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The '''diesis''' (/ˈdaɪəsɪs/ ''DY-ə-sis''; plural ''dieses'') ~~most~~ commonly refers to:

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''.

* The ~~difference between two enharmonic notes in a scale. See~~ ''[[~~Diesis (scale theory)~~]]~~'';~~

== History ==

* [[~~128~~/~~125~~]] ~~(≈ 41¢)~~, ~~the ''augmented comma'' or ''lesser diesis'';~~

The earliest usage of the term ''diesis'' was due to {{w|Philolaus}} in [[ancient Greek music]] to refer to an interval that is known as the [[256/243|limma]] today. Another usage, also in ancient Greek theory and notably used by {{w|Aristoxenus}}, would refer to a number of [[quartertone]]-sized intervals. In more recent times, {{w|Marchetto da Padova}} used it for fifth-tones, and finally, the modern diesis centered around [[31edo|1\31]] is due to [[Adriaan Fokker]].

** rarely, [[~~648/625~~]], the ~~''diminished comma'' or ''greater~~ diesis~~'';~~

* A melodic unit of about that size. See ''[[~~Diesis (melodic unit)~~]]''.

~~Intervals with~~ the ~~word in~~ the ~~name~~:

== 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]] {{monzo| 27 -17 }} and half of the [[19-comma]] {{monzo| -30 19 }}, about 33.4{{c}} to 68.6{{c}}<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.

* [[~~Magic~~ comma]], or ''~~small~~ diesis''

=== Just intervals ===

* [[~~Porcupine~~ comma]], or ''~~maximal~~ diesis''

Some just intervals have been named according to this sense of diesis.

* [[Tetracot comma]], or ''minimal diesis''

* [[Porcupine comma]], or ''maximal diesis'' (49.2{{c}})

* [[49/48]], the ''large septimal diesis''

* [[Magic comma]], or ''small diesis'' (29.6{{c}})

* [[Tetracot comma]], or ''minimal diesis'' (27.7{{c}})

* [[49/48]], the ''large septimal diesis'' (35.7{{c}})

* [[50/49]], the ''small septimal diesis'' or ''septimal tritonic diesis'' (35.0{{c}})

This is not to be confused with the related sense of the same term introduced next, for which the [[648/625|major diesis (648/625)]] was named while being way wider than the "maximal diesis".

=== Generalization ===

For someone looking for what sets these just intervals apart from others in the size range, it might be observed that almost all just intervals that have been 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]]{{idio}}, as characteristic of intervals of a size between that of the minimal diesis (27.66{{c}}) and maximal diesis (49.17{{c}}).

* Equating a ''short'' stack of one [[LCJI]] interval with some other LCJI interval (never more than 4 or 5). 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.

As a result, when tempered out and in a tuning that makes the other simple interval of interest pure, all dieses incur a noticeable amount of damage on the interval being stacked. 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{{c}}), as more than 7 cents of damage is too much for most complex intervals and is essentially a flexibility afforded by LCJI's temperability. 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.

Therefore, if we are interested only in how a stack of 2 to 4<ref group="note">Setting it to 5 only gives us one additional diesis, (2⋅(10/9))/(7/6)5, and the rest of the dieses only get more ways of reaching them at best. If the interval we want to reach with our stack is in the 7-odd-limit, the set of dieses is the same.</ref> 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) <ref group="note">Idiosyncratically agreed upon to be called the ''aberdiesis'' for the relation it has to the ''aberschisma'', (10/7)/(9/8)3.</ref>, 3125/3072, 50/49, [[5103/5000]] = (7/5)/(10/9)3 = (14/9)/(10/9)4 <ref group="note">Called a diesis in a theory of [[Lériendil]]'s that uses yet another definition of diesis.</ref>, 49/48, [[12288/12005]] = (8/7)4/(5/3), 128/125, [[33614/32805]] = (2⋅(7/5))/(9/7)4, [[19683/19208]] = (9/7)4/(4/3), [[16807/16384]] = (7/4)/(8/7)4, 36/35, 250/243

A few definitions conveniently happen to give an equivalent list; the set of LCJI intervals we are 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. To tweak with parameters yourself, you can use [[User:Godtone #My Python 3 code|Godtone's code]], which has no dependencies other than [https://www.python.org/ Python 3]; use <code>diesis</code> with <code>show=True</code>.)

(Also interesting is that the maximal diesis of 250/243 isn't only maximum in terms of size in cents but in terms of damage implied in a tuning that makes the target (6/5) pure, meaning half the damage is on each of two 10/9's.)

Finally, in regards to the specific set of equivalent definitions discussed, it should be noted that they are also equivalent without the requirement on a minimum size in [[cent]]s for the comma, instead allowing the minimum damage to impose a minimum size. 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 seventeen 7-limit dieses. In regards to 875/864, one might note that it is similar to the standard diesis of 128/125 (([[2/1]])/([[5/4]])3, S-expression [[16/15|S4]]/[[25/24|S5]]) as it is equal to ([[7/4]])/([[6/5]])3 and has the S-expression [[25/24|S5]]/[[36/35|S6]], so it is 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 [[5L 2s|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 fifth]]s, 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 [[comma]]s 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 {{nowrap| {{!}}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 ''n''L 1s scales, it is the double-diminished mosstep or inverse thereof since the small step itself is diminished. Except for ''n''L 1s scales, it is the diminished mosstep in [[soft]] {{nowrap|(L:s < 2:1)}} scales and the inverse diminished mosstep in [[hard]] {{nowrap|(L:s > 2:1)}} scales. It vanishes in basic {{nowrap|(L:s {{=}} 2:1)}} scales.

== Normal diesis ==

The '''normal diesis''' (≈ 38.7¢) is an [[interval size measure]] defined as one step of [[31edo]].

The '''normal diesis''' (~38.7{{c}}) is an [[interval size measure]] defined as one step of [[31edo]].

== See also ==

* [[Chroma]]

* [[Shimmerstep]]

== External links ==

* [http://tonalsoft.com/enc/d/diesis.aspx Tonalsoft Encyclopedia | ''Diesis'']

== Notes ==

~~{{Disambiguation}}~~

== References ==

[[Category:Terms]]

[[Category:MOS scale]]

[[Category:Interval size measures]]

@@ Line 1: / Line 1: @@
 {{Wikipedia}}
-The '''diesis''' (/ˈdaɪəsɪs/ ''DY-ə-sis''; plural ''dieses'') most commonly refers to:
+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''.
-* The difference between two enharmonic notes in a scale. See ''[[Diesis (scale theory)]]'';
+== History ==
-* [[128/125]] (≈ 41¢), the ''augmented comma'' or ''lesser diesis'';
+The earliest usage of the term ''diesis'' was due to {{w|Philolaus}} in [[ancient Greek music]] to refer to an interval that is known as the [[256/243|limma]] today. Another usage, also in ancient Greek theory and notably used by {{w|Aristoxenus}}, would refer to a number of [[quartertone]]-sized intervals. In more recent times, {{w|Marchetto da Padova}} used it for fifth-tones, and finally, the modern diesis centered around [[31edo|1\31]] is due to [[Adriaan Fokker]].
-** rarely, [[648/625]], the ''diminished comma'' or ''greater diesis'';
-* A melodic unit of about that size. See ''[[Diesis (melodic unit)]]''.
-Intervals with the word in the name:
+== 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]] {{monzo| 27 -17 }} and half of the [[19-comma]] {{monzo| -30 19 }}, about 33.4{{c}} to 68.6{{c}}<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.
-* [[Magic comma]], or ''small diesis''
+=== Just intervals ===
-* [[Porcupine comma]], or ''maximal diesis''
+Some just intervals have been named according to this sense of diesis.
-* [[Tetracot comma]], or ''minimal diesis''
+* [[Porcupine comma]], or ''maximal diesis'' (49.2{{c}})
-* [[49/48]], the ''large septimal diesis''
+* [[Magic comma]], or ''small diesis'' (29.6{{c}})
+* [[Tetracot comma]], or ''minimal diesis'' (27.7{{c}})
+* [[49/48]], the ''large septimal diesis'' (35.7{{c}})
+* [[50/49]], the ''small septimal diesis'' or ''septimal tritonic diesis'' (35.0{{c}})
+This is not to be confused with the related sense of the same term introduced next, for which the [[648/625|major diesis (648/625)]] was named while being way wider than the "maximal diesis".
+=== Generalization ===
+For someone looking for what sets these just intervals apart from others in the size range, it might be observed that almost all just intervals that have been 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]]{{idio}}, as characteristic of intervals of a size between that of the minimal diesis (27.66{{c}}) and maximal diesis (49.17{{c}}).
+* Equating a ''short'' stack of one [[LCJI]] interval with some other LCJI interval (never more than 4 or 5). 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.
+As a result, when tempered out and in a tuning that makes the other simple interval of interest pure, all dieses incur a noticeable amount of damage on the interval being stacked. 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{{c}}), as more than 7 cents of damage is too much for most complex intervals and is essentially a flexibility afforded by LCJI's temperability. 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.
+Therefore, if we are interested only in how a stack of 2 to 4<ref group="note">Setting it to 5 only gives us one additional diesis, (2⋅(10/9))/(7/6)<sup>5</sup>, and the rest of the dieses only get more ways of reaching them at best. If the interval we want to reach with our stack is in the 7-odd-limit, the set of dieses is the same.</ref> 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)<sup>3</sup>/(7/5) <ref group="note">Idiosyncratically agreed upon to be called the ''aberdiesis'' for the relation it has to the ''aberschisma'', (10/7)/(9/8)<sup>3</sup>.</ref>, 3125/3072, 50/49, [[5103/5000]] = (7/5)/(10/9)<sup>3</sup> = (14/9)/(10/9)<sup>4</sup> <ref group="note">Called a diesis in a theory of [[Lériendil]]'s that uses yet another definition of diesis.</ref>, 49/48, [[12288/12005]] = (8/7)<sup>4</sup>/(5/3), 128/125, [[33614/32805]] = (2⋅(7/5))/(9/7)<sup>4</sup>, [[19683/19208]] = (9/7)<sup>4</sup>/(4/3), [[16807/16384]] = (7/4)/(8/7)<sup>4</sup>, 36/35, 250/243
+A few definitions conveniently happen to give an equivalent list; the set of LCJI intervals we are 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. To tweak with parameters yourself, you can use [[User:Godtone #My Python 3 code|Godtone's code]], which has no dependencies other than [https://www.python.org/ Python 3]; use <code>diesis</code> with <code>show=True</code>.)
+(Also interesting is that the maximal diesis of 250/243 isn't only maximum in terms of size in cents but in terms of damage implied in a tuning that makes the target (6/5) pure, meaning half the damage is on each of two 10/9's.)
+Finally, in regards to the specific set of equivalent definitions discussed, it should be noted that they are also equivalent without the requirement on a minimum size in [[cent]]s for the comma, instead allowing the minimum damage to impose a minimum size. 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 seventeen 7-limit dieses. In regards to 875/864, one might note that it is similar to the standard diesis of 128/125 (([[2/1]])/([[5/4]])<sup>3</sup>, S-expression [[16/15|S4]]/[[25/24|S5]]) as it is equal to ([[7/4]])/([[6/5]])<sup>3</sup> and has the S-expression [[25/24|S5]]/[[36/35|S6]], so it is 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 [[5L 2s|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 fifth]]s, 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 [[comma]]s 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 {{nowrap| {{!}}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 ''n''L 1s scales, it is the double-diminished mosstep or inverse thereof since the small step itself is diminished. Except for ''n''L 1s scales, it is the diminished mosstep in [[soft]] {{nowrap|(L:s < 2:1)}} scales and the inverse diminished mosstep in [[hard]] {{nowrap|(L:s > 2:1)}} scales. It vanishes in basic {{nowrap|(L:s {{=}} 2:1)}} scales.
 == Normal diesis ==
-The '''normal diesis''' (≈ 38.7¢) is an [[interval size measure]] defined as one step of [[31edo]].
+The '''normal diesis''' (~38.7{{c}}) is an [[interval size measure]] defined as one step of [[31edo]].
+== See also ==
+* [[Chroma]]
+* [[Shimmerstep]]
+== External links ==
+* [http://tonalsoft.com/enc/d/diesis.aspx Tonalsoft Encyclopedia | ''Diesis'']
+== Notes ==
+<references group="note"/>
-{{Disambiguation}}
+== References ==
+<references/>
 [[Category:Terms]]
+[[Category:MOS scale]]
+[[Category:Interval size measures]]