MATH, MUSIC AND LOGIC

(1978?)

Quite a while back--950 years ago, in fact--an Italian monk, Guido d'Arezzo, devised a system for teaching singing. It was really thorough: interlocking hexachords, a way of swapping the notes onto the finger-joints and--what interests us here--a set of six syllables representing the relations among the tones. Whether this had anything to do with the Ancient Greek tetrachordal sol fa, ta tn tw te, is a matter of conjecture; so are the allegations that Guido got his idea from this or that exotic sources. Guido's method didn't need a seventh syllable at the time of its invention, but as music evolved, the syllable si was added to the system for the seventh degree of the scale or 'leading-tone.' The 19th-century English Tonic Sol-Fa system changed this ti (te in their English spelling) so that one could abbreviate the syllables d r m f s l t but si is still standard in Europe.

Ut, the original first syllable was changed later on to do, to be more singable to conform with the open structure with the other syllables. Nevertheless, the persistence of the syllables re mi fa sol la right up to the present day through all the violent events in history, and the drastic changes in languages and, for that matter, in music itself, is an exceptional unparalleled world-wide success story!

True, there is some division between the Germanic and English-speaking countries, which have retained Guido's moveable-do concept--i.e., the syllables denote relative, not absolute pitches; as against the Latin countries which later abandoned the letter-names for absolute pitches and now use la to mean our A and sol to mean our G, and so on; which compelled them to adopt numerals 1 through 8 to be the moveable-do substitute. (See the staves above.) Our relative-pitch use of the syllables is like the use of letters in algebra, where certain expressions will remain true regardless of what numerical values are substituted for the letters a b c n x y z &c.

While the mathematical aspect of music is usually considered in terms of frequencies and their ratios, it is equally useful to consider intervals as logical relations in the style of Bertrand Russel's a (logical and symbol) b. That is,t he experience of hearing the interval of a Fourth in a practical musical context is more like "Do is a Fourth from Fa' than "The frequency or string-length for C is in the ratio 3:4 to that of F." Arithmetic maybe necessary to work out the tuning and construction of musical instruments, but musical compositions and the way they sound involve relations similar to those of symbolic logic. A good composition makes sense, has form, exemplifies relations.

Some time ago it occurred to me that this solfeggio-syllable principle could be applied to mathematics and formal logic, creating a speakable, pronouncable notation, in almost perfect one-to-one correspondence with standard written international mathematical notation. The Guidonian syllables have been tested for nine and a half centuries through all the ravages of time, so I need not waste any time now proving that they work! Accordingly, I constructed the Numaudo (numbers made audible) Coding System, and its typewriter-keyboard-compatible phonemic transcription, Numalittera, adaptable to peripherals and computer printouts. The Numaudo System is not a language. The 'language' is the ordinary notation, which Numaudo pronounces and Numalittera typewrites within the limitations of the ordinary alphabet.