A New Integral Ratio Chromatic Scale

Abstract

The IRC scale presented herein is neither a Pythagorean scale, nor a just scale, nor an equally tempered scale, as those terms are defined in Acoustical Terminology, but it is shown to possess the major advantages of all three. In a D-D octave, all of the notes are interconnected by a perfectly symmetrical arrangement of just fourths, and the note frequencies of the octave containing C4=256 cps are 144, 153, 162, 171, 180, 192, 204, 216, 228, 240, 256, 272, 288. A distinguishing characteristic of the IRC scale is that four ascending octaves commencing with a C embrace exactly four successive just fifths (C-G-D-A-E) immediately followed by four successive just fourths (E-A-D-G-C). The IRC scale, like the conventional just scale, is based on alternately conjunct and disjunct just tetrachords, but the IRC tetrachord intervals are the smoothly decreasing ones of 9:8, 10:9, 16:15, instead of the just scale irregularly arranged intervals of 16:15, 9:8, 10:9. The major and minor tones of the IRC scale are divided by the known superparticular proportion, giving, in addition to the just semitone 16:15, semitone ratios of 17:16, 18:17, 19:18, and 20:19. These five semitones are located in smoothly progressing order, forming three perfect harmonic series embracing a just major third, fourth, and minor third. The IRC scale is shown to be superior to the just and Pythagorean scales and to excel the equally tempered scale in modulation. If, as the author believes, Helmholtz was correct in his thesis that just intervals are musically better than equal tempered intervals, then the IRC scale is also superior to the equally tempered scale, possessing, among many other advantages, greater ease of tuning pianos and organs, and the ability to tune pianos, violins, and cellos exactly alike and play them in the same temperament.