Extended meantone notation: Difference between revisions

Line 11:

! colspan="2" | Symbol

! rowspan="2" | Interval

! rowspan="2~~" style="width: 75px;~~" | Number of fifths

! rowspan="2" | Number of fifths

|-

! Raise

Line 39:

{| class="wikitable center-all"

! rowspan="2~~" style="width: 50px;~~" | Notes per octave

! rowspan="2" | Notes per octave

! rowspan="2~~" style="width: 150px;~~" | Approximate [[81/80|syntonic comma]] fraction

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

! colspan="4" | Steps

! rowspan="2~~" style="width: 275px;~~" | Explanation

! rowspan="2" | Explanation

|-

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

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

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

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

! Diesis

! Kleisma

|-

| [[7edo|7]]

| [[7edo]]

|

| 0

Line 55:

| 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 B♯ = C♭ and E♯ = F♭.</ref>

| 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 B♯ = C♭ and E♯ = F♭.</ref>

|-

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

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

| {{frac|11}} comma

| 1

Line 63:

| 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<ref group="note" name="diesis_note" />

|-

| [[19edo~~|19~~]]

| [[19edo]]

| {{frac|3}} comma

| 1

Line 71:

| 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<ref group="note" name="kleisma_note" />

|-

| [[26edo~~|26~~]]

| [[26edo]]

|

| 1

Line 79:

| 2

| −1

| rowspan="2" | ~~Diesis~~ is ~~larger~~ than ~~chromatic semitone~~, kleisma is negative<ref group="note" name="kleisma_note" />

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

|-

| [[33edo#Theory|33]] (c mapping)

| [[33edo#Theory|33edo]] (c mapping)

| {{frac|2}} comma

| 1

Line 88:

| −2

|-

| [[31edo~~|31~~]]

| [[31edo]]

| {{frac|4}} comma

| 2

Line 96:

| Diesis is equal to kleisma

|-

| [[43edo~~|43~~]]

| [[43edo]]

| {{frac|5}} comma

| 3

Line 104:

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

|-

| [[55edo~~|55~~]]

| [[55edo]]

| {{frac|6}} comma

| 4

Line 111:

| 3

|-

| [[50edo~~|50~~]]

| [[50edo]]

| {{frac|2|7}} comma

| 3

Line 170:

|}

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 ~~(see [[meantone vs meanpop]])~~.

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.

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

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{{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|s=thin|d=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.

@@ Line 11: / Line 11: @@
 ! colspan="2" | Symbol
 ! rowspan="2" | Interval
-! rowspan="2" style="width: 75px;" | Number of fifths
+! rowspan="2" | Number of<br>fifths
 |-
 ! Raise
@@ Line 39: / Line 39: @@
 {| class="wikitable center-all"
-! rowspan="2" style="width: 50px;" | Notes per octave
+! rowspan="2" | Notes per octave
-! rowspan="2" style="width: 150px;" | Approximate [[81/80|syntonic comma]] fraction
+! rowspan="2" | Approximate<br>[[81/80|syntonic<br>comma]]<br>fraction
 ! colspan="4" | Steps
-! rowspan="2" style="width: 275px;" | Explanation
+! rowspan="2" | Explanation
 |-
-! style="width: 90px;" | Chromatic semitone {{nowrap|(e.g. C&ndash;C♯)}}
+! style="width: 90px;" | Chromatic<br>semitone<br>(e.g.&nbsp;C&ndash;C♯)
-! style="width: 90px;" | Diatonic semitone {{nowrap|(e.g. C&ndash;D♭)}}
+! style="width: 90px;" | Diatonic<br>semitone<br>(e.g.&nbsp;C&ndash;D♭)
 ! Diesis
 ! Kleisma
 |-
-| [[7edo|7]]
+| [[7edo]]
 |
 | 0
@@ Line 55: / Line 55: @@
 | 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>, 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 B♯ = C♭ and E♯ = F♭.</ref>
+| 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♯ 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 B♯ = C♭ and E♯ = F♭.</ref>
 |-
-| [[12edo|12 (standard tuning)]]
+| [[12edo|12edo<br>(standard tuning)]]
 | {{frac|11}}&nbsp;comma
 | 1
@@ Line 63: / Line 63: @@
 | 0
 | 1
-| Chromatic semitone is equal to kleisma, diesis is tempered out<ref group="note" name="diesis_note" />
+| Chromatic semitone is equal to kleisma,<br>diesis is tempered out<ref group="note" name="diesis_note" />
 |-
-| [[19edo|19]]
+| [[19edo]]
 | {{frac|3}}&nbsp;comma
 | 1
@@ Line 71: / Line 71: @@
 | 1
 | 0
-| Chromatic semitone is equal to diesis, kleisma is tempered out<ref group="note" name="kleisma_note" />
+| Chromatic semitone is equal to diesis,<br>kleisma is tempered out<ref group="note" name="kleisma_note" />
 |-
-| [[26edo|26]]
+| [[26edo]]
 |
 | 1
@@ Line 79: / Line 79: @@
 | 2
 | &minus;1
-| rowspan="2" | Diesis is larger than chromatic semitone, kleisma is negative<ref group="note" name="kleisma_note" />
+| rowspan="2" | Chromatic semitone is smaller than diesis,<br>kleisma is negative<ref group="note" name="kleisma_note" />
 |-
-| [[33edo#Theory|33]] (c&nbsp;mapping)
+| [[33edo#Theory|33edo]]<br>(c&nbsp;mapping)
 | {{frac|2}}&nbsp;comma
 | 1
@@ Line 88: / Line 88: @@
 | &minus;2
 |-
-| [[31edo|31]]
+| [[31edo]]
 | {{frac|4}}&nbsp;comma
 | 2
@@ Line 96: / Line 96: @@
 | Diesis is equal to kleisma
 |-
-| [[43edo|43]]
+| [[43edo]]
 | {{frac|5}}&nbsp;comma
 | 3
@@ Line 104: / Line 104: @@
 | rowspan="2" | Diesis is smaller than kleisma
 |-
-| [[55edo|55]]
+| [[55edo]]
 | {{frac|6}}&nbsp;comma
 | 4
@@ Line 111: / Line 111: @@
 | 3
 |-
-| [[50edo|50]]
+| [[50edo]]
 | {{frac|2|7}}&nbsp;comma
 | 3
@@ Line 170: / Line 170: @@
 |}
-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 (see [[meantone vs meanpop]]).
+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.
 == True half-sharps and half-flats ==
@@ Line 184: / Line 184: @@
 {{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|s=thin|d=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.