scale of mathematical intonation

Either the one or the other must be at fault. Had the dictates of the mathematicians and the scale of mathematical intonation wholly ruled, the advent of the great masters would have been impossible. It was well said by one writing in The Choir - "Theory should be made from music, and not music from theory . . . the final judge of music is the Ear." The Great Masters are the exponent artists of what is true in the Science of Music, though it may differ from what has been taught by the merely mathematical-intonation advocates of music science. It should not be forgotten that the science of the mathematical theorists is one thing, and that of the composers is another. Schubert, Beethoven, Mozart, Haydin, Mendelssohn, and such inspired musicians, who walked in the liberty wherewith Nature made them free, are sufficient authority against the bondage of the one-law theorists who would tie us down to the mathematical command which comes from without, but who know nothing of the life within music which is the law unto itself.1
With twelve divisions in the Octave, each note is adapted to serve in any capacity, and does serve in every capacity by turns. It is quite clear that this cannot be said of the mathematically perfect notes. And this is where it is seen that what is perfect in mathematical ratios becomes imperfect in the Musical System. Indeed, the mathematical intonation does not give a boundary within which to constitute a System at all, but goes off into never-ending cycles.
In music, Nature begins by producing the Diatonic Octave of seven notes, derived by the mathematical ratios2; [Scientific Basis and Build of Music, page 34]

Twelve divisions in the Octave serve all the purposes of music. This is the master-stroke of Nature in putting so much into seemingly so little. Twelve is the most genial of all numbers; it is nature's representative of the social order in music. It is the manifold divisibility of twelve which makes the chromatic system in music possible. The equalized scale of twelve semitones in the octave and the chromatic system of music are indissolubly connected. With this the scale of mathematical intonation has neither part nor lot.
The life-force of the notes from the law of position gives them a versatility which they could never have had from fixed ratios, however numerous. If the interval of the octave be excepted, there are no two notes together in a chord, nor succeeding each other in the octave scale, having the same amount of specific levity or gravity; consequently each note has an expression and [Scientific Basis and Build of Music, page 35]

See Also

chromatic scale
Law of Mathematical Ratios
mathematical intonation
Mathematical Relations are Constant - page 125
mathematical scales
mathematical system
On the Partial Differential Equations of Mathematical Physics
Ramsay - PLATE VIII - The Mathematical Table of Majors and Minors and their Ratio Numbers
Ramsay - PLATE XXII - Mathematical Table of the Twelve Major Scales and their relative Minors
Ramsay - PLATE XXIII - The Mathematical and Tempered Scales
Ramsay - PLATE XXVII - The Mathematical Scale of Thirty two notes in Commas, Sharps and Flats
scale of mathematical intonation
semitonic progression
tempered key
tempered scale
tempered system
three mathematical primes

Created by Dale Pond. Last Modification: Tuesday November 3, 2020 04:56:57 MST by Dale Pond.