Extended meantone notation: Difference between revisions

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

[[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}} ...

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]].

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]].

{| class="wikitable center-all"

|-

! colspan="2" | Symbol

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

! rowspan="2" |Examples

! rowspan="2" | Examples

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

|-

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| ♭

| Chromatic semitone

|Augmented unison (A1)

| Augmented unison (A1)

|C–C♯ E♭–E

| C–C♯ E♭–E

| +7

|-

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| ↓

| Diesis

|Diminished 2nd (d2)

| Diminished 2nd (d2)

|C♯–D♭ D♯–E

| C♯–D♭ D♯–E

| -12

| −12

|-

| +

| −

| Kleisma

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

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

|C♭ – B♯ F♭ – E♯

| C♭ – B♯ F♭ – E♯

| +19

|}

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

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

An octave is made up of:

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

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

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

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

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

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

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:

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

* B♯↑ and B𝄪− are equal to C

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* 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 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♭}}.

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

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

|-

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

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

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

! colspan="4" | Steps

! rowspan="3" | Relative sizes of the

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

chromatic semitone,

diesis and kleisma

|-

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

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! Kleisma

|-

!A1

! A1

!m2

! m2

!d2

! d2

!-dd2

! −dd2

|-

| [[12edo]]

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| 0

| 1

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

| 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

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

|-

| [[26edo]]

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| 2

| −1

| Chromatic semitone is smaller than diesis, kleisma is negative

| Chromatic semitone is smaller than diesis, kleisma is negative

|-

|[[31edo]]

| [[31edo]]

|{{frac|4}} comma

| {{frac|4}} comma

|2

| 2

|3

| 3

|1

| 1

|1

| 1

|Diesis is equal to kleisma

| Diesis is equal to kleisma

|-

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

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| 3

| −2

|Chromatic semitone is smaller than diesis, kleisma is negative

| Chromatic semitone is smaller than diesis, kleisma is negative

|-

| [[43edo]]

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|}

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

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

| B A↓

| G♭ F♯↓

| G♭ F♯↓

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

| colspan="2" | B♭

| F♭ E↑

| F♭ E↑

| G♯ A♭↓

| G♯ A♭↓

|-

! Just interval

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

-->

@@ Line 1: / Line 1: @@
-Meantone can be notated with a [[chain of fifths]] consisting of the 7 natural notes along with sharps and flats:
+[[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}} ...
-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]].
+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]].
 {| class="wikitable center-all"
 |-
 ! colspan="2" | Symbol
 ! colspan="2" rowspan="2" | Interval
-! rowspan="2" |Examples
+! rowspan="2" | Examples
 ! rowspan="2" | [[Fifthspan]]
 |-
@@ Line 17: / Line 18: @@
 | &#x266D;
 | Chromatic<br>semitone
-|Augmented<br>unison (A1)
+| Augmented<br>unison (A1)
-|C&ndash;C&#x266F;<br>E&#x266D;&ndash;E
+| C&ndash;C&#x266F;<br>E&#x266D;&ndash;E
 | +7
 |-
@@ Line 24: / Line 25: @@
 | &darr;
 | Diesis
-|Diminished 2nd (d2)
+| Diminished 2nd (d2)
-|C&#x266F;&ndash;D&#x266D;<br>D&#x266F;&ndash;E
+| C&#x266F;&ndash;D&#x266D;<br>D&#x266F;&ndash;E
-| -12
+| &minus;12
 |-
 | +
 | &minus;
 | Kleisma
-|[[Negative interval|Negative]] double-<br>diminished 2nd (-dd2)
+| [[Negative interval|Negative]] double-<br>diminished 2nd (-dd2)
-|C&#x266D;&#x200A;&ndash;&#x200A;B&#x266F;<br>F&#x266D;&#x200A;&ndash;&#x200A;E&#x266F;
+| C&#x266D;&#x200A;&ndash;&#x200A;B&#x266F;<br>F&#x266D;&#x200A;&ndash;&#x200A;E&#x266F;
 | +19
 |}
-Because 19 - 12 = 7,  d2 + -dd2 = A1, and a diesis plus a kleisma equals a chromatic semitone.
+Because {{nowrap|19 &minus; 12 {{=}} 7}}, {{nowrap|d2 + &minus;dd2 {{=}} A1}}, and a diesis plus a kleisma equals a chromatic semitone.
 An octave is made up of:
-* 7 diatonic semitones and 5 chromatic semitones = 7 m2 + 5 A1 = 12 steps
+* 7 diatonic semitones and 5 chromatic semitones {{nowrap|{{=}} 7 m2 + 5 A1}} {{nowrap|{{=}} 12 steps}}
-* 12 chromatic semitones and 7 dieses = 12 A1 + 7 d2 = 19 steps
+* 12 chromatic semitones and 7 dieses {{nowrap|{{=}} 12 A1 + 7 d2}} {{nowrap|{{=}} 19 steps}}
-* 19 dieses and 12 kleismas = 19 d2 + 12 -dd2 = 31 steps
+* 19 dieses and 12 kleismas {{nowrap|{{=}} 19 d2 + 12 &minus;dd2}} {{nowrap|{{=}} 31 steps}}
 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:
+The [[Enharmonic unison|enharmonic unisons]] &darr;d2 and &minus;&darr;A1 create various notational equivalences:
 * B&#x266F;&uarr; and B&#x1D12A;&minus; are equal to C
@@ Line 52: / Line 53: @@
 * 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 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;}}.
 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
+|+ style="font-size: 105%;" | Various EDOs that support meantone
 |-
 ! rowspan="3" | [[EDO]]
-! rowspan="3" | Approximate<br>[[81/80|syntonic<br>comma]]<br>fraction
+! rowspan="3" | Approximate<br />[[81/80|syntonic<br />comma]]<br />fraction
 ! colspan="4" | Steps
-! rowspan="3" | Relative sizes of the
+! rowspan="3" | Relative sizes of the<br />chromatic semitone,<br />diesis, and kleisma
-chromatic semitone,
-diesis and kleisma
 |-
 ! style="width: 90px;" | Chromatic<br>semitone
@@ Line 72: / Line 70: @@
 ! Kleisma
 |-
-!A1
+! A1
-!m2
+! m2
-!d2
+! d2
-!-dd2
+! &minus;dd2
 |-
 | [[12edo]]
@@ Line 83: / Line 81: @@
 | 0
 | 1
-| Chromatic semitone is equal to kleisma,<br>diesis is tempered out
+| Chromatic semitone is equal to kleisma,<br />diesis is tempered out
 |-
 | [[19edo]]
@@ Line 91: / Line 89: @@
 | 1
 | 0
-| Chromatic semitone is equal to diesis,<br>kleisma is tempered out
+| Chromatic semitone is equal to diesis,<br />kleisma is tempered out
 |-
 | [[26edo]]
@@ Line 99: / Line 97: @@
 | 2
 | &minus;1
-| Chromatic semitone is smaller than diesis,<br>kleisma is negative
+| Chromatic semitone is smaller than diesis,<br />kleisma is negative
 |-
-|[[31edo]]
+| [[31edo]]
-|{{frac|4}}&nbsp;comma
+| {{frac|4}}&nbsp;comma
-|2
+| 2
-|3
+| 3
-|1
+| 1
-|1
+| 1
-|Diesis is equal to kleisma
+| Diesis is equal to kleisma
 |-
 | [[33edo#Theory|33c-edo]]
@@ Line 115: / Line 113: @@
 | 3
 | &minus;2
-|Chromatic semitone is smaller than diesis,<br>kleisma is negative
+| Chromatic semitone is smaller than diesis,<br />kleisma is negative
 |-
 | [[43edo]]
@@ Line 141: / Line 139: @@
 |}
-In 33c-edo, 5/4 is mapped to 10\33 = 364¢ instead of 11\33 = 400¢.
+In 33c-edo, 5/4 is mapped to {{nowrap|10\33 {{=}} 364{{c}}}} instead of {{nowrap|11\33 {{=}} 400{{c}}}}.
 [[9-odd-limit]] intervals and their notation relative to C:
@@ Line 155: / Line 153: @@
 | 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<br>D&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;
+| F&#x266D;<br />E&uarr;
-| G&#x266F; <br>A&#x266D;&darr;
+| G&#x266F; <br />A&#x266D;&darr;
 |-
 ! Just interval
@@ Line 211: / Line 209: @@
 (This section needs considerable re-wording, I'm commenting it out for now) - ArrowHead294
 -->
 {{Navbox notation}}