Remarks on Just Intonation and Musical Scales

Any keyboard musical instrument may be tuned so that the intonation is just for any one key. Unfortunately, however, the intonation is no longer just if the instrument is played in any other key. In order to preserve just intonation, it is necessary to add two new notes in each octave for each sharp or flat which is added to the key signature. Thus, in order to be able to play a keyboard instrument so that it is justly tuned in each of the fifteen major keys which are recognized by musicians, it is necessary to provide 7 + 14 × 2, or 35 notes in each octave. It will be shown how it is possible to reduce the number of notes from 35 to 24 per octave if one will permit an error of 0.057 percent (0.98 cent) in some of the intervals. Thus, it is possible to obtain a musical scale with 24 notes to the octave in which any musical interval in any key is in error by not more than 0.98 cent. For the sake of comparison, it may be noted that errors in the major and minor thirds in the equally tempered scale are 13.7 and 15.6 cents, respectively. A tabular method is described which permits the rapid determination of the frequencies of the notes which must be added as sharps and flats are added to the key signature. The tabulation makes evident a striking lack of symmetry of the customary (Ptolemaic) just intonation. It is shown that the asymmetry is due to the fact that Ptolemy's just intonation is only one of two possible intonations which are equally just. The second was discovered by Robert Smith two hundred years ago, and was rediscovered about twenty years ago by the late John Redfield.