Chain-of-fifths notation - Revision history

FloraC: /* Expansions */ update

2026-05-12T11:07:09Z

Expansions: update

← Older revision		Revision as of 11:07, 12 May 2026
Line 394:		Line 394:

	== Expansions ==		== Expansions ==
			* [[Kite's ups and downs notation]] – built on chain-of-fifths notation
			* [[Stein–Zimmermann–Gould notation]] – built on neutral chain-of-fifths notation
	* [[Syntonic–rastmic subchroma notation]] – built on neutral chain-of-fifths notation		* [[Syntonic–rastmic subchroma notation]] – built on neutral chain-of-fifths notation
	* [[Ups and downs notation]] – built on chain-of-fifths notation
	** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]])
	* [[Diamond-mos notation]] – built on neutral chain-of-fifths notation
	* [[Sagittal notation]] (''evo flavor'') – built on chain-of-fifths notation or neutral chain-of-fifths notation		* [[Sagittal notation]] (''evo flavor'') – built on chain-of-fifths notation or neutral chain-of-fifths notation
	* [[Extended meantone notation]] – built on chain-of-fifths notation or neutral chain-of-fifths notation		* [[Extended meantone notation]] – built on chain-of-fifths notation or neutral chain-of-fifths notation
	* [[Fractional sharp notation]] – ~~built on~~ chain-of-fifths notation		* [[Fractional sharp notation]] – generalizes neutral chain-of-fifths notation to other fractions
			* [[Diamond-mos notation]] – generalizes neutral chain-of-fifths notation to non-diatonic scales

	== See also ==		== See also ==

FloraC: +1

2026-05-12T09:39:34Z

← Older revision		Revision as of 09:39, 12 May 2026
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	\|+ style="font-size: 105%;" \| Accidentals in (neutral) chain-of-fifths notation		\|+ style="font-size: 105%;" \| Accidentals in (neutral) chain-of-fifths notation
	\|-		\|-
	! Style \ ~~Offset~~		! Style \ offset
	! −2		! −2
	! ~~−1½~~		! −1½
	! −1		! −1
	! ~~−½~~		! −½
	! 0		! 0
	! +~~½~~		! +½
	! +1		! +1
	! +~~1½~~		! +1½
	! +2		! +2
	\|-		\|-
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	\|- style="vertical-align: top;"		\|- style="vertical-align: top;"
	\| style="vertical-align: middle;" \| Standard accidentals<br>+ Stein–Zimmermann accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/stein-zimmermann-accidentals-24-edo.html Standard Music Font Layout \| Stein-Zimmermann accidentals (24-EDO)]</ref>		\| style="vertical-align: middle;" \| Standard accidentals<br>+ Stein–Zimmermann accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/stein-zimmermann-accidentals-24-edo.html Standard Music Font Layout \| Stein-Zimmermann accidentals (24-EDO)]</ref>
	\| {{flat2}}<br/>(U+E264)		\| {{flat2}}<br>(U+E264)
	\| {{sesquiflat}}<br>(U+E281)		\| {{sesquiflat}}<br>(U+E281)
	\| {{flat}}<br>(U+E260)		\| {{flat}}<br>(U+E260)
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	\|-		\|-
	\| Xen Wiki [[:Category: Character templates\|character templates]]		\| Xen Wiki [[:Category: Character templates\|character templates]]
	\| {{tlx\|flat2\|plaincode}}		\| <small>{{tlx\|flat2\|plaincode}}</small>
	\| {{tlx\|sesquiflat\|plaincode}}<br>{{tlx\|sesquiflat2\|plaincode}}		\| <small>{{tlx\|sesquiflat\|plaincode}}<br>{{tlx\|sesquiflat2\|plaincode}}</small>
	\| {{tlx\|flat\|plaincode}}		\| <small>{{tlx\|flat\|plaincode}}</small>
	\| {{tlx\|demiflat\|plaincode}}<br>{{tlx\|demiflat2\|plaincode}}		\| <small>{{tlx\|demiflat\|plaincode}}<br>{{tlx\|demiflat2\|plaincode}}</small>
	\| {{tlx\|natural\|plaincode}}		\| <small>{{tlx\|natural\|plaincode}}</small>
	\| {{tlx\|demisharp\|plaincode}}<br>{{tlx\|demisharp2\|plaincode}}		\| <small>{{tlx\|demisharp\|plaincode}}<br>{{tlx\|demisharp2\|plaincode}}</small>
	\| {{tlx\|sharp\|plaincode}}		\| <small>{{tlx\|sharp\|plaincode}}</small>
	\| {{tlx\|sesquisharp\|plaincode}}<br>{{tlx\|sesquisharp2\|plaincode}}		\| <small>{{tlx\|sesquisharp\|plaincode}}<br>{{tlx\|sesquisharp2\|plaincode}}</small>
	\| {{tlx\|sharp2\|plaincode}}		\| <small>{{tlx\|sharp2\|plaincode}}</small>
	\|}		\|}

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	\|+ style="font-size: 105%;" \| Alternative accidentals in (neutral) chain-of-fifths notation		\|+ style="font-size: 105%;" \| Alternative accidentals in (neutral) chain-of-fifths notation
	\|-		\|-
	! Style \ ~~Offset~~		! Style \ offset
	! −2		! −2
	! −1½		! −1½

FloraC: Various style improvements

2026-05-12T09:36:33Z

Various style improvements

← Older revision		Revision as of 09:36, 12 May 2026
Line 3:		Line 3:
	Chain-of-fifths notation can cover all notes only in [[ring number\|single-ring]] edos. Some tunings have multiple mutually-exclusive circles of fifths, such as [[24edo]] which has two, and [[36edo]] which has three. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes will run out of order. For example, 17edo's notes would be {{dash\|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C\|hair\|med}}. If the fifth is flatter than 685.714{{cent}}, the order of the sharps and flats will be inverted. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated using [[subset notation]], such as in the case of 13edo, which can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the '''native fifth notation'''.		Chain-of-fifths notation can cover all notes only in [[ring number\|single-ring]] edos. Some tunings have multiple mutually-exclusive circles of fifths, such as [[24edo]] which has two, and [[36edo]] which has three. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes will run out of order. For example, 17edo's notes would be {{dash\|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C\|hair\|med}}. If the fifth is flatter than 685.714{{cent}}, the order of the sharps and flats will be inverted. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated using [[subset notation]], such as in the case of 13edo, which can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the '''native fifth notation'''.

	The '''neutral chain-of-fifths notation''' (a.k.a. '''chain-of-half-fifths notation''', '''chain-of-neutral-thirds notation''', or less accurately, '''quartertone notation''') uses an extended accidental set including '''half-sharps''' and '''half-flats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a [[pergen]] of (P8, P5/2), such as the [[mohaha]] temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be [[Ring number #Generalizations\|single-ring]] with respect to the half-fifth. All edos with sharpness 2 or −2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, in 41edo, which is sharp-4, the notes within a (major) whole tone are {{dash\|C, D{{~~sesquiflat2~~}}, C{{~~demisharp2~~}}, D♭, C♯, D{{~~demiflat2~~}}, C{{~~sesquisharp2~~}}, D\|hair\|med}}.		The '''neutral chain-of-fifths notation''' (a.k.a. '''chain-of-half-fifths notation''', '''chain-of-neutral-thirds notation''', or less accurately, '''quartertone notation''') uses an extended accidental set including '''half-sharps''' and '''half-flats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a [[pergen]] of (P8, P5/2), such as the [[mohaha]] temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be [[Ring number #Generalizations\|single-ring]] with respect to the half-fifth. All edos with sharpness 2 or −2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, in 41edo, which is sharp-4, the notes within a (major) whole tone are {{dash\|C, D{{sesquiflat}}, C{{demisharp}}, D{{flat}}, C{{sharp}}, D{{demiflat}}, C{{sesquisharp}}, D\|hair\|med}}.

	Finer divisions (chain-of-third-fifths, chain-of-quarter-fifths, and beyond) are also theoretical possibilities. In practice, ups and downs are usually used when sharps raise by three or more steps.		Finer divisions (chain-of-third-fifths, chain-of-quarter-fifths, and beyond) are also theoretical possibilities. In practice, ups and downs are usually used when sharps raise by three or more steps.

	== Accidentals ==		== Accidentals ==
	The [https://w3c.github.io/smufl/latest/ Standard Music Font Layout (SMuFL)] specification provides Unicode codepoints for the standard accidentals of chain-of-fifths notation and for the Stein–Zimmermann accidentals of neutral chain-of-fifths notation. Some fonts may not include all symbols, so fonts designed for musical notation, such as Bravura or Leland<ref>[https://www.smufl.org/fonts/ SMuFL \| Introducing SMuFL]</ref>, are recommended.		The [https://w3c-cg.github.io/smufl/latest/ Standard Music Font Layout (SMuFL)] specification provides Unicode codepoints for the standard accidentals of chain-of-fifths notation and for the Stein–Zimmermann accidentals of neutral chain-of-fifths notation. Some fonts may not include all symbols, so fonts designed for musical notation, such as Bravura or Leland<ref>[https://www.smufl.org/fonts/ SMuFL \| Introducing SMuFL]</ref>, are recommended.

	In circumstances where the fonts or codepoints are not quickly accessible, ASCII substitute symbols are used instead of the regular symbols. In addition, the Xenharmonic Wiki provides [[:Category:~~character~~ templates\|character templates]] to enter these symbols easily in wiki pages. The following table includes these equivalences.		In circumstances where the fonts or codepoints are not quickly accessible, ASCII substitute symbols are used instead of the regular symbols. In addition, the Xenharmonic Wiki provides [[:Category: Character templates\|character templates]] to enter these symbols easily in wiki pages. The following table includes these equivalences.

	{\| class="wikitable center-all left-1"		{\| class="wikitable center-all left-1"
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	\| Double sharp		\| Double sharp
	\|- style="vertical-align: top;"		\|- style="vertical-align: top;"
	\| style="vertical-align: middle;" \| Standard accidentals<ref>[https://~~www~~.w3.~~org~~/~~2021~~/~~03/smufl14~~/tables/standard-accidentals-12-edo.html Standard Music Font Layout \| Standard accidentals (12-EDO)]</ref>		\| style="vertical-align: middle;" \| Standard accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/standard-accidentals-12-edo.html Standard Music Font Layout \| Standard accidentals (12-EDO)]</ref>
	\| 𝄫<br>(U+1D12B)		\| 𝄫<br>(U+1D12B)
	\|		\|
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	\| 𝄪<br>(U+1D12A)		\| 𝄪<br>(U+1D12A)
	\|- style="vertical-align: top;"		\|- style="vertical-align: top;"
	\| style="vertical-align: middle;" \| Standard accidentals<br>+ Stein–Zimmermann accidentals<ref>[https://~~www~~.w3.~~org~~/~~2021~~/~~03/smufl14~~/tables/stein-zimmermann-accidentals-24-edo.html Standard Music Font Layout \| Stein-Zimmermann accidentals (24-EDO)]</ref>		\| style="vertical-align: middle;" \| Standard accidentals<br>+ Stein–Zimmermann accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/stein-zimmermann-accidentals-24-edo.html Standard Music Font Layout \| Stein-Zimmermann accidentals (24-EDO)]</ref>
	\| {{flat2}}<br/>(U+E264)		\| {{flat2}}<br/>(U+E264)
	\| {{sesquiflat}}<br>(U+E281)		\| {{sesquiflat}}<br>(U+E281)
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	\| x		\| x
	\|-		\|-
	\| Xen Wiki [[:Category:~~character~~ templates\|character templates]]		\| Xen Wiki [[:Category: Character templates\|character templates]]
	\| {{tlx\|flat2\|plaincode}}		\| {{tlx\|flat2\|plaincode}}
	\| {{tlx\|sesquiflat\|plaincode}}<br>{{tlx\|sesquiflat2\|plaincode}}		\| {{tlx\|sesquiflat\|plaincode}}<br>{{tlx\|sesquiflat2\|plaincode}}
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	! Style \ Offset		! Style \ Offset
	! −2		! −2
	! ~~−1½~~		! −1½
	! −1		! −1
	! ~~−½~~		! −½
	! 0		! 0
	! +~~½~~		! +½
	! +1		! +1
	! +~~1½~~		! +1½
	! +2		! +2
	\|- style="vertical-align: top;"		\|- style="vertical-align: top;"
	\| style="vertical-align: middle;" \| Gould arrow quartertone accidentals<ref>[https://~~www~~.w3.~~org~~/~~2021~~/~~03/smufl14~~/tables/gould-arrow-quartertone-accidentals-24-edo.html Standard Music Font Layout \| Gould arrow quartertone accidentals (24-EDO)]</ref><ref group="note">Symbols for five-quarter-tones accidentals are also available.</ref>		\| style="vertical-align: middle;" \| Gould arrow quartertone accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/gould-arrow-quartertone-accidentals-24-edo.html Standard Music Font Layout \| Gould arrow quartertone accidentals (24-EDO)]</ref><ref group="note">Symbols for five-quarter-tones accidentals are also available.</ref>
	\| {{Flat2}}		\| {{Flat2}}
	\| {{Bravura\|}}<br>(U+E271)<br>(U+1D12D)<br>{{Bravura\|}}<br>(U+E278)		\| {{Bravura\|}}<br>(U+E271)<br>(U+1D12D)<br>{{Bravura\|}}<br>(U+E278)
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	\| {{Sharp2}}		\| {{Sharp2}}
	\|- style="vertical-align: top;"		\|- style="vertical-align: top;"
	\| style="vertical-align: middle;" \| Persian accidentals<ref>[https://~~www~~.w3.~~org~~/~~2021~~/~~03/smufl14~~/tables/persian-accidentals.html Standard Music Font Layout \| Persian accidentals]</ref>		\| style="vertical-align: middle;" \| Persian accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/persian-accidentals.html Standard Music Font Layout \| Persian accidentals]</ref>
	\|		\|
	\|		\|
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	\|		\|
	\|- style="vertical-align: top;"		\|- style="vertical-align: top;"
	\| style="vertical-align: middle;" \| [[Sagittal]] accidentals<ref>[https://~~www~~.w3.~~org/2021~~/03/~~smufl14~~/tables/spartan-sagittal-single-shaft-accidentals.html Standard Music Font Layout \| Spartan Sagittal single-shaft accidentals]</ref><ref>[https://~~www~~.w3.~~org/2021~~/03/~~smufl14~~/tables/spartan-sagittal-multi-shaft-accidentals.html Standard Music Font Layout \| Spartan Sagittal multi-shaft accidentals]</ref><ref group="note">In mixed Sagittal notation, standard sharps and flats may be used instead of sagittal sharps and flats, and sagittal accidentals may be used to the left of those to alter them. Also, Sagittal notation includes many more accidentals besides those included in the table.</ref>		\| style="vertical-align: middle;" \| [[Sagittal]] accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/spartan-sagittal-single-shaft-accidentals.html Standard Music Font Layout \| Spartan Sagittal single-shaft accidentals]</ref><ref>[https://w3c-cg.github.io/smufl/latest/tables/spartan-sagittal-multi-shaft-accidentals.html Standard Music Font Layout \| Spartan Sagittal multi-shaft accidentals]</ref><ref group="note">In mixed Sagittal notation, standard sharps and flats may be used instead of sagittal sharps and flats, and sagittal accidentals may be used to the left of those to alter them. Also, Sagittal notation includes many more accidentals besides those included in the table.</ref>
	\| {{Bravura\|}}<br>(U+E335)		\| {{Bravura\|}}<br>(U+E335)
	\| {{Bravura\|}}<br>(U+E327)		\| {{Bravura\|}}<br>(U+E327)
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	\| {{Bravura\|}}<br>(U+E334)		\| {{Bravura\|}}<br>(U+E334)
	\|- style="vertical-align: top;"		\|- style="vertical-align: top;"
	\| style="vertical-align: middle;" \| [[Wyschnegradsky]] accidentals<ref>[https://~~www~~.w3.~~org~~/~~2021~~/~~03/smufl14~~/tables/wyschnegradsky-accidentals-72-edo.html Standard Music Font Layout \| Wyschnegradsky accidentals (72-EDO)]</ref><ref group="note">Wyschnegradsky accidentals also include twelfth-tone ([[72edo]]) accidentals.</ref>		\| style="vertical-align: middle;" \| [[Wyschnegradsky]] accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/wyschnegradsky-accidentals-72-edo.html Standard Music Font Layout \| Wyschnegradsky accidentals (72-EDO)]</ref><ref group="note">Wyschnegradsky accidentals also include twelfth-tone ([[72edo]]) accidentals.</ref>
	\| {{flat2}}		\| {{flat2}}
	\| {{Bravura\|}}<br>(U+E433)		\| {{Bravura\|}}<br>(U+E433)

ArrowHead294: /* Expansions */

2025-05-27T13:19:49Z

Expansions

← Older revision		Revision as of 13:19, 27 May 2025
Line 396:		Line 396:
	* [[Syntonic–rastmic subchroma notation]] – built on neutral chain-of-fifths notation		* [[Syntonic–rastmic subchroma notation]] – built on neutral chain-of-fifths notation
	* [[Ups and downs notation]] – built on chain-of-fifths notation		* [[Ups and downs notation]] – built on chain-of-fifths notation
	** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]])		** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]])
	* [[Diamond-mos notation]] – built on neutral chain-of-fifths notation		* [[Diamond-mos notation]] – built on neutral chain-of-fifths notation
	* [[Sagittal notation]] (''evo flavor'') – built on chain-of-fifths notation or neutral chain-of-fifths notation		* [[Sagittal notation]] (''evo flavor'') – built on chain-of-fifths notation or neutral chain-of-fifths notation

FloraC: No non-breaking space after e.g.

2025-04-16T14:32:52Z

No non-breaking space after *e.g.*

← Older revision		Revision as of 14:32, 16 April 2025
Line 87:		Line 87:
	Note that certain symbols may be very similar or identical to standard or Stein–Zimmermann accidentals despite having different Unicode codepoints.		Note that certain symbols may be very similar or identical to standard or Stein–Zimmermann accidentals despite having different Unicode codepoints.

	A particular case is [[ups and downs notation]], which uses [[arrow]]s placed to the left of accidentals (e.g.~~ ~~^#) or note names (e.g.~~ ~~^C#). Since different tuning systems associate a different number arrows to different offsets, they are not included below, but the most basic notation can be found at [[24edo #Notation]].		A particular case is [[ups and downs notation]], which uses [[arrow]]s placed to the left of accidentals (e.g. ^#) or note names (e.g. ^C#). Since different tuning systems associate a different number arrows to different offsets, they are not included below, but the most basic notation can be found at [[24edo #Notation]].

	{\| class="wikitable center-all left-1"		{\| class="wikitable center-all left-1"

ArrowHead294: /* See also */

2025-04-15T13:34:44Z

ArrowHead294 at 13:34, 15 April 2025

2025-04-15T13:34:27Z

← Older revision		Revision as of 13:34, 15 April 2025
Line 3:		Line 3:
	Chain-of-fifths notation can cover all notes only in [[ring number\|single-ring]] edos. Some tunings have multiple mutually-exclusive circles of fifths, such as [[24edo]] which has two, and [[36edo]] which has three. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes will run out of order. For example, 17edo's notes would be {{dash\|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C\|hair\|med}}. If the fifth is flatter than 685.714{{cent}}, the order of the sharps and flats will be inverted. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated using [[subset notation]], such as in the case of 13edo, which can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the '''native fifth notation'''.		Chain-of-fifths notation can cover all notes only in [[ring number\|single-ring]] edos. Some tunings have multiple mutually-exclusive circles of fifths, such as [[24edo]] which has two, and [[36edo]] which has three. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes will run out of order. For example, 17edo's notes would be {{dash\|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C\|hair\|med}}. If the fifth is flatter than 685.714{{cent}}, the order of the sharps and flats will be inverted. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated using [[subset notation]], such as in the case of 13edo, which can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the '''native fifth notation'''.

	The '''neutral chain-of-fifths notation''' (a.k.a. '''chain-of-half-fifths notation''', '''chain-of-neutral-thirds notation''', or less accurately, '''quartertone notation''') uses an extended accidental set including '''half-sharps''' and '''half-flats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a [[pergen]] of (P8, P5/2), such as the [[mohaha]] temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be [[Ring number #Generalizations\|single-ring]] with respect to the half-fifth. All edos with sharpness 2 or −2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, in 41edo, which is sharp-4, the notes within a (major) whole tone are {{dash\|C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D\|hair\|med}}.		The '''neutral chain-of-fifths notation''' (a.k.a. '''chain-of-half-fifths notation''', '''chain-of-neutral-thirds notation''', or less accurately, '''quartertone notation''') uses an extended accidental set including '''half-sharps''' and '''half-flats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a [[pergen]] of (P8, P5/2), such as the [[mohaha]] temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be [[Ring number #Generalizations\|single-ring]] with respect to the half-fifth. All edos with sharpness 2 or −2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, in 41edo, which is sharp-4, the notes within a (major) whole tone are {{dash\|C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D\|hair\|med}}.

	Finer divisions (chain-of-third-fifths, chain-of-quarter-fifths, and beyond) are also theoretical possibilities. In practice, ups and downs are usually used when sharps raise by three or more steps.		Finer divisions (chain-of-third-fifths, chain-of-quarter-fifths, and beyond) are also theoretical possibilities. In practice, ups and downs are usually used when sharps raise by three or more steps.
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	Note that certain symbols may be very similar or identical to standard or Stein–Zimmermann accidentals despite having different Unicode codepoints.		Note that certain symbols may be very similar or identical to standard or Stein–Zimmermann accidentals despite having different Unicode codepoints.

	A particular case is [[ups and downs notation]], which uses [[arrow]]s placed to the left of accidentals (e.g. ^#) or note names (e.g. ^C#). Since different tuning systems associate a different number arrows to different offsets, they are not included below, but the most basic notation can be found at [[24edo #Notation]].		A particular case is [[ups and downs notation]], which uses [[arrow]]s placed to the left of accidentals (e.g. ^#) or note names (e.g. ^C#). Since different tuning systems associate a different number arrows to different offsets, they are not included below, but the most basic notation can be found at [[24edo #Notation]].

	{\| class="wikitable center-all left-1"		{\| class="wikitable center-all left-1"

ArrowHead294 at 16:07, 20 March 2025

2025-03-20T16:07:33Z

← Older revision		Revision as of 16:07, 20 March 2025
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	The '''chain-of-fifths notation''', also known as '''extended Pythagorean notation''', is a [[musical notation]] system that supports a variety of [[tuning system]]s which are [[octave]]-repeating and generated by the [[3/2\|fifth]] ([[just]] or [[tempered]]). A good number of [[edo]]s and [[regular temperament]]s can be notated this way, as it generalizes the classical notation system for [[Pythagorean tuning]] and [[meantone]] tunings (including [[12edo]]). It uses the seven natural notes of the [[diatonic]] scale (A to G) and accidentals (♯, ♭, and their multiples) to sharpen and flatten these seven notes by the [[~~chromatic semitone\|~~chromatic semitone]]. Any regular rank-2 temperament generated by the octave and fifth (i.e. one with the unsplit [[pergen]]) can be notated this way. For [[equal divisions of the octave]] in particular, this becomes the familiar ''circle of fifths''.		The '''chain-of-fifths notation''', also known as '''extended Pythagorean notation''', is a [[musical notation]] system that supports a variety of [[tuning system]]s which are [[octave]]-repeating and generated by the [[3/2\|fifth]] ([[just]] or [[tempered]]). A good number of [[edo]]s and [[regular temperament]]s can be notated this way, as it generalizes the classical notation system for [[Pythagorean tuning]] and [[meantone]] tunings (including [[12edo]]). It uses the seven natural notes of the [[diatonic]] scale (A to G) and accidentals (♯, ♭, and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone]]. Any regular rank-2 temperament generated by the octave and fifth (i.e. one with the unsplit [[pergen]]) can be notated this way. For [[equal divisions of the octave]] in particular, this becomes the familiar ''circle of fifths''.

	Chain-of-fifths notation can cover all notes only in [[ring number\|single-ring]] edos. Some tunings have multiple mutually-exclusive circles of fifths, such as [[24edo]] which has two, and [[36edo]] which has three. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes will run out of order. For example, 17edo's notes would be {{dash\|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C\|hair\|med}}. If the fifth is flatter than 685.714{{cent}}, the order of the sharps and flats will be inverted. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated using [[subset notation]], such as in the case of 13edo, which can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the '''native fifth notation'''.		Chain-of-fifths notation can cover all notes only in [[ring number\|single-ring]] edos. Some tunings have multiple mutually-exclusive circles of fifths, such as [[24edo]] which has two, and [[36edo]] which has three. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes will run out of order. For example, 17edo's notes would be {{dash\|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C\|hair\|med}}. If the fifth is flatter than 685.714{{cent}}, the order of the sharps and flats will be inverted. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated using [[subset notation]], such as in the case of 13edo, which can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the '''native fifth notation'''.

ArrowHead294: /* Expansions */

2025-02-23T15:11:48Z

Expansions

← Older revision		Revision as of 15:11, 23 February 2025
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	== Expansions ==		== Expansions ==
	* [[~~Syntonic-rastmic~~ subchroma notation]] – built on neutral chain-of-fifths notation		* [[Syntonic–rastmic subchroma notation]] – built on neutral chain-of-fifths notation
	* [[Ups and downs notation]] – built on chain-of-fifths notation		* [[Ups and downs notation]] – built on chain-of-fifths notation
	** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]])		** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]])

FloraC: Style

2025-01-20T14:34:35Z

Style

Show changes

← Older revision		Revision as of 13:34, 15 April 2025
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	* [[Fifthspan]]		* [[Fifthspan]]
	* [[Pythagorean tuning]]		* [[Pythagorean tuning]]
	* [[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]")		* [[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]")

	== Notes ==		== Notes ==

Chain-of-fifths notation - Revision history

FloraC: /* Expansions */ update

FloraC: +1

FloraC: Various style improvements

ArrowHead294: /* Expansions */

FloraC: No non-breaking space after *e.g.*

ArrowHead294: /* See also */

ArrowHead294 at 13:34, 15 April 2025

ArrowHead294 at 16:07, 20 March 2025

ArrowHead294: /* Expansions */

FloraC: Style

FloraC: No non-breaking space after e.g.