Ascending and descending circles of fifths produce different pitches

Evidence

If the circle of fifths were established by descending by perfect fifths instead of ascending, the resulting note 12 fifths below the starting notes would be flatter than seven octaves by 1.0136433, and every note of the descending circle would be slightly different from the members of the ascending circle. Figure 3.18 shows this effect and the manner in which the notes can be notated. For example, notes such as D# and E♭, A# and B♭, B♭♭ and A are not the same and are known as “enharmonics,” giving rise to the pairs of intervals such as major third and diminished fourth and major seventh and diminished octave shown in Figure 3.15. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1015

Categorization

• DEFINE: Explains the concerete meaning of enharmonic equivalence — that D# and E♭ are only enharmonically, not acoustically, the same.
• corroborated: 2 independent passages support this claim — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1015
• XREF: connects to: Twelve stacked perfect fifths fail to close the octave exactly; Consecutive intervals of equal magnitude and opposite sign create repeated tones in pitch sequences

Broader Connections

• Consecutive intervals of equal magnitude and opposite sign create repeated tones in pitch sequences #expands (0.80)