The Musical Scale of Just Intonation

Abstract

The musical scale of Just Intonation in one tonality (key) as evolved since early Greek times, may be defined as consisting of four tonal notes: subdominant, tonic, dominant and supertonic at intervals of fifths (3/2) apart with four major notes, each an interval of the major third (5/4) above these tonal notes and with four minor notes, each an interval of the major third (5/4) below these tonal notes, all brought within the limits of an octave (2/1). Assuming the frequency of the keynote of C equals 260.74 cycles per second, we obtain the following scale of frequencies: Frequency Frequency Note Name of Interval Ratio to Prime Cycles/Second C Prime (Tonic) 1/1 1.0000 260.74 D♭ Subdominant Minor Sixth 16/15 1.0667 278.13 D Second (Supertonic) 9/8 1.1250 293.33 E♭ Minor Third 6/5 1.2000 312.89 E Major Third 5/4 1.2500 325.93 F Fourth (Subdominant) 4/3 1.3333 347.65 F♯ Dominant Major Seventh 45/32 1.4063 366.69 G Fifth (Dominant) 3/2 1.5000 391.11 A♭ Minor Sixth 8/5 1.6000 417.18 A Major Sixth 5/3 1.6667 434.57 B♭ Minor Seventh 9/5 1.8000 469.33 B Major Seventh 15/8 1.8750 488.89 C Octave (Tonic) 2/1 2.0000 521.48 In connection with research and experimental work on an electronic type piano, the writer devised the Electronichord as described in the American Physics Teacher, February, 1934. Using two Electronichords, it was possible to study the relations that exist between the harmonic overtones of two string tones tuned any harmonic or tempered interval apart. These experiments, among other things, audibly demonstrated the importance of the harmonic intervals to satisfy the discriminating musical ear. The purpose of this paper is to offer a mathematical approach in demonstrating that the musical scale of just intonation is a fundamental acoustical measurement.