Extended meantone notation: Difference between revisions

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~~:''This page is currently reworked at~~ [[~~User:PiotrGrochowski/Extended meantone notation~~]]~~, see also~~ [[~~{{TALKPAGENAME}} #under construction|Talk:Extended meantone notation #under construction~~]]~~''.~~

[[Meantone]] can be notated with a [[chain of fifths]] consisting of the 7 natural notes along with sharps and flats:

~~== Extending the chain of fifths ==~~

... {{dash|F𝄫, C𝄫, G𝄫, D𝄫, A𝄫, E𝄫, B𝄫, F♭, C♭, G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, E♯, B♯, F𝄪, C𝄪, G𝄪, D𝄪, A𝄪, E𝄪, B𝄪|hair|long}} ...

~~[[Circle-of-fifths notation~~|~~Standard meantone notation]] uses 7 base note letters~~, ~~plus sharps and flats~~.

~~... F𝄫 C𝄫 G𝄫 D𝄫 A𝄫 E𝄫 B𝄫 F♭ C♭ G♭ D♭ A♭ E♭ B♭ F~~ C G D ~~A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯ F𝄪 C𝄪 G𝄪 D𝄪 A𝄪 E𝄪 B𝄪~~ ...

The chain is theoretically infinite, and C♯ and D♭ are not (in general) equivalent. When meantone is extended beyond 12 notes, it may require double-sharps, double-flats, and beyond. To avoid this, two new accidental pairs are introduced that raise/lower by the [[diesis]] and the [[kleisma]].

~~However, when transferred onto a 31edo scale, it looks like this:~~

{| class="wikitable center-all"

|-

! colspan="2" | Symbol

! colspan="2" rowspan="2" | Interval

! rowspan="2" | Examples

! rowspan="2" | [[Fifthspan]]

|-

! Raise

! Lower

|-

| ♯

| ♭

| Chromatic semitone

| Augmented unison (A1)

| C–C♯ E♭–E

| +7

|-

| ↑

| ↓

| Diesis

| Diminished 2nd (d2)

| C♯–D♭ D♯–E

| −12

|-

| +

| −

| Kleisma

| [[Negative interval|Negative]] double- diminished 2nd (-dd2)

| C♭ – B♯ F♭ – E♯

| +19

|}

~~C D𝄫 C♯ D♭ C𝄪 D E𝄫 D♯ E♭ D𝄪 E F♭ E♯ F G𝄫 F♯ G♭ F𝄪 G A𝄫 G♯ A♭ G𝄪 A B𝄫 A♯ B♭ A𝄪 B C♭ B♯ C~~

Because {{nowrap|19 − 12 {{=}} 7}}, {{nowrap|d2 + −dd2 {{=}} A1}}, and a diesis plus a kleisma equals a chromatic semitone.

~~Note that the base note letters alternate.~~

An octave is made up of:

~~The 31edo sharp can be split in half, so in 31edo this is solved by semisharps~~ and ~~semiflats, sometimes notated with [[ups~~ and ~~Downs Notation~~|~~ups~~ and ~~downs]].~~

* 7 diatonic semitones and 5 chromatic semitones {{nowrap|{{=}} 7 m2 + 5 A1}} {{nowrap|{{=}} 12 steps}}

* 12 chromatic semitones and 7 dieses {{nowrap|{{=}} 12 A1 + 7 d2}} {{nowrap|{{=}} 19 steps}}

* 19 dieses and 12 kleismas {{nowrap|{{=}} 19 d2 + 12 −dd2}} {{nowrap|{{=}} 31 steps}}

The meantone ~~circle of fifths however, has no single semisharp~~/~~semiflat~~. In ~~extended~~ meantone ~~notation~~, ~~a sharp~~ is ~~split into 2 different parts that can be added to produce a sharp:~~

The diesis represents the [[just intonation|just]] intervals [[128/125]] and [[648/625]] among others, while the meantone kleisma represents [[15625/15552]] = [-6 -5 6⟩ and [[3125/3072]] = [-10 -1 5⟩ among others. In [[septimal meantone]], where 7/4 is an augmented sixth, the diesis also represents [[36/35]], [[50/49]], and [[64/63]], while the kleisma also represents [[49/48]] and [[245/243]].

~~<pre>~~

The [[Enharmonic unison|enharmonic unisons]] ↓d2 and −↓A1 create various notational equivalences:

~~♯ — sharpen by meantone chromatic semitone, 7 fifths up~~

~~♭ — flatten by meantone chromatic semitone, 7 fifths down~~

~~↑ — sharpen by meantone diesis, 12 fifths down~~

~~↓ — flatten by meantone diesis, 12 fifths up~~

~~+ — sharpen by meantone kleisma, 19 fifths up~~

~~− — flatten by meantone kleisma, 19 fifths down~~

~~</pre>~~

~~A diesis plus a kleisma, added together, equals a meantone chromatic semitone. Note that in most meantone tunings,~~ the ~~diesis~~ and ~~kleisma~~ are ~~roughly a quarter tone.~~

* B♯↑ and B𝄪− are equal to C

* C+↑ is equal to C♯ (because the two semisharps add up)

* D𝄫↓ and D♭♭♭− are equal to C

~~Unlike~~ a ~~single semisharp/semiflat~~, ~~this can be generalized to other meantone tunings:~~

If the fifth is wider than {{nowrap|7\12 {{=}} 700{{c}}}}, C♯ is higher in pitch than D♭ and the diesis becomes a descending pythagorean comma. In 12edo, the tempering out of the diesis means that {{nowrap|C♯ {{=}} D♭}}.

*[[7edo]] (chromatic semitone is ~~tempered out~~, ~~diesis~~ is ~~positive,~~ and kleisma ~~is negative)~~

If the fifth is narrower than 11\19 = ~695¢, B♯ is lower in pitch than C♭ and the kleisma becomes a descending double-diminished 2nd. In 19edo, the tempering out of the kleisma means that {{nowrap|B♯ {{=}} C♭}}.

*[[12edo]] (chromatic semitone is equal to kleisma, diesis is tempered out)

*[[19edo~~]] (chromatic semitone is equal to diesis~~, ~~kleisma is tempered~~ out)

*[[26edo]] (chromatic semitone is smaller than diesis, kleisma is negative)

*[[31edo]] (diesis is equal to kleisma)

*[[43edo]] and [[55edo]] (diesis is smaller than kleisma)

*[[50edo]] (diesis is larger than kleisma)

~~There are~~ of ~~course notational equivalences.~~

{| class="wikitable center-all"

|+ style="font-size: 105%;" | Various EDOs that support meantone

|-

! rowspan="3" | [[EDO]]

! rowspan="3" | Approximate [[81/80|syntonic comma]] fraction

! colspan="4" | Steps

! rowspan="3" | Relative sizes of the chromatic semitone, diesis, and kleisma

|-

! style="width: 90px;" | Chromatic semitone

! style="width: 90px;" | Diatonic semitone

! Diesis

! Kleisma

|-

! A1

! m2

! d2

! −dd2

|-

| [[12edo]]

| {{frac|11}} comma

| 1

| 0

| 1

| Chromatic semitone is equal to kleisma, diesis is tempered out

|-

| [[19edo]]

| {{frac|3}} comma

| 1

| 2

| 1

| 0

| Chromatic semitone is equal to diesis, kleisma is tempered out

|-

| [[26edo]]

|

| 1

| 3

| 2

| −1

| Chromatic semitone is smaller than diesis, kleisma is negative

|-

| [[31edo]]

| {{frac|4}} comma

| 2

| 3

| 1

| Diesis is equal to kleisma

|-

| [[33edo#Theory|33c-edo]]

| {{frac|2}} comma

| 1

| 4

| 3

| −2

| Chromatic semitone is smaller than diesis, kleisma is negative

|-

| [[43edo]]

| {{frac|5}} comma

| 3

| 4

| 1

| 2

| rowspan="2" | Diesis is smaller than kleisma

|-

| [[55edo]]

| {{frac|6}} comma

| 4

| 5

| 1

| 3

|-

| [[50edo]]

| {{frac|2|7}} comma

| 3

| 5

| 2

| 1

| Diesis is larger than kleisma

|}

*B♯↑ and B𝄪− are equal to C

In 33c-edo, 5/4 is mapped to {{nowrap|10\33 {{=}} 364{{c}}}} instead of {{nowrap|11\33 {{=}} 400{{c}}}}.

*C+↑ is ~~equal to C♯ (because the two semisharps add up)~~

*D𝄫v and D♭♭♭− are equal to C

~~Assuming~~ [[~~septimal meantone~~]]~~, the meantone diesis can be considered~~ to ~~be [[36/35]], [[50/49]], [[64/63]], [[128/125]], or [[648/625]], while the meantone kleisma is [[49/48]], [[245/243]], [[3125/3072]], or [[15625/15552]]. An octave is made of 19 dieses and 12 kleisma.~~

[[9-odd-limit]] intervals and their notation relative to C:

~~[[9–odd–limit]] intervals and their notation relative to~~ C:

{| class="wikitable center-all"

|-

! Note

| C

| style="border-left: 5px solid black;" | G

| style="border-right: 5px solid black;" | F

| style="border-left: 5px solid black;" | E

| A

| E♭

| A♭

| style="border-left: 5px solid black;" | A♯ B♭↓

| D♯ E♭↓

| F♯ G♭↓

| E D↓

| B A↓

| G♭ F♯↓

| colspan="2" style="border-left: 5px solid black;" | D

| colspan="2" | B♭

| F♭ E↑

| G♯ A♭↓

|-

! Just interval

| {{sfrac|1|1}}

| style="border-left: 5px solid black;" | {{sfrac|3|2}}

| {{sfrac|4|3}}

| style="border-left: 5px solid black;" | {{sfrac|5|4}}

| {{sfrac|5|3}}

| {{sfrac|6|5}}

| style="border-right: 5px solid black;" | {{sfrac|8|5}}

| style="border-left: 5px solid black;" | {{sfrac|7|4}}

| {{sfrac|7|6}}

| {{sfrac|7|5}}

| {{sfrac|8|7}}

| {{sfrac|12|7}}

| {{sfrac|10|7}}

| style="border-left: 5px solid black;" | {{sfrac|9|8}}

| {{sfrac|10|9}}

| {{sfrac|9|5}}

| {{sfrac|16|9}}

| {{sfrac|9|7}}

| {{sfrac|14|9}}

|}

~~<pre>~~

Two dieses or two kleismas cannot be stacked to produce a chromatic semitone except in 31edo, and notation for [[11-limit]] and [[13-limit]] intervals (intervals involving the [[11/8|11th harmonic]] and [[13/8|13th harmonic]]) can vary.

1/~~1 — C~~

~~3/2 — G~~

== True half-sharps and half-flats ==

~~4/3 — F~~

If sharps raise by an even number of edosteps, such as [[24-tone equal temperament]] (quarter tones) and [[31-tone equal temperament]] (approximately extended [[quarter-comma meantone]]), they (along with flats) can be split in half. Thus, some notes can be notated using semisharps and semiflats, or with [[ups and downs notation|ups and downs]].

~~5/4 — E~~

For example, in 31 equal, the chromatic scale becomes:

~~5/3 — A~~

~~8/5 — A♭~~

~~6/5 — E♭~~

~~7/4 — A♯~~, ~~or B♭↓~~

{{dash|C, D𝄫, C♯, D♭, C𝄪, D, E𝄫, D♯, E♭, D𝄪, E, F♭, E♯, F, G𝄫, F♯, G♭, F𝄪, G, A𝄫, G♯, A♭, G𝄪, A, B𝄫, A♯, B♭, A𝄪, B, C♭, B♯, C|hair|long}}

~~7/6 — D♯~~, ~~or E♭↓~~

~~7/5 — F♯~~, ~~or G♭↓~~

~~8/7 — E𝄫~~, ~~or D↑~~

~~12/7 — B𝄫~~, ~~or A↑~~

~~10/7 — G♭~~, ~~or F♯↑~~

~~9/8 — D~~

Note that the base note letters alternate.

~~9/5 — B♭~~

~~9/7 — F♭, or E↑~~

~~16/9 — B♭~~

~~10/9 — D~~

~~14/9 — G♯, or A♭↓~~

~~</pre>~~

~~Two dieses or two kleismas cannot be stacked to produce a chromatic semitone. 11–limit~~ and ~~13–limit notation~~ can ~~vary (see [[meantone vs meanpop]]).~~

Using semisharps and semiflats, this can be re-written as:

~~Extended~~ meantone ~~notation was created as~~ a ~~way to notate [[43edo]]~~ with ~~only~~ a ~~base letter with one symbol~~.

{{dash|C, C{{demisharp2}}, C♯, D♭, D{{demiflat2}}, D, D{{demisharp2}}, D♯, E♭, E{{demiflat2}}, E, E{{demisharp2}}, F{{demiflat2}}, F, F{{demisharp2}}, F♯, G♭, G{{demiflat2}}, G, G{{demisharp2}}, G♯, A♭, A{{demiflat2}}, A, A{{demisharp2}}, A♯, B♭, B{{demiflat2}}, B, B{{demisharp2}}, C{{demiflat2}}, C|hair|long}}

<!--

If true half-sharps and true half-flats are desired, which exactly bisect the chromatic semitone, the meantone fifth is split in half. This creates a new tuning system consisting of a two-dimensional lattice generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class.

~~== True~~ half-sharps and half-flats ==

While real-world Arabic and Persian music often involve many very fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and are very difficult to notate exactly as a result, they are commonly notated using half-sharps and half-flats. If we take these to be exactly equal to one-half of a chromatic semitone, then mathematically, this notation system results in the aforementioned two-dimensional lattice. If notes separated by a diminished second, such as C♯ and D♭, are made enharmonically equivalent, this lattice degenerates further into 24edo, which is often suggested as a simplified framework and tuning system for notating and playing Arabic and Persian music. But the usual written notation typically lets musicians and composers treat adjacent sharps and flats as two distinct entities if it is decided that they should be different.

If true ~~half~~-~~sharps and half-flats are desired~~, ~~which exactly bisect~~ the ~~chromatic semitone~~, ~~this mathematically implies~~ that the ~~meantone~~ fifth ~~is split~~ in ~~half~~. ~~This creates~~ a ~~2D new tuning system which is generated~~ by a ~~chain~~ of ~~neutral thirds~~, ~~with meantone existing as every other note in the generator chain. This adds true~~ half-~~sharps~~ and ~~half-flats, and creates a "neutral" version of each interval class~~.

The chain-of-neutral-thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any just interval mapping to it in the 7-limit. But, if we bring in the 11th harmonic, and decide that there should only be a single neutral second (resulting in 11/10 and 12/11, the greater and lesser neutral seconds, both being mapped to a single equally-tempered interval), we obtain [[mohajira]], a very accurate 11-limit temperament. The neutral third approximates 11/9, and two of them make a perfect fifth, resulting in [[243/242]] being tempered out. Furthermore, flattening a minor third by a half-flat results in an approximation of 7/6, while sharpening a major third by a half-sharp gives an approximation of 9/7. Mohajira is supported very well by 24edo and 31edo.

~~While Middle Eastern maqam music is far too complex in real life to be represented by either of these temperaments~~, ~~one can certainly try, see [[Maqamat in maqamic temperament]]~~)~~, it is commonly notated using half~~-~~sharps and~~-half flats. If we take these to be exactly equal to 1/2 of a chromatic semitone, then mathematically, this notation system results in a 2D lattice that is generated by a neutral third and an octave. If we furthermore decide that C# and Db are enharmonically equal, this 2D lattice collapses further to the 1D lattice of [[24edo]], which has sometimes been suggested as a simplified framework for maqam music. But the usual written notation typically lets you notate them as two distinct entities if you want, so if we instead decide to leave them unequal, we get the 2D lattice above.

(This section needs considerable re-wording, I'm commenting it out for now) - ArrowHead294

-->

The chain-of-neutral thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any JI interval mapping to it in the 7-limit. But, if we go to the 11-limit, and add 121/120 to the kernel, we obtain [[mohajira]], an exceptionally good 11-limit temperament. The neutral third becomes equal to 11/9, and two of them make 3/2. Furthermore, if you take a minor third and ''flatten'' it by a half-flat, you obtain a good representation of 7/6. Conversely if you take a major third and ''sharpen'' it by a half-sharp, you obtain a good representation for 9/7. [[31edo]] is a very good tuning for mohajira.

Although mohajira may not be a great tuning to reflect the way maqam music is played in practice, which often uses multiple unequal neutral thirds and exhibits significant regional variations, it is a highly interesting regular temperament of its own merit, and deserves further study.

[[Category:Meantone]]

[[Category:Notation]]

@@ Line 1: / Line 1: @@
-:''This page is currently reworked at [[User:PiotrGrochowski/Extended meantone notation]], see also [[{{TALKPAGENAME}} #under construction|Talk:Extended meantone notation #under construction]]''.
+[[Meantone]] can be notated with a [[chain of fifths]] consisting of the 7 natural notes along with sharps and flats:
-== Extending the chain of fifths ==
+... {{dash|F&#x1D12B;, C&#x1D12B;, G&#x1D12B;, D&#x1D12B;, A&#x1D12B;, E&#x1D12B;, B&#x1D12B;, F&#x266D;, C&#x266D;, G&#x266D;, D&#x266D;, A&#x266D;, E&#x266D;, B&#x266D;, F, C, G, D, A, E, B, F&#x266F;, C&#x266F;, G&#x266F;, D&#x266F;, A&#x266F;, E&#x266F;, B&#x266F;, F&#x1D12A;, C&#x1D12A;, G&#x1D12A;, D&#x1D12A;, A&#x1D12A;, E&#x1D12A;, B&#x1D12A;|hair|long}} ...
-[[Circle-of-fifths notation|Standard meantone notation]] uses 7 base note letters, plus sharps and flats.
-... F𝄫 C𝄫 G𝄫 D𝄫 A𝄫 E𝄫 B𝄫 F♭ C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯ F𝄪 C𝄪 G𝄪 D𝄪 A𝄪 E𝄪 B𝄪 ...
+The chain is theoretically infinite, and C&#x266F; and D&#x266D; are not (in general) equivalent. When meantone is extended beyond 12 notes, it may require double-sharps, double-flats, and beyond. To avoid this, two new accidental pairs are introduced that raise/lower by the [[diesis]] and the [[kleisma]].
-However, when transferred onto a 31edo scale, it looks like this:
+{| class="wikitable center-all"
+|-
+! colspan="2" | Symbol
+! colspan="2" rowspan="2" | Interval
+! rowspan="2" | Examples
+! rowspan="2" | [[Fifthspan]]
+|-
+! Raise
+! Lower
+|-
+| &#x266F;
+| &#x266D;
+| Chromatic<br>semitone
+| Augmented<br>unison (A1)
+| C&ndash;C&#x266F;<br>E&#x266D;&ndash;E
+| +7
+|-
+| &uarr;
+| &darr;
+| Diesis
+| Diminished 2nd (d2)
+| C&#x266F;&ndash;D&#x266D;<br>D&#x266F;&ndash;E
+| &minus;12
+|-
+| +
+| &minus;
+| Kleisma
+| [[Negative interval|Negative]] double-<br>diminished 2nd (-dd2)
+| C&#x266D;&#x200A;&ndash;&#x200A;B&#x266F;<br>F&#x266D;&#x200A;&ndash;&#x200A;E&#x266F;
+| +19
+|}
-C D𝄫 C♯ D♭ C𝄪 D E𝄫 D♯ E♭ D𝄪 E F♭ E♯ F G𝄫 F♯ G♭ F𝄪 G A𝄫 G♯ A♭ G𝄪 A B𝄫 A♯ B♭ A𝄪 B C♭ B♯ C
+Because {{nowrap|19 &minus; 12 {{=}} 7}}, {{nowrap|d2 + &minus;dd2 {{=}} A1}}, and a diesis plus a kleisma equals a chromatic semitone.
-Note that the base note letters alternate.
+An octave is made up of:
-The 31edo sharp can be split in half, so in 31edo this is solved by semisharps and semiflats, sometimes notated with [[ups and Downs Notation|ups and downs]].
+* 7 diatonic semitones and 5 chromatic semitones {{nowrap|{{=}} 7 m2 + 5 A1}} {{nowrap|{{=}} 12 steps}}
+* 12 chromatic semitones and 7 dieses {{nowrap|{{=}} 12 A1 + 7 d2}} {{nowrap|{{=}} 19 steps}}
+* 19 dieses and 12 kleismas {{nowrap|{{=}} 19 d2 + 12 &minus;dd2}} {{nowrap|{{=}} 31 steps}}
-The meantone circle of fifths however, has no single semisharp/semiflat. In extended meantone notation, a sharp is split into 2 different parts that can be added to produce a sharp:
+The diesis represents the [[just intonation|just]] intervals [[128/125]] and [[648/625]] among others, while the meantone kleisma represents [[15625/15552]] = [-6 -5 6⟩ and [[3125/3072]] = [-10 -1 5⟩ among others. In [[septimal meantone]], where 7/4 is an augmented sixth, the diesis also represents [[36/35]], [[50/49]], and [[64/63]], while the kleisma also represents [[49/48]] and [[245/243]].
-<pre>
+The [[Enharmonic unison|enharmonic unisons]] &darr;d2 and &minus;&darr;A1 create various notational equivalences:
-♯ — sharpen by meantone chromatic semitone, 7 fifths up
-♭ — flatten by meantone chromatic semitone, 7 fifths down
-↑ — sharpen by meantone diesis, 12 fifths down
-↓ — flatten by meantone diesis, 12 fifths up
-+ — sharpen by meantone kleisma, 19 fifths up
-− — flatten by meantone kleisma, 19 fifths down
-</pre>
-A diesis plus a kleisma, added together, equals a meantone chromatic semitone. Note that in most meantone tunings, the diesis and kleisma are roughly a quarter tone.
+* B&#x266F;&uarr; and B&#x1D12A;&minus; are equal to C
+* C+&uarr; is equal to C&#x266F; (because the two semisharps add up)
+* D&#x1D12B;&darr; and D&#x266D;&#x266D;&#x266D;&minus; are equal to C
-Unlike a single semisharp/semiflat, this can be generalized to other meantone tunings:
+If the fifth is wider than {{nowrap|7\12 {{=}} 700{{c}}}}, C&#x266F; is higher in pitch than D&#x266D; and the diesis becomes a descending pythagorean comma. In 12edo, the tempering out of the diesis means that  {{nowrap|C&#x266F; {{=}} D&#x266D;}}.
-*[[7edo]] (chromatic semitone is tempered out, diesis is positive, and kleisma is negative)
+If the fifth is narrower than 11\19 = ~695¢, B&#x266F; is lower in pitch than C&#x266D; and the kleisma becomes a descending double-diminished 2nd. In 19edo, the tempering out of the kleisma means that {{nowrap|B&#x266F; {{=}} C&#x266D;}}.
-*[[12edo]] (chromatic semitone is equal to kleisma, diesis is tempered out)
-*[[19edo]] (chromatic semitone is equal to diesis, kleisma is tempered out)
-*[[26edo]] (chromatic semitone is smaller than diesis, kleisma is negative)
-*[[31edo]] (diesis is equal to kleisma)
-*[[43edo]] and [[55edo]] (diesis is smaller than kleisma)
-*[[50edo]] (diesis is larger than kleisma)
-There are of course notational equivalences.
+{| class="wikitable center-all"
+|+ style="font-size: 105%;" | Various EDOs that support meantone
+|-
+! rowspan="3" | [[EDO]]
+! rowspan="3" | Approximate<br />[[81/80|syntonic<br />comma]]<br />fraction
+! colspan="4" | Steps
+! rowspan="3" | Relative sizes of the<br />chromatic semitone,<br />diesis, and kleisma
+|-
+! style="width: 90px;" | Chromatic<br>semitone
+! style="width: 90px;" | Diatonic<br>semitone
+! Diesis
+! Kleisma
+|-
+! A1
+! m2
+! d2
+! &minus;dd2
+|-
+| [[12edo]]
+| {{frac|11}}&nbsp;comma
+| 1
+| 1
+| 0
+| 1
+| Chromatic semitone is equal to kleisma,<br />diesis is tempered out
+|-
+| [[19edo]]
+| {{frac|3}}&nbsp;comma
+| 1
+| 2
+| 1
+| 0
+| Chromatic semitone is equal to diesis,<br />kleisma is tempered out
+|-
+| [[26edo]]
+|
+| 1
+| 3
+| 2
+| &minus;1
+| Chromatic semitone is smaller than diesis,<br />kleisma is negative
+|-
+| [[31edo]]
+| {{frac|4}}&nbsp;comma
+| 2
+| 3
+| 1
+| 1
+| Diesis is equal to kleisma
+|-
+| [[33edo#Theory|33c-edo]]
+| {{frac|2}}&nbsp;comma
+| 1
+| 4
+| 3
+| &minus;2
+| Chromatic semitone is smaller than diesis,<br />kleisma is negative
+|-
+| [[43edo]]
+| {{frac|5}}&nbsp;comma
+| 3
+| 4
+| 1
+| 2
+| rowspan="2" | Diesis is smaller than kleisma
+|-
+| [[55edo]]
+| {{frac|6}}&nbsp;comma
+| 4
+| 5
+| 1
+| 3
+|-
+| [[50edo]]
+| {{frac|2|7}}&nbsp;comma
+| 3
+| 5
+| 2
+| 1
+| Diesis is larger than kleisma
+|}
-*B♯↑ and B𝄪− are equal to C
+In 33c-edo, 5/4 is mapped to {{nowrap|10\33 {{=}} 364{{c}}}} instead of {{nowrap|11\33 {{=}} 400{{c}}}}.
-*C+↑ is equal to C♯ (because the two semisharps add up)
-*D𝄫v and D♭♭♭− are equal to C
-Assuming [[septimal meantone]], the meantone diesis can be considered to be [[36/35]], [[50/49]], [[64/63]], [[128/125]], or [[648/625]], while the meantone kleisma is [[49/48]], [[245/243]], [[3125/3072]], or [[15625/15552]]. An octave is made of 19 dieses and 12 kleisma.
+[[9-odd-limit]] intervals and their notation relative to C:
-[[9–odd–limit]] intervals and their notation relative to C:
+{| class="wikitable center-all"
+|-
+! Note
+| C
+| style="border-left: 5px solid black;" | G
+| style="border-right: 5px solid black;" | F
+| style="border-left: 5px solid black;" | E
+| A
+| E&#x266D;
+| A&#x266D;
+| style="border-left: 5px solid black;" | A&#x266F; <br />B&#x266D;&darr;
+| D&#x266F;<br />E&#x266D;&darr;
+| F&#x266F;<br />G&#x266D;&darr;
+| E<br>D&darr;
+| B<br>A&darr;
+| G&#x266D;<br />F&#x266F;&darr;
+| colspan="2" style="border-left: 5px solid black;" | D
+| colspan="2" | B&#x266D;
+| F&#x266D;<br />E&uarr;
+| G&#x266F; <br />A&#x266D;&darr;
+|-
+! Just interval
+| {{sfrac|1|1}}
+| style="border-left: 5px solid black;" | {{sfrac|3|2}}
+| {{sfrac|4|3}}
+| style="border-left: 5px solid black;" | {{sfrac|5|4}}
+| {{sfrac|5|3}}
+| {{sfrac|6|5}}
+| style="border-right: 5px solid black;" | {{sfrac|8|5}}
+| style="border-left: 5px solid black;" | {{sfrac|7|4}}
+| {{sfrac|7|6}}
+| {{sfrac|7|5}}
+| {{sfrac|8|7}}
+| {{sfrac|12|7}}
+| {{sfrac|10|7}}
+| style="border-left: 5px solid black;" | {{sfrac|9|8}}
+| {{sfrac|10|9}}
+| {{sfrac|9|5}}
+| {{sfrac|16|9}}
+| {{sfrac|9|7}}
+| {{sfrac|14|9}}
+|}
-<pre>
+Two dieses or two kleismas cannot be stacked to produce a chromatic semitone except in 31edo, and notation for [[11-limit]] and [[13-limit]] intervals (intervals involving the [[11/8|11th harmonic]] and [[13/8|13th harmonic]]) can vary.
-/1 — C
-/2 — G
+== True half-sharps and half-flats ==
-/3 — F
+If sharps raise by an even number of edosteps, such as [[24-tone equal temperament]] (quarter tones) and [[31-tone equal temperament]] (approximately extended [[quarter-comma meantone]]), they (along with flats) can be split in half. Thus, some notes can be notated using semisharps and semiflats, or with [[ups and downs notation|ups and downs]].
-/4 — E
+For example, in 31 equal, the chromatic scale becomes:
-/3 — A
-/5 — A♭
-/5 — E♭
-/4 — A♯, or B♭↓
+{{dash|C, D&#x1D12B;, C&#x266F;, D&#x266D;, C&#x1D12A;, D, E&#x1D12B;, D&#x266F;, E&#x266D;, D&#x1D12A;, E, F&#x266D;, E&#x266F;, F, G&#x1D12B;, F&#x266F;, G&#x266D;, F&#x1D12A;, G, A&#x1D12B;, G&#x266F;, A&#x266D;, G&#x1D12A;, A, B&#x1D12B;, A&#x266F;, B&#x266D;, A&#x1D12A;, B, C&#x266D;, B&#x266F;, C|hair|long}}
-/6 — D♯, or E♭↓
-/5 — F♯, or G♭↓
-/7 — E𝄫, or D↑
-/7 — B𝄫, or A↑
-/7 — G♭, or F♯↑
-/8 — D
+Note that the base note letters alternate.
-/5 — B♭
-/7 — F♭, or E↑
-/9 — B♭
-/9 — D
-/9 — G♯, or A♭↓
-</pre>
-Two dieses or two kleismas cannot be stacked to produce a chromatic semitone. 11–limit and 13–limit notation can vary (see [[meantone vs meanpop]]).
+Using semisharps and semiflats, this can be re-written as:
-Extended meantone notation was created as a way to notate [[43edo]] with only a base letter with one symbol.
+{{dash|C, C{{demisharp2}}, C&#x266F;, D&#x266D;, D{{demiflat2}}, D, D{{demisharp2}}, D&#x266F;, E&#x266D;, E{{demiflat2}}, E, E{{demisharp2}}, F{{demiflat2}}, F, F{{demisharp2}}, F&#x266F;, G&#x266D;, G{{demiflat2}}, G, G{{demisharp2}}, G&#x266F;, A&#x266D;, A{{demiflat2}}, A, A{{demisharp2}}, A&#x266F;, B&#x266D;, B{{demiflat2}}, B, B{{demisharp2}}, C{{demiflat2}}, C|hair|long}}
+<!--
+If true half-sharps and true half-flats are desired, which exactly bisect the chromatic semitone, the meantone fifth is split in half. This creates a new tuning system consisting of a two-dimensional lattice generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class.
-== True half-sharps and half-flats ==
+While real-world Arabic and Persian music often involve many very fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and are very difficult to notate exactly as a result, they are commonly notated using half-sharps and half-flats. If we take these to be exactly equal to one-half of a chromatic semitone, then mathematically, this notation system results in the aforementioned two-dimensional lattice. If notes separated by a diminished second, such as C&#x266F; and D&#x266D;, are made enharmonically equivalent, this lattice degenerates further into 24edo, which is often suggested as a simplified framework and tuning system for notating and playing Arabic and Persian music. But the usual written notation typically lets musicians and composers treat adjacent sharps and flats as two distinct entities if it is decided that they should be different.
-If true half-sharps and half-flats are desired, which exactly bisect the chromatic semitone, this mathematically implies that the meantone fifth is split in half. This creates a 2D new tuning system which is generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class.
+The chain-of-neutral-thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any just interval mapping to it in the 7-limit. But, if we bring in the 11th harmonic, and decide that there should only be a single neutral second (resulting in 11/10 and 12/11, the greater and lesser neutral seconds, both being mapped to a single equally-tempered interval), we obtain [[mohajira]], a very accurate 11-limit temperament. The neutral third approximates 11/9, and two of them make a perfect fifth, resulting in [[243/242]] being tempered out. Furthermore, flattening a minor third by a half-flat results in an approximation of 7/6, while sharpening a major third by a half-sharp gives an approximation of 9/7. Mohajira is supported very well by 24edo and 31edo.
-While Middle Eastern maqam music is far too complex in real life to be represented by either of these temperaments, one can certainly try, see [[Maqamat in maqamic temperament]]), it is commonly notated using half-sharps and-half flats. If we take these to be exactly equal to 1/2 of a chromatic semitone, then mathematically, this notation system results in a 2D lattice that is generated by a neutral third and an octave. If we furthermore decide that C# and Db are enharmonically equal, this 2D lattice collapses further to the 1D lattice of [[24edo]], which has sometimes been suggested as a simplified framework for maqam music. But the usual written notation typically lets you notate them as two distinct entities if you want, so if we instead decide to leave them unequal, we get the 2D lattice above.
+(This section needs considerable re-wording, I'm commenting it out for now) - ArrowHead294
+-->
-The chain-of-neutral thirds tuning system is not a true "temperament," because it is [[contorted]]: the neutral third does not have any JI interval mapping to it in the 7-limit. But, if we go to the 11-limit, and add 121/120 to the kernel, we obtain [[mohajira]], an exceptionally good 11-limit temperament. The neutral third becomes equal to 11/9, and two of them make 3/2. Furthermore, if you take a minor third and ''flatten'' it by a half-flat, you obtain a good representation of 7/6. Conversely if you take a major third and ''sharpen'' it by a half-sharp, you obtain a good representation for 9/7. [[31edo]] is a very good tuning for mohajira.
+{{Navbox notation}}
-Although mohajira may not be a great tuning to reflect the way maqam music is played in practice, which often uses multiple unequal neutral thirds and exhibits significant regional variations, it is a highly interesting regular temperament of its own merit, and deserves further study.
 [[Category:Meantone]]
 [[Category:Notation]]