Diesis - Alternative Definitions

Alternative Definitions

In any tuning system, the deviation of an octave from three major thirds, however large that is, is typically referred to as a diminished second. The diminished second is an interval between pairs of enharmonically equivalent notes; for instance the interval between E and F♭. As mentioned above, the term diesis most commonly refers to the diminished second in quarter-comma meantone temperament. Less frequently and less strictly, the same term is also used to refer to a diminished second of any size. In third-comma meantone, the diminished second is typically denoted as a greater diesis (see below).

In quarter-comma meantone, since major thirds are justly tuned, the width of the diminished second coincides with the above mentioned value of 128:125. Notice that 128:125 is larger than a unison (1:1). This means that, for instance, C' is sharper than B♯. In other tuning systems, the diminished second has different widths, and may be smaller than a unison (e.g. C' may be flatter than B♯):

• a greater diesis above unison (648:625) for third-comma meantone temperament (see below),
• a diaschisma above unison (2048:2025) for sixth-comma,
• a schisma below unison (32768:32805) for twelfth-comma, and
• a Pythagorean comma below unison (524288:531441) for Pythagorean tuning.

In eleventh-comma meantone, the diminished second is within 1/716 (0.0014) of a cent above unison, so it closely resembles the 1:1 unison ratio of twelve-tone equal temperament.

The word diesis has also been used to describe a large number of intervals, of varying sizes, but typically around 50 cents. Philolaus used it to describe the interval now usually called a limma, that of a justly tuned perfect fourth (4:3) minus two whole tones (9:8), equal to 256:243 or about 90.22 cents. Other theorists have used it for various other intervals.