Extended meantone notation: Difference between revisions

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~~Most musicians are familiar~~ with ~~the~~ [[~~circle~~ of fifths]]~~. This is a way of organizing and showing relationships between pitches as a sequence~~ of ~~[[3/2|fifths]], and applies to any tuning system that can be generated by fifths and octaves. The generalized chain of fifths involves~~ the 7 ~~base note letters of the C major scale,~~ along with sharps~~, double-sharps, flats,~~ and ~~double-~~flats ~~(and beyond)~~:

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

... {{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}} ...

~~In a general meantone tuning, a sharp~~ is ~~split into 2 different parts~~, ~~the diesis and the kleisma.~~

The chain is theoretically infinite, and C♯ and D♭ are not 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]].

~~== Generalizing accidentals ==~~

Most musicians are familiar with single and double sharps and flats—these denote raising and lowering by one or two chromatic semitones, respectively. In a general meantone tuning, there are two additional intervals: the diesis, which is the difference between adjacent accidentals (e.g. C♯–D♭ and ~~D♯–E♭),<ref group="note" name="diesis_note">Having~~ C♯ and D♭ ~~be enharmonically~~ equivalent is ~~what most musicians would expect, but this is only true in equal temperament tunings where the number of~~ notes ~~is a multiple of 12. In most tuning systems~~, ~~there are no enharmonic equivalents involving only~~ sharps and ~~flats~~.</~~ref>~~ and the kleisma~~, which is the amount by which B♯ exceeds C♭ and E♯ exceeds F♭ (that is, C♭ – B♯ and F♭ – E♯)~~.

{| class="wikitable center-all"

|-

! colspan="2" | Symbol

! rowspan="2" | Interval

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

! rowspan="2" | ~~Number of fifths~~

! rowspan="2" |Examples

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

|-

! Raise

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| ♯

| ♭

| Chromatic semitone

| Chromatic semitone

| 7

|Augmented unison (A1)

|C–C♯ E♭–E

| +7

|-

| ↑

| ↓

| Diesis

| 12

|Diminished 2nd (d2)

|C♯–D♭ D♯–E

| -12

|-

| +

| −

| Kleisma

| 19

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

|C♭ – B♯ F♭ – E♯

| +19

|}

A meantone chromatic semitone consists of one diesis and one kleisma. The diesis represents the [[just intonation|just]] intervals [[128/125]] and [[648/625]], while the meantone kleisma represents [[15625/15552]] or [[3125/3072]]. 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]]~~.

Because 19 - 12 = 7, d2 + -dd2 = A1, and a diesis plus a kleisma equals a chromatic semitone.

An octave is made up of:

~~An octave is made of~~ 19 dieses and 12 kleismas.

* 7 diatonic semitones and 5 chromatic semitones = 7 m2 + 5 A1 = 12 steps

* 12 chromatic semitones and 7 dieses = 12 A1 + 7 d2 = 19 steps

* 19 dieses and 12 kleismas = 19 d2 + 12 -dd2 = 31 steps

~~Unlike~~ semisharps and ~~semiflats~~, the diesis and kleisma ~~can be generalized to other tunings:~~

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]].

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

* 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

If the fifth is wider than 7\12 = 700¢, 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♭}}.

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♭}}.

{| class="wikitable center-all"

|+Various EDOs that support meantone

|-

! rowspan="2" | ~~Notes per octave~~

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

! rowspan="2" | Approximate [[81/80|syntonic comma]] fraction

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

! colspan="4" | Steps

! rowspan="2" | ~~Explanation~~

! rowspan="3" | Relative sizes of the

chromatic semitone,

diesis and kleisma

|-

! style="width: 90px;" | Chromatic semitone~~ (e.g. C–C♯)~~

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

! style="width: 90px;" | Diatonic semitone~~ (e.g. C–D♭)~~

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

! Diesis

! Kleisma

|-

~~| [[7edo]]~~

!A1

|

!m2

~~| 0~~

!d2

~~| 1~~

!-dd2

~~| 1~~

~~| −1~~

| Chromatic semitone is tempered out<ref group="note" name="chroma_note">In 7-tone equal temperament, the tempering out of the chromatic semitone means that sharps and flats are redundant (in the sense that they cannot alter the pitch).</ref>, diesis is positive, and kleisma is negative<ref group="note" name="kleisma_note">A negative kleisma means that B♯ is lower in pitch than C♭ and E♯ is lower in pitch than F♭. Conversely, a positive kleisma means B♯ sits higher than C♭ and E♯ sits higher than F♭. In 19-~~tone equal temperament, the tempering out of the kleisma means that {{nowrap|B♯ {{=}} C♭}} and {{nowrap|E♯ {{=}} F♭}}.</ref>~~

|-

| [[12edo~~|12edo (standard tuning)~~]]

| [[12edo]]

| {{frac|11}} comma

| 1

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| 0

| 1

| Chromatic semitone is equal to kleisma, diesis is tempered out~~<ref group="note" name="diesis_note" />~~

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

|-

| [[19edo]]

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| 1

| 0

| Chromatic semitone is equal to diesis, kleisma is tempered out~~<ref group="note" name="kleisma_note" />~~

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

|-

| [[26edo]]

|

| 1

| 3

| 2

| −1

~~| rowspan="2"~~ | Chromatic semitone is smaller than diesis, kleisma is negative~~<ref group="note" name="kleisma_note" />~~

| Chromatic semitone is smaller than diesis, kleisma is negative

|-

|[[31edo]]

|{{frac|4}} comma

|2

|3

|1

|Diesis is equal to kleisma

|-

| [[33edo#Theory|~~33edo~~]]~~ (c mapping)~~

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

| {{frac|2}} comma

| 1

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| 3

| −2

|-

|Chromatic semitone is smaller than diesis, kleisma is negative

~~| [[31edo]]~~

~~| {{frac|4}} comma~~

~~| 2~~

~~| 3~~

~~| 1~~

~~| Diesis~~ is ~~equal to~~ kleisma

|-

| [[43edo]]

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|}

~~There are of course notational equivalences:~~

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

* 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

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

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| E♭

| A♭

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

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

| D♯ E♭↓

| D♯ E♭↓

| F♯ G♭↓

| F♯ G♭↓

| E~~𝄫~~ D↓

| E D↓

| B~~𝄫~~ A↓

| B A↓

| G♭ F♯↓

| G♭ F♯↓

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

| colspan="2" | B♭

| F♭ E↑

| G♯ A♭↓

| G♯ A♭↓

|-

! Just interval

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

If sharps raise by an even number of ~~steps~~, such as [[24-tone equal temperament]] (quarter tones) and [[31-tone equal temperament]] (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]].

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]].

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

Line 197:

Line 212:

-->

~~== Notes ==~~

~~<references group="note" />~~

@@ Line 1: / Line 1: @@
-Most musicians are familiar with the [[circle of fifths]]. This is a way of organizing and showing relationships between pitches as a sequence of [[3/2|fifths]], and applies to any tuning system that can be generated by fifths and octaves. The generalized chain of fifths involves the 7 base note letters of the C major scale, along with sharps, double-sharps, flats, and double-flats (and beyond):
+Meantone can be notated with a [[chain of fifths]] consisting of the 7 natural notes along with sharps and flats:
 ... {{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}} ...
-In a general meantone tuning, a sharp is split into 2 different parts, the diesis and the kleisma.
+The chain is theoretically infinite, and C&#x266F; and D&#x266D; are not 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]].
-== Generalizing accidentals ==
-Most musicians are familiar with single and double sharps and flats&mdash;these denote raising and lowering by one or two chromatic semitones, respectively. In a general meantone tuning, there are two additional intervals: the diesis, which is the difference between adjacent accidentals (e.g. C&#x266F;&ndash;D&#x266D; and D&#x266F;&ndash;E&#x266D;),<ref group="note" name="diesis_note">Having C&#x266F; and D&#x266D; be enharmonically equivalent is what most musicians would expect, but this is only true in equal temperament tunings where the number of notes is a multiple of 12. In most tuning systems, there are no enharmonic equivalents involving only sharps and flats.</ref> and the kleisma, which is the amount by which B&#x266F; exceeds C&#x266D; and E&#x266F; exceeds F&#x266D; (that is, C&#x266D;&#x200A;&ndash;&#x200A;B&#x266F; and F&#x266D;&#x200A;&ndash;&#x200A;E&#x266F;).
 {| class="wikitable center-all"
 |-
 ! colspan="2" | Symbol
-! rowspan="2" | Interval
+! colspan="2" rowspan="2" | Interval
-! rowspan="2" | Number of<br>fifths
+! rowspan="2" |Examples
+! rowspan="2" | [[Fifthspan]]
 |-
 ! Raise
@@ Line 19: / Line 16: @@
 | &#x266F;
 | &#x266D;
-| Chromatic semitone
+| Chromatic<br>semitone
-| 7
+|Augmented<br>unison (A1)
+|C&ndash;C&#x266F;<br>E&#x266D;&ndash;E
+| +7
 |-
 | &uarr;
 | &darr;
 | Diesis
-| 12
+|Diminished 2nd (d2)
+|C&#x266F;&ndash;D&#x266D;<br>D&#x266F;&ndash;E
+| -12
 |-
 | +
 | &minus;
 | Kleisma
-| 19
+|[[Negative interval|Negative]] double-<br>diminished 2nd (-dd2)
+|C&#x266D;&#x200A;&ndash;&#x200A;B&#x266F;<br>F&#x266D;&#x200A;&ndash;&#x200A;E&#x266F;
+| +19
 |}
-A meantone chromatic semitone consists of one diesis and one kleisma. The diesis represents the [[just intonation|just]] intervals [[128/125]] and [[648/625]], while the meantone kleisma represents [[15625/15552]] or [[3125/3072]]. 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]].
+Because 19 - 12 = 7,  d2 + -dd2 = A1, and a diesis plus a kleisma equals a chromatic semitone.
+An octave is made up of:
-An octave is made of 19 dieses and 12 kleismas.
+* 7 diatonic semitones and 5 chromatic semitones = 7 m2 + 5 A1 = 12 steps
+* 12 chromatic semitones and 7 dieses = 12 A1 + 7 d2 = 19 steps
+* 19 dieses and 12 kleismas = 19 d2 + 12 -dd2 = 31 steps
-Unlike semisharps and semiflats, the diesis and kleisma can be generalized to other tunings:
+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]].
+The [[Enharmonic unison|enharmonic unisons]] &darr;d2 and -&darr;A1 create various notational equivalences:
+* 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
+If the fifth is wider than 7\12 = 700¢, 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;}}.
+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;}}.
 {| class="wikitable center-all"
+|+Various EDOs that support meantone
 |-
-! rowspan="2" | Notes per octave
+! rowspan="3" | [[EDO]]
-! rowspan="2" | Approximate<br>[[81/80|syntonic<br>comma]]<br>fraction
+! rowspan="3" | Approximate<br>[[81/80|syntonic<br>comma]]<br>fraction
 ! colspan="4" | Steps
-! rowspan="2" | Explanation
+! rowspan="3" | Relative sizes of the
+chromatic semitone,
+diesis and kleisma
 |-
-! style="width: 90px;" | Chromatic<br>semitone<br>(e.g.&nbsp;C&ndash;C&#x266F;)
+! style="width: 90px;" | Chromatic<br>semitone
-! style="width: 90px;" | Diatonic<br>semitone<br>(e.g.&nbsp;C&ndash;D&#x266D;)
+! style="width: 90px;" | Diatonic<br>semitone
 ! Diesis
 ! Kleisma
 |-
-| [[7edo]]
+!A1
-|
+!m2
-| 0
+!d2
-| 1
+!-dd2
-| 1
-| &minus;1
-| Chromatic semitone is tempered out<ref group="note" name="chroma_note">In 7-tone equal temperament, the tempering out of the chromatic semitone means that sharps and flats are redundant (in the sense that they cannot alter the pitch).</ref>,<br>diesis is positive, and kleisma is negative<ref group="note" name="kleisma_note">A negative kleisma means that B&#x266F; is lower in pitch than C&#x266D; and E&#x266F; is lower in pitch than F&#x266D;. Conversely, a positive kleisma means B&#x266F; sits higher than C&#x266D; and E&#x266F; sits higher than F&#x266D;. In 19-tone equal temperament, the tempering out of the kleisma means that {{nowrap|B&#x266F; {{=}} C&#x266D;}} and {{nowrap|E&#x266F; {{=}} F&#x266D;}}.</ref>
 |-
-| [[12edo|12edo<br>(standard tuning)]]
+| [[12edo]]
 | {{frac|11}}&nbsp;comma
 | 1
@@ Line 65: / Line 83: @@
 | 0
 | 1
-| Chromatic semitone is equal to kleisma,<br>diesis is tempered out<ref group="note" name="diesis_note" />
+| Chromatic semitone is equal to kleisma,<br>diesis is tempered out
 |-
 | [[19edo]]
@@ Line 73: / Line 91: @@
 | 1
 | 0
-| Chromatic semitone is equal to diesis,<br>kleisma is tempered out<ref group="note" name="kleisma_note" />
+| Chromatic semitone is equal to diesis,<br>kleisma is tempered out
 |-
 | [[26edo]]
 |
 | 1
 | 3
 | 2
 | &minus;1
-| rowspan="2" | Chromatic semitone is smaller than diesis,<br>kleisma is negative<ref group="note" name="kleisma_note" />
+| 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|33edo]]<br>(c&nbsp;mapping)
+| [[33edo#Theory|33c-edo]]
 | {{frac|2}}&nbsp;comma
 | 1
@@ Line 89: / Line 115: @@
 | 3
 | &minus;2
-|-
+|Chromatic semitone is smaller than diesis,<br>kleisma is negative
-| [[31edo]]
-| {{frac|4}}&nbsp;comma
-| 2
-| 3
-| 1
-| 1
-| Diesis is equal to kleisma
 |-
 | [[43edo]]
@@ Line 122: / Line 141: @@
 |}
-There are of course notational equivalences:
+In 33c-edo, 5/4 is mapped to 10\33 = 364¢ instead of 11\33 = 400¢.
-* 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
 [[9-odd-limit]] intervals and their notation relative to C:
@@ Line 140: / Line 155: @@
 | E&#x266D;
 | A&#x266D;
-| style="border-left: 5px solid black;" | A&#x266F;<br>B&#x266D;&darr;
+| style="border-left: 5px solid black;" | A&#x266F; <br>B&#x266D;&darr;
-| D&#x266F;<br>E&#x266D;&darr;
+| D&#x266F; <br>E&#x266D;&darr;
-| F&#x266F;<br>G&#x266D;&darr;
+| F&#x266F;  <br>G&#x266D;&darr;
-| E&#x1D12B;<br>D&darr;
+| E<br>D&darr;
-| B&#x1D12B;<br>A&darr;
+| B<br>A&darr;
-| G&#x266D;<br>F&#x266F;&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;
+| G&#x266F; <br>A&#x266D;&darr;
 |-
 ! Just interval
@@ Line 176: / Line 191: @@
 == True half-sharps and half-flats ==
-If sharps raise by an even number of steps, such as [[24-tone equal temperament]] (quarter tones) and [[31-tone equal temperament]] (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]].
+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]].
 For example, in 31 equal, the chromatic scale becomes:
@@ Line 197: / Line 212: @@
 -->
-== Notes ==
-<references group="note" />
 {{Navbox notation}}