5. Spacing the Spheres – the protractions of scale building 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. [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. 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. This theory falls far short, of course. Certainly better fractional relations can be devised to approximate ET if we so desired. next 6. The Devil's Playground Not even Partch goes this far in preserving these extended relationships in his socalled ‘monophony’, see below.
