full potential may sound strange and hard to accept, but it is no more so than any other music which transgressed the bounds of what was harmonically "acceptable" at the time.

5 Notation

Looking first at the notational style of ET, and then at the table of 43 identities, Partch found himself confronted with a major headache. His first option, if he were to write music for players to play, was to adapt the existing notation in respect of pitch representation. If each identity were to have a line/space to itself, this would imply a stave of around 27 lines, bearing in mind that JI does not enjoy the symmetry of ET which invites the convenience of #, =, and >. To have a stave with few enough lines to be practical, a substantial and complex symbology for the identities would be required. But, where would you put the symbol? The obvious space is taken up by notation of duration.

Partch's solution (to cut a long story short) was supremely pragmatic: forget the stave to represent pitch, he would use the ratios themselves, with duration noted as a "suffix".

It quickly became apparent that his general notation, although fine for composing, was too complicated and clumsy for performance use. This he resolved by developing, for each instrument, a specialised notation which was less complicated and thus easier to read.

 

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