Comma and diesis: Difference between revisions

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"~~Comma~~" and "diesis" are two terms used to refer to intervals that are less than about 60 ~~cents~~ in size.

: ''This article is about "comma" and "diesis" as interval regions. For other senses of these two words, see [[comma]] and [[diesis]].''

{{Infobox interval region

| Name=Comma, diesis

| Cents lower=0

| Cents upper=40

| Cents upper wide=60

| JI intervals=81/80, 128/125

| Complement=(Imperfect) [[octave]]

| Lower region=[[Unison]]

| Higher region=[[Semitone (interval region)|Semitone]]

}}

'''Comma''' and '''diesis''' are two terms used to refer to intervals that are less than about 60{{cent}} in size. In terms of [[interval region]]s, "comma" refers to an interval flatter than about 30{{cent}}, and "diesis" refers to an interval between about 30 and 60{{cent}}. In [[Sagittal notation]], a comma is specifically defined as between half of the [[Pythagorean comma]] {{monzo| -19 12}} and half of the Pythagorean 17-fifths diesis {{monzo| 27 -17}}, about 11.7{{c}} to 33.4{{c}}, and a diesis is defined as between the comma upper bound and half of the Pythagorean 19-fifths apotome-plus-comma {{monzo| -30 19}}, about 68.6{{c}}.

"Comma" also refers to an interval that is tempered out by any given [[temperament]]~~, but that is not the sense of "comma" that this article covers~~.

"[[Comma]]" also refers to an interval that is tempered out by any given [[temperament]].

~~Generally~~, "comma" ~~refers~~ to ~~an interval flatter than about 30 cents~~, and "diesis" ~~refers~~ to an interval ~~between about 30 and 60 cents~~.

The range of dieses largely overlaps with the range of [[quartertone]]s (between 40 and 60{{c}}, reasonably mapped to 1/24edo), which, according to systems that determine consonance in terms of proximity to simple just ratios, is one of the most dissonant interval regions. This also corresponds to an [[interseptimal]] interval range. However, quarter tones are still covered here to provide a resource for them in the same format as the other interval region pages.

In the diatonic scale, the analogous concepts are '''subchromatic''' and '''enharmonic''' steps. A subchromatic step (a "comma") does not change the interval category (for example, in most just notation systems, if you flatten the major third [[81/64]] by an [[81/80]] comma to produce [[5/4]], the latter is still considered a major third). Diatonically, subchromatic steps are '''perfect unisons (P1)''', and there are none that are not a unison in a rank-2 diatonic tuning. An enharmonic step (a "diesis", although this is controversial) changes the interval category to an enharmonic interval (for example, a major third to a diminished fourth, or a chromatic semitone to a diatonic semitone). Similarly, enharmonic steps are ascending or descending '''diminished seconds (d2)'''.

== In just intonation ==

In just intonation, commas are often seen as the difference between two similar intervals, so it is hard to find intervals within this range that are treated as steps in their own right. The 3-limit interval in this range is the ~~'''~~Pythagorean comma~~'''~~ of [[531441/524288]], which can be considered an augmented seventh (octave-reduced), and is about 23 ~~cents~~.

=== By prime limit ===

In just intonation, commas are often seen as the difference between two similar intervals, so it is hard to find intervals within this range that are treated as steps in their own right. The 3-limit interval in this range is the Pythagorean comma of [[531441/524288]], which can be considered an augmented seventh (octave-reduced), and is about 23{{c}}.

For the remainder of this list, intervals are provided that are ''not'' mostly treated as commas (in the temperament sense). Higher-limit intervals in the comma and diesis range are:

* The 5-limit '''augmented diesis''' is a ratio of 128/125, and is about 41{{c}}.

** There is also the 5-limit '''magic comma''' of 3125/3072, which is about 30{{c}}.

* The 7-limit '''slendro diesis''' is a ratio of 49/48, and is about 36{{c}}.

* The 11-limit '''quarter tone''' is a ratio of 33/32, and is about 53{{c}}.

* The 13-limit '''minor diesis''' is a ratio of 40/39, and is about 43{{c}}.

~~For the remainder of this list, I have tried to choose intervals that are '''not''' mostly treated as commas (in the temperament sense). Higher-limit intervals in the~~ comma and diesis ~~range are~~:

=== By delta ===

As comma and diesis is the smallest interval class, it may be represented by:

* ~~The 5~~-~~limit '''augmented diesis''' is a ratio of 128/125, and is about 41 cents~~.

* Any delta-1 (i.e. superparticular) interval smaller than 29/28

** There is also the 5-limit '''magic comma''' of 3125/3072, which is about 30 cents.

* Any delta-2 interval smaller than 57/55

* The 7-limit '''slendro diesis''' is a ratio of 49/~~48, and is about 36 cents.~~

* Any delta-3 interval smaller than 88/85

* ~~The 11~~-~~limit '''quarter tone''' is a ratio of 33~~/~~32, and is about 53 cents.~~

* ~~The 13~~-~~limit '''minor diesis''' is a ratio of 40~~/~~39, and is about 43 cents.~~

== In EDOs ==

The following table lists the best tuning of 128/125, and other dieses or commas if present, in various significant [[EDOs]]. Not included are EDOs (i.e. those smaller than 15) where the best tuning is the unison, or 0c, or those where the best tuning is sharper than 60 ~~cents~~ (i.e. not a diesis or comma).

The following table lists the best tuning of 128/125, and other dieses or commas if present, in various significant [[edos|EDOs]]. Not included are EDOs (i.e. those smaller than 15) where the best tuning is the unison, or 0{{c}}, or those where the best tuning is sharper than 60{{c}} (i.e. not a diesis or comma). Note that this does not depend on how each EDO tunes the intervals that 128/125 might be derived from, only on which edostep is closest to 128/125's size.

{| class="wikitable"

|+

~~!EDO~~

~~!128/125~~

~~!Other commas and dieses~~

|-

~~|22~~

! EDO

~~|54c~~

! 128/125

|

! Other commas and dieses

|-

|24

| 22

|~~50c~~

| 54{{c}}

|

|-

|25

| 24

|~~48c~~

| 50{{c}}

|

| 50¢ ≈ 33/32

|-

|26

| 25

|~~46c~~

| 48{{c}}

|

|-

|27

| 26

|~~44c~~

| 46{{c}}

|

|-

|29

| 27

|~~41c~~

| 44{{c}}

|

|-

|31

| 29

|~~39c~~

| 41{{c}}

|

|-

|34

| 31

|~~35c~~

| 39{{c}}

|

|-

|41

| 34

|~~29c~~

| 35{{c}}

|~~59c ≈ 33/32~~

|

|-

|53

| 41

|~~45c~~

| 29{{c}}

|~~22c~~ ≈ 81/80

| {{nowrap|59{{c}} ≈ 33/32}}

|-

| 53

| 45{{c}}

| {{nowrap|22{{c}} ≈ 81/80}}

|}

== In regular temperaments ==

The role of commas and dieses in regular temperaments is often as the intervals that are tempered out (i.e. equated to 0 cents). Discussing that is not within the scope of this article; you may learn more at [[~~Temperament~~]].

The role of commas and dieses in regular temperaments is often as the intervals that are tempered out (i.e. equated to 0 cents). Discussing that is not within the scope of this article; you may learn more at [[Regular temperament]].

However, there are, rarely, temperaments generated by commas. One example is [[slender]], where ten [[49/48]]~~<nowiki/>~~s ~~equal~~ [[5/4]].

However, there are, rarely, temperaments generated by commas. One example is [[slender]], where a stack of ten [[49/48]]'s equals [[5/4]].

== In MOS scales ==

Intervals less than 100{{c}} generate the following [[mos|MOS]] scales:

These tables start from the last monolarge MOS generated by the interval range.

Scales with more than 12 notes are not included.

{| class="wikitable"

|-

! Range

! MOS

|-

| 0–100{{c}}

| [[1L 11s]]

|}

@@ Line 1: / Line 1: @@
-"Comma" and "diesis" are two terms used to refer to intervals that are less than about 60 cents in size.
+: ''This article is about "comma" and "diesis" as interval regions. For other senses of these two words, see [[comma]] and [[diesis]].''
+{{Infobox interval region
+| Name=Comma, diesis
+| Cents lower=0
+| Cents upper=40
+| Cents upper wide=60
+| JI intervals=81/80, 128/125
+| Complement=(Imperfect) [[octave]]
+| Lower region=[[Unison]]
+| Higher region=[[Semitone (interval region)|Semitone]]
+}}
+'''Comma''' and '''diesis''' are two terms used to refer to intervals that are less than about 60{{cent}} in size. In terms of [[interval region]]s, "comma" refers to an interval flatter than about 30{{cent}}, and "diesis" refers to an interval between about 30 and 60{{cent}}. In [[Sagittal notation]], a comma is specifically defined as between half of the [[Pythagorean comma]] {{monzo| -19 12}} and half of the Pythagorean 17-fifths diesis {{monzo| 27 -17}}, about 11.7{{c}} to 33.4{{c}}, and a diesis is defined as between the comma upper bound and half of the Pythagorean 19-fifths apotome-plus-comma {{monzo| -30 19}}, about 68.6{{c}}.
-"Comma" also refers to an interval that is tempered out by any given [[temperament]], but that is not the sense of "comma" that this article covers.
+"[[Comma]]" also refers to an interval that is tempered out by any given [[temperament]].
-Generally, "comma" refers to an interval flatter than about 30 cents, and "diesis" refers to an interval between about 30 and 60 cents.
+The range of dieses largely overlaps with the range of [[quartertone]]s (between 40 and 60{{c}}, reasonably mapped to 1/24edo), which, according to systems that determine consonance in terms of proximity to simple just ratios, is one of the most dissonant interval regions. This also corresponds to an [[interseptimal]] interval range. However, quarter tones are still covered here to provide a resource for them in the same format as the other interval region pages.
+In the diatonic scale, the analogous concepts are '''subchromatic''' and '''enharmonic''' steps. A subchromatic step (a "comma") does not change the interval category (for example, in most just notation systems, if you flatten the major third [[81/64]] by an [[81/80]] comma to produce [[5/4]], the latter is still considered a major third). Diatonically, subchromatic steps are '''perfect unisons (P1)''', and there are none that are not a unison in a rank-2 diatonic tuning. An enharmonic step (a "diesis", although this is controversial) changes the interval category to an enharmonic interval (for example, a major third to a diminished fourth, or a chromatic semitone to a diatonic semitone). Similarly, enharmonic steps are ascending or descending '''diminished seconds (d2)'''.
 == In just intonation ==
-In just intonation, commas are often seen as the difference between two similar intervals, so it is hard to find intervals within this range that are treated as steps in their own right. The 3-limit interval in this range is the '''Pythagorean comma''' of [[531441/524288]], which can be considered an augmented seventh (octave-reduced), and is about 23 cents.
+=== By prime limit ===
+In just intonation, commas are often seen as the difference between two similar intervals, so it is hard to find intervals within this range that are treated as steps in their own right. The 3-limit interval in this range is the Pythagorean comma of [[531441/524288]], which can be considered an augmented seventh (octave-reduced), and is about 23{{c}}.
+For the remainder of this list, intervals are provided that are ''not'' mostly treated as commas (in the temperament sense). Higher-limit intervals in the comma and diesis range are:
+* The 5-limit '''augmented diesis''' is a ratio of 128/125, and is about 41{{c}}.
+** There is also the 5-limit '''magic comma''' of 3125/3072, which is about 30{{c}}.
+* The 7-limit '''slendro diesis''' is a ratio of 49/48, and is about 36{{c}}.
+* The 11-limit '''quarter tone''' is a ratio of 33/32, and is about 53{{c}}.
+* The 13-limit '''minor diesis''' is a ratio of 40/39, and is about 43{{c}}.
-For the remainder of this list, I have tried to choose intervals that are '''not''' mostly treated as commas (in the temperament sense). Higher-limit intervals in the comma and diesis range are:
+=== By delta ===
+As comma and diesis is the smallest interval class, it may be represented by:
-* The 5-limit '''augmented diesis''' is a ratio of 128/125, and is about 41 cents.
+* Any delta-1 (i.e. superparticular) interval smaller than 29/28
-** There is also the 5-limit '''magic comma''' of 3125/3072, which is about 30 cents.
+* Any delta-2 interval smaller than 57/55
-* The 7-limit '''slendro diesis''' is a ratio of 49/48, and is about 36 cents.
+* Any delta-3 interval smaller than 88/85
-* The 11-limit '''quarter tone''' is a ratio of 33/32, and is about 53 cents.
-* The 13-limit '''minor diesis''' is a ratio of 40/39, and is about 43 cents.
 == In EDOs ==
-The following table lists the best tuning of 128/125, and other dieses or commas if present, in various significant [[EDOs]]. Not included are EDOs (i.e. those smaller than 15) where the best tuning is the unison, or 0c, or those where the best tuning is sharper than 60 cents (i.e. not a diesis or comma).
+The following table lists the best tuning of 128/125, and other dieses or commas if present, in various significant [[edos|EDOs]]. Not included are EDOs (i.e. those smaller than 15) where the best tuning is the unison, or 0{{c}}, or those where the best tuning is sharper than 60{{c}} (i.e. not a diesis or comma). Note that this does not depend on how each EDO tunes the intervals that 128/125 might be derived from, only on which edostep is closest to 128/125's size.
 {| class="wikitable"
-|+
-!EDO
-!128/125
-!Other commas and dieses
 |-
-|22
+! EDO
-|54c
+! 128/125
-|
+! Other commas and dieses
 |-
-|24
+| 22
-|50c
+| 54{{c}}
 |
 |-
-|25
+| 24
-|48c
+| 50{{c}}
-|
+| 50¢ ≈ 33/32
 |-
-|26
+| 25
-|46c
+| 48{{c}}
 |
 |-
-|27
+| 26
-|44c
+| 46{{c}}
 |
 |-
-|29
+| 27
-|41c
+| 44{{c}}
 |
 |-
-|31
+| 29
-|39c
+| 41{{c}}
 |
 |-
-|34
+| 31
-|35c
+| 39{{c}}
 |
 |-
-|41
+| 34
-|29c
+| 35{{c}}
-|59c ≈ 33/32
+|
 |-
-|53
+| 41
-|45c
+| 29{{c}}
-|22c ≈ 81/80
+| {{nowrap|59{{c}} ≈ 33/32}}
+|-
+| 53
+| 45{{c}}
+| {{nowrap|22{{c}} ≈ 81/80}}
 |}
 == In regular temperaments ==
-The role of commas and dieses in regular temperaments is often as the intervals that are tempered out (i.e. equated to 0 cents). Discussing that is not within the scope of this article; you may learn more at [[Temperament]].
+The role of commas and dieses in regular temperaments is often as the intervals that are tempered out (i.e. equated to 0 cents). Discussing that is not within the scope of this article; you may learn more at [[Regular temperament]].
-However, there are, rarely, temperaments generated by commas. One example is [[slender]], where ten [[49/48]]<nowiki/>s equal [[5/4]].
+However, there are, rarely, temperaments generated by commas. One example is [[slender]], where a stack of ten [[49/48]]'s equals [[5/4]].
+== In MOS scales ==
+Intervals less than 100{{c}} generate the following [[mos|MOS]] scales:
+These tables start from the last monolarge MOS generated by the interval range.
+Scales with more than 12 notes are not included.
+{| class="wikitable"
+|-
+! Range
+! MOS
+|-
+| 0–100{{c}}
+| [[1L&nbsp;11s]]
+|}
 {{Navbox intervals}}