Diesis
The diesis (/ˈdaɪəsɪs/ DY-ə-sis; plural dieses) is a small interval that has several related definitions. Most commonly, it refers to 128/125, the augmented comma a.k.a. lesser diesis, though rarely and if the context is clear, it can refer to 648/625, the diminished comma a.k.a. greater diesis.
History
The earliest usage of the term diesis was due to Philolaus in ancient Greek music to refer to an interval that is known as the limma today. Another usage, also in ancient Greek theory and notably used by Aristoxenus, would refer to a number of quartertone-sized intervals. In more recent times, Marchetto da Padova used it for fifth-tones, and finally, the modern diesis centered around 1\31 is due to Adriaan Fokker.
As an interval region
As an interval region, the diesis is a small melodic unit of about an augmented comma's size. The specific range varies considerably among musicians, but is generally agreed to be roughly 30–60 cents. In Sagittal notation, a diesis is specifically defined as between half of the 17-comma [27 -17⟩ and half of the 19-comma [-30 19⟩, about 33.4 ¢ to 68.6 ¢[1].
Just intervals
Some just intervals have been named according to this sense of diesis.
- Porcupine comma, or maximal diesis (49.2 ¢)
- Magic comma, or small diesis (29.6 ¢)
- Tetracot comma, or minimal diesis (27.7 ¢)
- 49/48, the large septimal diesis (35.7 ¢)
- 50/49, the small septimal diesis or septimal tritonic diesis (35.0 ¢)
This is not to be confused with the related sense of the same term introduced next, for which the major diesis (648/625) was named while being way wider than the "maximal diesis".
Generalization
For someone looking for what sets these (and a few others) apart from other commas in the size range, it might be worth noting be noted that (with the exception of 648/625 being slightly larger than 250/243), almost all just intervals commonly called dieses have a few properties in common that might be used to derive a definition that expands the set of commas called dieses to something closer to the spirit of the term as a whole:
- Being an awkward xenmelodic size (as characteristic of intervals of a size between that of the minimal diesis (27.66 ¢) and maximal diesis (49.17 ¢)). Relatedly:
- Equating a short stack of one LCJI interval with some other LCJI interval (and it appears it's never more than 4 or 5, as suggested by the unusual historical diesis of 256/243 = (4/3)5 / 4 = (4/3)4 / 3). Note that due to the minimal and maximal diesis both using a stack of ~10/9's, one could argue that at most, one is looking at how a short stack of some 9-odd-limit interval relates to some other simple interval of interest.
- Due to the last two constraints, when tempered out and in a tuning that makes the other simple interval of interest pure, all dieses incur a not-unnoticeable amount of damage on the interval being stacked*. This is arguably what truly makes them feel awkward in JI, as they are also small enough to feel like potential commas without being very efficient to temper out. *Notably, some of these are more debatable than others in terms of damage, so one should clarify that the minimum damage logically is that of the minimal diesis (6.9 ¢), as more than 7 cents of damage is not insignificant for most intervals and is essentially a flexibility afforded by LCJI's temperability.
Therefore, if we are interested only in how a stack of 2 to 4 or 2 to 5 of a 9-odd-limit interval differs from another 9-odd-limit interval under these constraints, we get the following list of 7-limit dieses, with new things categorized as such linked:
20000/19683, 3645/3584 = (9/8)3 / (7/5) (*), 3125/3072, 50/49, 5103/5000 = (7/5) / (10/9)3 = (14/9) / (10/9)4 (**), 49/48, 12288/12005 = (8/7)4 / (5/3), 128/125, 19683/19208 = (9/7)4 / (4/3) = (9/7)5 / (12/7), 16807/16384 = (7/4) / (8/7)4 = (2/1) / (8/7)5, 36/35, 250/243
(*which might be the tritonic diesis by contrast with the septimal tritonic diesis of 50/49)
(**called a diesis in a theory of Lériendil's that uses a definition of diesis currently not documented on this page)
A few definitions conveniently happen to give an equivalent list; the set of LCJI intervals we're interested in the stack being near to could be the 7-odd-limit instead, and whether we choose a 2 to 4 or 2 to 5 range only changes the number of expressions for some of the dieses, so this appears to be an algorithmically significant result at the very least, evidencing a possible computational basis for the intuitive properties of the notion. (A more general parametrization might only use the 2 to 5 range to look for alternate expressions but 2 to 4 to avoid overcomplex expressions, while having some stack of 9-odd-limit equal a 13-odd-limit interval, but it might be preferred to use definitions that keep the set elegant.)
Finally, in regards to the specific set of 4 equivalent definitions discussed, it should be noted that they are also equivalent if we don't require a minimum size in cents for the comma, instead allowing the minimum damage to impose a minimum size (which is arguably more relevant). This causes 81/80, 64/63, 875/864 and 245/243 to also be considered dieses, which arguably is not so unexpected as they all share the intuitively-motivated properties discussed above, for a total of 16 7-limit dieses. (In regards to 875/864, one might note that according to S-expressions, it's similar to the standard diesis of 128/125 = S4/S5 = (6/3)/(5/4)3 as it's equal to 875/864 = S5/S6 = (7/4)/(6/5)3, so it's in some sense a 7-limit analogue of the 5-limit standard diesis, and might be named based on this.)
As a diatonic interval category
In the diatonic scale, the diesis, more specifically enharmonic diesis, is a diminished second or inverse diminished second, whichever is positive in size. An example of a diesis is the interval between C♯ and D♭. The diesis spans twelve perfect fifths, and is observed in any tuning whose perfect fifth is not the same as 12edo's. Notes related by the diesis are said to be enharmonic to each other.
Just intervals
If the fifth represents the just interval 3/2, the diesis or inverse diesis represents the Pythagorean comma. In meantone, the diesis approximates a class of commas separated by the syntonic comma (81/80), among which 128/125, the augmented comma, is notable for being tuned pure in quarter-comma meantone. Therefore diesis traditionally refers to the augmented comma by default. Other dieses according to this definition are
- 648/625, the major diesis, tuned pure in 1/3-comma meantone.
- 2048/2025, the diaschisma, tuned pure in 1/6-comma meantone.
- 32805/32768, the schisma, tuned pure in 1/12-comma meantone.
This is not to be confused with the related sense of the same term introduced in the section above, for which a number of other intervals are named despite not being reached through twelve fifths.
Generalization
The diesis can be generalized to any mos scale as the mosdiesis, defined as |L - 2s|, i.e. the difference between a large step and two small steps. In terms of stepspan, it is usually the diminished mosstep or inverse diminished mosstep, whichever is positive. However, in nL 1s scales, it is the double-diminished mosstep or inverse thereof since the small step itself is diminished. Except for nL 1s scales, it is the diminished mosstep in soft (L:s < 2:1) scales and the inverse diminished mosstep in hard (L:s > 2:1) scales. It vanishes in basic (L:s = 2:1) scales.
Normal diesis
The normal diesis (~38.7 ¢) is an interval size measure defined as one step of 31edo.