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[[file:sagittal_sample.gif]]
[[file:sagittal_sample.gif]]


'''Sagittal notation''' is a [[musical notation]] system capable of notating almost any conceivable tuning. It was developed by [[Dave Keenan]] and [[George Secor]] with significant contributions from numerous others.  
'''Sagittal notation''' is a [[musical notation]] system capable of notating almost any conceivable tuning while preserving, as much as possible, the notation of harmonies across different tunings. It uses arrow-like symbols made up of four simple components whose visual size is proportional to their alteration and whose alterations sum. It was developed by [[Dave Keenan]] and [[George Secor]] with significant contributions from numerous others.  


== Flavors of Sagittal notation ==
== Flavors of Sagittal notation ==
Line 10: Line 10:
The '''Evo''' flavor (short for "evolutionary", previously called "mixed") uses only single-shaft Sagittal symbols, e.g. {{sagittal| /| }} {{sagittal| \! }} {{sagittal| |) }} {{sagittal| !) }}, alone or in combination with conventional sharps and flats and their doubles. Only the large variant of the double sharp {{sagittal| x }} (U+E47D) is considered to be stylistically-compatible with Sagittal symbols. Evo is much easier to learn, but it results in a greater number of symbols on the sheet, which can give it a more cluttered appearance, particularly with chords, and it may be confusing when two symbols alter the same note in opposite directions.
The '''Evo''' flavor (short for "evolutionary", previously called "mixed") uses only single-shaft Sagittal symbols, e.g. {{sagittal| /| }} {{sagittal| \! }} {{sagittal| |) }} {{sagittal| !) }}, alone or in combination with conventional sharps and flats and their doubles. Only the large variant of the double sharp {{sagittal| x }} (U+E47D) is considered to be stylistically-compatible with Sagittal symbols. Evo is much easier to learn, but it results in a greater number of symbols on the sheet, which can give it a more cluttered appearance, particularly with chords, and it may be confusing when two symbols alter the same note in opposite directions.


A sub-flavor of Evo is '''Evo-SZ''' (Evo with Stein–Zimmermann). This is where any sagittals that are notating exactly half the alteration of a sharp or flat (most often {{sagittal| /|\ }} {{sagittal| \!/ }}) are replaced by the Stein–Zimmermann semisharp {{sagittal| > }} and narrow reversed flat {{sagittal| < }}, and the corresponding combinations (most often {{sagittal| /|\ }}{{sagittal| # }} and {{sagittal| \!/ }}{{sagittal| b }}) are replaced by {{sagittal| ># }} and {{sagittal| <b }}. The narrow variants of the fractional flats {{sagittal| < }} (U+E284) and {{sagittal| <b }} (U+E285) are preferred because they preserve the Sagittal principle that the visual size of a symbol should indicate the relative size of its alteration and they reduce left-right confusion.
A sub-flavor of Evo is '''Evo-SZ''' (Evo with Stein–Zimmermann). This is where, in tempered systems, any sagittals that are notating exactly half the alteration of a sharp or flat (most often {{sagittal| /|\ }} {{sagittal| \!/ }}) are replaced by the Stein–Zimmermann semisharp {{sagittal| > }} and narrow reversed flat {{sagittal| < }}, and the corresponding combinations (most often {{sagittal| /|\ }}{{sagittal| # }} and {{sagittal| \!/ }}{{sagittal| b }}) are replaced by {{sagittal| ># }} and {{sagittal| <b }}. The narrow variants of the fractional flats {{sagittal| < }} (U+E284) and {{sagittal| <b }} (U+E285) are preferred because they preserve the Sagittal principle that the visual size of a symbol should indicate the relative size of its alteration and they reduce left-right confusion.


In tempered systems, Evo-SZ {{sagittal|>}} {{sagittal|<}} {{sagittal|>#}} {{sagittal|<b}} accidentals ''may instead'' be distinct from Sagittals like {{sagittal| /|\ }}{{sagittal| \!/ }}, when the apotome can be split in two equal parts as in [[Hemipyth|Hemipythagorean]] as literal half-sharps and half-flats, with a distinction between mapped neutral thirds like [[11/9|~11/9]] or  ~[[16/13]] and [[Sqrt(3/2)|hemififths]] which might be mapped to another interval like [[49/40]] in particularly fine-grained EDOs like [[270edo|270]] or [[311edo|311]]. There is no need for this in coarser EDOs like [[24edo|24]], [[31edo|31]] or [[41edo|41]], where in this case the [[243/242|rastma]] is tempered out, and thus the mappings of the irrational hemififth and [[11/9]] coincide.
=== Revo ===
The '''Revo''' flavor (short for "revolutionary", previously called "pure") only requires one accidental per note. Revo therefore takes up less space on the sheet and presents a cleaner appearance, and it clearly indicates the direction of the overall alteration. It discards the conventional sharps and flats and their doubles and replaces them with these multi-shaft arrow-like symbols: {{sagittal| /||\ }} {{sagittal| \!!/ }} {{sagittal| /X\ }} {{sagittal| \Y/ }}. Adding a sharp or flat to a Sagittal is achieved by adding two more shafts, e.g. {{sagittal| /| }}{{sagittal| # }} becomes {{sagittal| /||| }} and {{sagittal| !) }}{{sagittal| b }} becomes {{sagittal| !!!) }}, as expected. [[2187/2048#Notation|''Apotome'']] ''complements'', that arise when the Sagital accidental alters in the opposite direction to the apotome, do not have a simple rule in Revo. One must simply memorize the complements one needs, as shown below. For example, {{sagittal| \! }}{{sagittal| # }} becomes {{sagittal| ||\ }} (flag swaps sides) while {{sagittal| !) }}{{sagittal| # }} becomes {{sagittal| ||) }} (flag stays on same side).


=== Revo ===
The '''Revo''' flavor (short for "revolutionary", previously called "pure") only requires one accidental per note. Revo therefore takes up less space on the sheet and presents a cleaner appearance, and it clearly indicates the direction of the overall alteration. It discards the conventional sharps and flats and their doubles and replaces them with these multi-shaft arrow-like symbols: {{sagittal| /||\ }} {{sagittal| \!!/ }} {{sagittal| /X\ }} {{sagittal| \Y/ }}. Adding a sharp or flat to a Sagittal is achieved by adding two more shafts, e.g. {{sagittal| /| }}{{sagittal| # }} becomes {{sagittal| /||| }} and {{sagittal| !) }}{{sagittal| b }} becomes {{sagittal| !!!) }}, as expected. [[2187/2048#Notation|''Apotome'']] ''complements'', that arise when the Sagital accidental alters in the opposite direction to the apotome, are not predictable. {{sagittal| \! }}{{sagittal| # }} becomes {{sagittal| ||\ }},  {{sagittal| |) }}{{sagittal| b }} becomes {{sagittal| !!) }}, {{sagittal| \!/ }}{{sagittal| # }} becomes {{sagittal|(|) }}, {{sagittal| \!)}}{{sagittal| # }} becomes {{sagittal|(|\}} ; one must learn these ''apotome complements'' by rote memorization.
{| class="wikitable"
{| class="wikitable"
|+Table of apotome complements<ref>https://sagittal.org/sagittal.pdf p. 24 Figure 13</ref>
|+Table of Spartan (most common) <br> apotome complements<ref>https://sagittal.org/sagittal.pdf p. 7 Figure 4</ref>
|Symbol
|<big>{{sagittal|h}}</big>
|<big>{{sagittal| |(}}</big>
|<big>{{sagittal|/|}}</big>
|<big>{{sagittal| |)}}</big>
|<big>{{sagittal|//| }}</big>
|<big>{{sagittal|/|)}}</big>
|<big>{{sagittal|/|\}}</big>
|-
|Complement
|<big>{{sagittal|/||\}}</big>
|<big>{{sagittal|/||)}}</big>
|<big>{{sagittal|||\}}</big>
|<big>{{sagittal|||)}}</big>
|<big>{{sagittal|)||(}}</big>
|<big>{{sagittal|(|\}}</big>
|<big>{{sagittal|(|)}}</big>
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+Table&nbsp;of&nbsp;all&nbsp;apotome&nbsp;complements<ref>Most of these are symbols are rarely used. {{sagittal| )/|\ }} is its own complement. https://sagittal.org/sagittal.pdf p. 24 Figure 13</ref>
|Symbol
|Symbol
|<big>{{sagittal|h}}</big>
|<big>{{sagittal|h}}</big>
Line 75: Line 95:
|<big>{{sagittal||\)}}</big>
|<big>{{sagittal||\)}}</big>
|}
|}
{{sagittal|)/|\}} doesn't have an assigned apotome complement, so in many tuning systems it is itself.


== Notation software support ==
== Notation software support ==
Line 108: Line 127:


== The symbol sets ==
== The symbol sets ==
Sagittal symbols are defined in 6 "official" symbol sets, each one being defined as spliting the [[2187/2048|apotome]] in psuedoequal or equal parts (latter being '''EDA'''). It is not necessary to learn all of the Sagittal notation sets to be able to compose with it. The '''Spartan''' and '''Athenian''' sets will be more than enough for most purposes.  
Sagittal symbols come in 7 sets of increasing size and resolution, each one being an approximately equal division of the [[2187/2048|apotome]] ('''EDA'''). It is not necessary to learn all of the Sagittal sets to be able to compose with it. It is like a natural language in that you don't need to know every word in the dictionary to speak the language. Some words are used extremely rarely, but they must be there to provide full coverage. The '''Spartan''' set covers 90% of what most people want to do, and the '''Athenian''' extension covers most of the rest.
Sagittal accidentals are not intended to be combined with one another, except in the Prime Factor JI notation, as symbols representing useful combinations and powers of primes are already provided. An accidental can often be used to represent alternative commas that differ by 2 cents or less. In such cases the intended comma ratio may be determined by the note to which it is applied, or by the musical context. Alternatively, accent marks (from the Herculean and subsequent extensions) may be added to distinguish these commas.
 
Sagittal accidentals are not intended to be combined with one another, except in the Prime Factor JI notation, as symbols representing useful combinations and powers of primes are already provided. An accidental can often be used to represent alternative commas that differ by 2 cents or less. In such cases the intended comma ratio may be determined by the pitch-class to which it is applied, or by the musical context. Alternatively, accent marks (from the Herculean and subsequent extensions) may be added to distinguish these commas.


[https://www.desmos.com/calculator/iehdworjko You can look up what the minimum precision is required to write an EDO in this Desmos graph.]
[https://www.desmos.com/calculator/iehdworjko This Desmos graph] shows the theoretical minimum Sagittal set required to notate an EDO. However the standard Sagittal EDO notations don't always adhere to this because there are many other considerations that go into choosing a good EDO notation from among the possible ones.


=== Sharps/Flats ===
=== Sharps/Flats ===
Using {{sagittal| # }} and {{sagittal| b }} (or {{sagittal|/||\}} {{sagittal|\!!/}} for Revo flavor) is still ''technically'' Sagittal notation, however, it's just a reskin of the usual chain-of-fifths notation. Ditto for Stein-Zimmermann half-sharps and half-flats in [[hemipythagorean]].  
Using {{sagittal| # }} and {{sagittal| b }} (or {{sagittal|/||\}} {{sagittal|\!!/}} for Revo flavor) is still ''technically'' Sagittal notation, however, it's just a reskin of the usual chain-of-fifths notation. Ditto for Stein-Zimmermann half-sharps and half-flats in [[hemipythagorean]].
 
[[File:SagittalEulerDiagram1.png|500px|thumb|right|Relationships between sagittal symbol subsets (excluding accents)]]


=== Spartan ===
=== Spartan ===
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=== Trojan (12-EDO relative) ===
=== Trojan (12-EDO relative) ===
This is a special extension to Spartan which adds 3 more symbol pairs, which is exclusively used to notate [[compton]] EDOs (12N EDOs up to [[312edo|312]]) in which the apotome is 100 cents. However, all of these EDOs lower than 168EDO can be notated using just Spartan, in which cases, depending on the composer's needs, may prove Athenian to be more useful.
This is a special set that combines some Spartans with some from the Athenian extension and adds 3 more symbol pairs, which can be used to notate any tuning relative to 12-EDO with medium precision (typical error ±2 cents). It also provides the exact standard notations for the [[compton]] or 12N EDOs up to [[192edo|192]], in which the apotome is 100 cents.


=== Promethean ===
=== Promethean ===
It adds 20 more symbols to Athenian symbol set, allowing for a maximum resolution of 47EDA. When used for JI, it defines the ''Standard High Precision JI'' capable of writing in the 23-limit or 41-limit, however, this JI notation is not recommended. It instead can be used to write in EDOs such as the [[Zeta peak edo|zeta edos]] [[270edo|270]] and [[311edo|311]], the latter to write music in the [[41-limit]].
It adds 20 more symbols to Athenian symbol set, allowing for a maximum resolution of 47EDA. When used for JI, it defines the ''Standard High Precision JI'' capable of writing in the 23-limit or 41-limit, however, this JI notation is not recommended. It instead can be used to write in EDOs such as the [[Zeta peak edo|zeta edos]] [[270edo|270]] and [[311edo|311]], the latter to write music in the [[41-limit]].
== The accent sets ==
Fine-grained Sagittal notations can use accents, also called diacritics, to the left of a symbol or a bare shaft, to indicate very subtle distinctions in pitch.
<div class="mw-collapsible mw-collapsed">
[[File:SagittalEulerDiagram2.png|500px|thumb|right|Relationships between sagittal symbol subsets including accents]]


=== Herculean ===
=== Herculean ===
It adds the [[schisma]] diacritic to the Promethean symbol set, which can be stacked with the remaining alterations, allowing for a maximum resolution of 58EDA, of which a great edo is [[612edo|612]]. When used for JI, it defines the ''Standard Ultra Precision JI'' capable of writing in the 23-limit with higher precision.
It adds the [[schisma]] accent (a diacritic) to the Promethean symbol set, which can be stacked with the remaining alterations, allowing for a maximum resolution of 58EDA, of which a great edo is [[612edo|612]]. When used for JI, it defines the ''Standard Ultra Precision JI'' capable of writing in the 23-limit with higher precision.


=== Olympian ===
=== Olympian ===
It adds the [[4096/4095|''mina'']] diacritic to the Herculean symbol set, able to be stacked up to twice with the schisma and the remaining alterations, allowing for a maximum resolution of 233EDA, The zeta peak [[2460edo]] has been used as a base to define the mina as an interval measure, and the Olympian set of intervals generally, due to its extremely precise 27-odd-limit palette. When used for JI, it defines the ''Standard Extreme Precision JI'' capable of writing in the [[47-limit]] with great precision. It also is the smallest precision level that has an "exact" mapping for prime 13, thanks to the mina's appearance. 13/8 is now written as a major sixth minus the 35 large diesis and a mina. From C, this would be C - A{{sagittal|,}}{{sagittal|(!/}} .
It adds the [[4096/4095|''mina'']] accent to the Herculean symbol set, able to be stacked up to twice with the schisma and the remaining alterations, allowing for a maximum resolution of 233EDA, The zeta peak [[2460edo]] has been used as a base to define the mina as an interval measure, and the Olympian set of intervals generally, due to its extremely precise 27-odd-limit palette. When used for JI, it defines the ''Standard Extreme Precision JI'' capable of writing in the [[47-limit]] with great precision. It also is the smallest precision level that has an "exact" mapping for prime 13, thanks to the mina's appearance. 13/8 is now written as a major sixth minus the 35 large diesis and a mina. From C, this would be C - A{{sagittal|,}}{{sagittal|(!/}} .


=== Magrathean ===
=== Magrathean ===
It adds the ''tina'' diacritic to the Olympian symbol set, able to be stacked up to thrice with any of the symbols (three tinas make a ~mina), allowing for a whopping maximum resolution of 809EDA. The strict zeta peak [[8539edo]] has been used to define the tina as an interval measure, due its insanely precise 27-odd-limit (and beyond) interval palette. When used for JI, it defines the ''Standard Insane Precision JI'' capable of writing in the 127-limit with astonishing precision. There is no level of precision higher than this, and it is unlikely that one will ever exist. Unless you want some hot sauce.<ref name=":0">https://forum.sagittal.org/viewtopic.php?p=2714&hilit=bomb#p2714  "A tina is approximately 1/809th of an apotome, 1/8539th of an octave (a [[Zeta peak edo|zeta-peak EDO]]), or 0.14 cents. The fractional-tina is generally half a tina but is intentionally arbitrary, because if you need any more precision than that, I have a bottle of Da' Bomb Beyond Insanity Hot Sauce with your name on it"</ref>
It adds the ''tina'' accent to the Olympian symbol set, able to be stacked up to thrice with any of the symbols (three tinas make a ~mina), allowing for a whopping maximum resolution of 809EDA. The strict zeta peak [[8539edo]] has been used to define the tina as an interval measure, due its insanely precise 27-odd-limit (and beyond) interval palette. When used for JI, it defines the ''Standard Insane Precision JI'' capable of writing in the 127-limit with astonishing precision. There is no level of precision higher than this, and it is unlikely that one will ever exist. Unless you want some hot sauce.<ref name=":0">https://forum.sagittal.org/viewtopic.php?p=2714&hilit=bomb#p2714  "A tina is approximately 1/809th of an apotome, 1/8539th of an octave (a [[Zeta peak edo|zeta-peak EDO]]), or 0.14 cents. The fractional-tina is generally half a tina but is intentionally arbitrary, because if you need any more precision than that, I have a bottle of Da' Bomb Beyond Insanity Hot Sauce with your name on it"</ref>
 
</div>
== Prime approximations ==
Here are some approximations to octave reduced primes from D, using the several precision sets available in JI. Values in parentheses are absolute error in cents from just; if exact, none is displayed.
{| class="wikitable"
|
|5
|7
|11
|13
|17
|19
|23
|29
|31
|-
|Spartan
| rowspan="5" |F {{sagittal|||\}}
| rowspan="5" |C {{sagittal|!)}}
| rowspan="5" |G {{sagittal|/|\}}
| rowspan="3" |B {{sagittal|(!/}}(0.42)
|D {{sagittal|)||| }} (2.971)
| rowspan="2" |F {{sagittal||(}} (2.380)
|A {{sagittal|\\!}} (3.008)
|C {{sagittal||)}} (6.223)
| rowspan="2" |D {{sagittal|\!/}}(1.691)
|-
|Athenian
| rowspan="4" |E {{sagittal|(!!(}}
|A {{sagittal|~!!(}}(1.009)
| rowspan="2" |C {{sagittal|(| }} (0.339)
|-
|Promethean
| rowspan="3" |F {{sagittal|)| }}
| rowspan="3" |A {{sagittal|)~!!}}
|D {{sagittal|(\!}}(0.436)
|-
|Olympian
| rowspan="2" |B {{sagittal|,}}{{sagittal|(!/}}
|C {{sagittal|`}}{{sagittal|(| }} (0.130)
| rowspan="2" |D {{sagittal|,}}{{sagittal|(\!}} (0.013)
|-
|Magrathean
|C {{sagittal|@2}}{{sagittal|(| }} (0.039)
|}
Tempered systems might yield better or worse approximations to each just interval. Used for JI, only some non-Pythagorean intervals in the 23-limit will ever be exact. The error gets increasingly small however, down to hundredths of a cent in Magrathean.


== Gallery ==
== Gallery of symbols ==
=== Spartan single-shaft ===
=== Spartan single-shaft ===
{{sge|  ¦( |  !( | n  | 5120/5103 | 7/5 kleisma }}
{{sge|  ¦( |  !( | n  | 5120/5103 | 7/5 kleisma }}
Line 223: Line 207:
{{sge| (¦¦¦) | (!!!) | jat |  729/704  | 11L }}
{{sge| (¦¦¦) | (!!!) | jat |  729/704  | 11L }}
{{sge| (¦¦¦\ | (!!!/ | jak | 8505/8192 | 35L }}
{{sge| (¦¦¦\ | (!!!/ | jak | 8505/8192 | 35L }}
{{sge| )X( | )Y( | ph  | 6561/6400 | 25S | mid=48px }}
{{sge| )X( | )Y( | ph  | 6561/6400 | 25S }}
{{sge|  X) |  Y) | t  |  64/63  | 7C   | mid=48px }}
{{sge|  X) |  Y) | t  |  64/63  | 7C }}
{{sge|  X\ |  Y/ | p  |  81/80  | 5C   | mid=48px }}
{{sge|  X\ |  Y/ | p  |  81/80  | 5C }}
{{sge| /X) | \Y) | n  | 5120/5103 | 7/5k | mid=48px }}
{{sge| /X) | \Y) | n  | 5120/5103 | 7/5k }}
{{sge| /X\ | \Y/ | mid=48px }}
{{sge| /X\ | \Y/ }}
</div>
</div>


Line 251: Line 235:
{{sge| (¦¦¦  | (!!!  | j  |45927/45056| 11/7C }}
{{sge| (¦¦¦  | (!!!  | j  |45927/45056| 11/7C }}
{{sge| (¦¦¦( | (!!!( | jan |  45/44  | 11/5S }}
{{sge| (¦¦¦( | (!!!( | jan |  45/44  | 11/5S }}
{{sge| ~X( | ~Y( | jan |  45/44  | 11/5S | mid=48px }}
{{sge| ~X( | ~Y( | jan |  45/44  | 11/5S }}
{{sge| )X~ | )Y~ | j  |45927/45056| 11/7C | mid=48px }}
{{sge| )X~ | )Y~ | j  |45927/45056| 11/7C }}
{{sge| /X  | \Y  | k  |  55/54  | 55C | mid=48px }}
{{sge| /X  | \Y  | k  |  55/54  | 55C }}
{{sge| (X( | (Y( | san | 4131/4096 | 17C | mid=48px }}
{{sge| (X( | (Y( | san | 4131/4096 | 17C }}
{{sge|//X  |\\Y  | ran |  896/891  | 11/7k | mid=48px }}
{{sge|//X  |\\Y  | ran |  896/891  | 11/7k }}
</div>
</div>


Line 269: Line 253:
{{sge|)/¦¦¦|)\!!!|pr|41553/40960 |19/5C}}
{{sge|)/¦¦¦|)\!!!|pr|41553/40960 |19/5C}}
{{sge|/¦¦¦~|\!!!~|paz|46/45|23/5S}}
{{sge|/¦¦¦~|\!!!~|paz|46/45|23/5S}}
{{sge|X~|Y~|paz|46/45|23S| mid=48px }}
{{sge|X~|Y~|paz|46/45|23S }}
{{sge|)X)|)Y)|pr|41553/40960 |19/5C| mid=48px }}
{{sge|)X)|)Y)|pr|41553/40960 |19/5C }}
{{sge|/X~|\Y~|z|736/729|23C| mid=48px }}
{{sge|/X~|\Y~|z|736/729|23C }}


=== Promethean extension single-shaft ===
=== Promethean extension single-shaft ===
Promethean symbols are rarely used, but they allow Sagittal to notate some EDOs with more than 400 pitches per octave.
{{sge| )¦  | )!  | r    |    513/512    | 19s    }}
{{sge| )¦  | )!  | r    |    513/512    | 19s    }}
{{sge| ~¦  | ~!  | s    |    2187/2176    | 17k    }}
{{sge| ~¦  | ~!  | s    |    2187/2176    | 17k    }}
Line 329: Line 315:
{{sge| (X~ | (Y~    | r  |  81/76  | 19s }}
{{sge| (X~ | (Y~    | r  |  81/76  | 19s }}


=== Herculean extension diacritics ===
== Gallery of accents ==
{{sge| ¦ | ! | &NoBreak; | 1/1 | unison }}
Fine-grained Sagittal notations can use accents, also called diacritics, to the left of a sagittal or a bare shaft, to indicate very subtle distinctions in pitch.
Ai and ao are bare shafts, used when there would otherwise be bare diacritics, as for the schisma below, and the minas and tinas of the following extensions.
<div class="mw-collapsible mw-collapsed">
=== Herculean extension accents ===
{{sge| ¦ | ! | &NoBreak; | 1/1 | shaft }}
Ai and ao are not accents but bare shafts which do not alter the pitch. They are used when there would otherwise be bare accents.
{| class="wikitable"
{| class="wikitable"
|- style="height: 40px; text-align: center;"
|- style="text-align: center;"
| {{sagittal|'|size=500%}}
| {{sagittal|'|size=500%}}
| {{sagittal|.|size=500%}}
| {{sagittal|.|size=500%}}
|- style="text-align: center;"
|- style="text-align: center;"
|bi
|bi
|bo
|bo
|- style="height: 64px;"
|-
|[[32805/32768 | 5s up]]
|[[32805/32768 | 5s up]]
|5s down
|5s down
|}
|}
The interval is called a "schisma". The accent's shape is called "tick" (up or down).


=== Olympian extension diacritics ===
=== Olympian extension accents ===
{| class="wikitable"
{| class="wikitable"
!{{sagittal|`}}
|- style="text-align: center;"
!{{sagittal|,}}
|{{sagittal|`|size=500%}}
!{{sagittal|``}}
|{{sagittal|,|size=500%}}
!{{sagittal|,,}}
|{{sagittal|``|size=500%}}
|-
|{{sagittal|,,|size=500%}}
|- style="text-align: center;"
|mi
|mi
|mo
|mo
Line 356: Line 347:
|momo
|momo
|-
|-
|[https://en.xen.wiki/w/4096/4095 35/13sx] up
|[[4096/4095 | 5⋅7⋅13n up]]
|35/13sx down
|5⋅7⋅13n down
|[https://en.xen.wiki/w/2080/2079 77/65sx] up
|[[2080/2079 | 77/65n up]]
|77/65sx down
|77/65n down
|}
|}
The diacritic is called "mina".
The average unit interval is called a "mina" (rhymes with ballerina) and is approximately 0.42 of a cent. The shapes are called "wing" and "bird".


=== Magrathean extension diacritics ===
=== Magrathean extension accents ===
{| class="wikitable"
{| class="wikitable"
!{{sagittal|@1}}
|- style="text-align: center;"
!{{sagittal|l1}}
|{{sagittal|@1|size=500%}}
!{{sagittal|@2}}
|{{sagittal|l1|size=500%}}
!{{sagittal|l2}}
|{{sagittal|@2|size=500%}}
!{{sagittal|`}}
|{{sagittal|l2|size=500%}}
!{{sagittal|,}}
|{{sagittal|@3|size=500%}}
|-
|{{sagittal|l3|size=500%}}
|- style="text-align: center;"
|qui
|qui
|quo
|quo
Line 379: Line 371:
|mo
|mo
|-
|-
|10241/10240 up
|[[10241/10240 | 7²⋅11⋅19/5n up]]
|10241/10240 up
|7²⋅11⋅19/5n down
|5832/5831 up
|[[5832/5831 | 7³⋅17n up]]
|5832/5831 down
|7³⋅17n down
|[https://en.xen.wiki/w/4096/4095 35/13sx] up
|[[4096/4095 | 5⋅7⋅13n up]]
|35/13sx down
|5⋅7⋅13n down
|}
|}
{| class="wikitable"
{| class="wikitable"
!{{sagittal|@4}}
|- style="text-align: center;"
!{{sagittal|l4}}
|{{sagittal|@4|size=500%}}
!{{sagittal|@5}}
|{{sagittal|l4|size=500%}}
!{{sagittal|l5}}
|{{sagittal|@5|size=500%}}
!{{sagittal|``}}
|{{sagittal|l5|size=500%}}
!{{sagittal|,,}}
|{{sagittal|@6|size=500%}}
|-
|{{sagittal|l6|size=500%}}
|- style="text-align: center;"
|quimi
|quimi
|quomo
|quomo
Line 401: Line 394:
|momo
|momo
|-
|-
|3025/3024 up
|[[3025/3024 | 5²⋅11²/7n up]]
|3025/3024 up
|5²⋅11²/7n down
|2401/2400 up
|[[2401/2400 | 7⁴/25n up]]
|2401/2400 down
|7⁴/25n down
|[https://en.xen.wiki/w/2080/2079 77/65sx] up
|[[2080/2079 | 77/65n up]]
|77/65sx down
|77/65n down
|}
|}
{| class="wikitable"
{| class="wikitable"
!{{sagittal|@7}}
|- style="text-align: center;"
!{{sagittal|l7}}
|{{sagittal|@7|size=500%}}
!{{sagittal|@8}}
|{{sagittal|l7|size=500%}}
!{{sagittal|l8}}
|{{sagittal|@8|size=500%}}
!{{sagittal|@9}}
|{{sagittal|l8|size=500%}}
!{{sagittal|l9}}
|{{sagittal|@9|size=500%}}
|-
|{{sagittal|l9|size=500%}}
|- style="text-align: center;"
|quimimi
|quimimi
|quomomo
|quomomo
Line 423: Line 417:
|momomo
|momomo
|-
|-
|1701/1700 up
|[[1701/1700 | 5²⋅17/7n up]]
|1701/1700 down
|5²⋅17/7n down
|382976/382725 up
|[[382976/382725 | 11⋅17/(5²⋅7)n up]]
|382976/382725 down
|11⋅17/(5²⋅7)n down
|131072/130977 up
|[[131072/130977 | 7²⋅11n up]]
|131072/130977 down
|7²⋅11n down
|}
|}
{| class="wikitable"
{| class="wikitable"
!{{sagittal|@.}}
|- style="text-align: center;"
!{{sagittal|l.}}
|{{sagittal|@.|size=500%}}
|-
|{{sagittal|l.|size=500%}}
|- style="text-align: center;"
|i
|i
|o
|o
|-
|-
|1515591/1515520 up
|[[1515591/1515520 | 77/(5⋅37)n up]]
|1515591/1515520  down
|77/(5⋅37)n down
|}
The average unit interval is called a "tina" (rhymes with ballerina) and is approximately 0.14 of a cent. The new shapes are called "horn" and "wedge". Notice how 3 tinas is ''approximately'' equal to one mina, so the system just equates the 3. Either way, this is an ''insane'' level of pitch precision. The "i/o" accent, whose shape is called "dot", represents some unit fraction of a tina, often a half as shown, but it is intentionally left to be defined by the user.<ref name=":0" />
</div>
 
== Prime approximations ==
Here are some approximations to primes from D, using the several precision sets available in JI. Values in parentheses are absolute error in cents from just; if none is shown, the notation is exact.
{| class="wikitable"
|
|5
|7
|11
|13
|17
|19
|23
|29
|31
|-
|Spartan
| rowspan="5" |F {{sagittal|||\}}
| rowspan="5" |C {{sagittal|!)}}
| rowspan="5" |G {{sagittal|/|\}}
| rowspan="3" |B {{sagittal|(!/}}(0.42)
|D {{sagittal|)||| }} (2.971)
| rowspan="2" |F {{sagittal||(}} (2.380)
|A {{sagittal|\\!}} (3.008)
|C {{sagittal||)}} (6.223)
| rowspan="2" |D {{sagittal|\!/}}(1.691)
|-
|Athenian
| rowspan="4" |E {{sagittal|(!!(}}
|A {{sagittal|~!!(}}(1.009)
| rowspan="2" |C {{sagittal|(| }} (0.339)
|-
|Promethean
| rowspan="3" |F {{sagittal|)| }}
| rowspan="3" |A {{sagittal|)~!!}}
|D {{sagittal|(\!}}(0.436)
|-
|Olympian
| rowspan="2" |B {{sagittal|,}}{{sagittal|(!/}}
|C {{sagittal|`}}{{sagittal|(| }} (0.130)
| rowspan="2" |D {{sagittal|,}}{{sagittal|(\!}}
|-
|Magrathean
|C {{sagittal|@2}}{{sagittal|(| }}
|}
|}
The diacritic is called "tina". Notice how 3 tinas is ''approximately'' equal to one mina, so the system just equates the 3. Either way, this is an ''insane'' level of pitch precision, down to less than tenths of a cent. The "i/o" diacritic is merely a dot and represents some fraction of a tina, often a half, but it is left arbitrary intentionally.<ref name=":0" />


== See also ==
== See also ==
Line 449: Line 489:


== External links ==
== External links ==
* [http://sagittal.org/ Official site]
* [http://sagittal.org/ Official site] (with an introductory video)
* [http://forum.sagittal.org Sagittal Forum]
* [http://forum.sagittal.org Sagittal Forum]
* [http://sagittal.org/sagittal.pdf The original Xenharmonikon article (updated)]
* [http://sagittal.org/sagittal.pdf The original Xenharmonikon article (updated)]

Latest revision as of 03:48, 2 June 2026

Sagittal notation is a musical notation system capable of notating almost any conceivable tuning while preserving, as much as possible, the notation of harmonies across different tunings. It uses arrow-like symbols made up of four simple components whose visual size is proportional to their alteration and whose alterations sum. It was developed by Dave Keenan and George Secor with significant contributions from numerous others.

Flavors of Sagittal notation

Sagittal notation comes in two mutually compatible flavors.

Evo

The Evo flavor (short for "evolutionary", previously called "mixed") uses only single-shaft Sagittal symbols, e.g. , alone or in combination with conventional sharps and flats and their doubles. Only the large variant of the double sharp (U+E47D) is considered to be stylistically-compatible with Sagittal symbols. Evo is much easier to learn, but it results in a greater number of symbols on the sheet, which can give it a more cluttered appearance, particularly with chords, and it may be confusing when two symbols alter the same note in opposite directions.

A sub-flavor of Evo is Evo-SZ (Evo with Stein–Zimmermann). This is where, in tempered systems, any sagittals that are notating exactly half the alteration of a sharp or flat (most often ) are replaced by the Stein–Zimmermann semisharp and narrow reversed flat , and the corresponding combinations (most often and ) are replaced by and . The narrow variants of the fractional flats (U+E284) and (U+E285) are preferred because they preserve the Sagittal principle that the visual size of a symbol should indicate the relative size of its alteration and they reduce left-right confusion.

Revo

The Revo flavor (short for "revolutionary", previously called "pure") only requires one accidental per note. Revo therefore takes up less space on the sheet and presents a cleaner appearance, and it clearly indicates the direction of the overall alteration. It discards the conventional sharps and flats and their doubles and replaces them with these multi-shaft arrow-like symbols: . Adding a sharp or flat to a Sagittal is achieved by adding two more shafts, e.g. becomes and becomes , as expected. Apotome complements, that arise when the Sagital accidental alters in the opposite direction to the apotome, do not have a simple rule in Revo. One must simply memorize the complements one needs, as shown below. For example, becomes (flag swaps sides) while becomes (flag stays on same side).

Table of Spartan (most common)
apotome complements[1]
Symbol
Complement
Table of all apotome complements[2]
Symbol
Complement

Notation software support

Sibelius

Sagibelius 2.0 – plugins for using Sagittal notation in Sibelius 4 and up. By Jacob Barton. Hosted on this wiki. Donationware.

Sagibelius_2.0.zip

Lilypond

Plugin for Sagittal notation in Lilypond by Graham Breed

MuseScore

Sagittal accidentals are available in MuseScore via the Bravura font which implements the SMuFL standard. They can be accessed by opening the Master Palette and finding them in the Symbols section at the end.

Scala

Sagittal notation is available in Scala.

Dorico

Because Dorico is built by Steinberg Media, the same company that maintains the SMuFL standard, it supports Sagittal.

Scores in Sagittal notation

The symbol sets

Sagittal symbols come in 7 sets of increasing size and resolution, each one being an approximately equal division of the apotome (EDA). It is not necessary to learn all of the Sagittal sets to be able to compose with it. It is like a natural language in that you don't need to know every word in the dictionary to speak the language. Some words are used extremely rarely, but they must be there to provide full coverage. The Spartan set covers 90% of what most people want to do, and the Athenian extension covers most of the rest.

Sagittal accidentals are not intended to be combined with one another, except in the Prime Factor JI notation, as symbols representing useful combinations and powers of primes are already provided. An accidental can often be used to represent alternative commas that differ by 2 cents or less. In such cases the intended comma ratio may be determined by the pitch-class to which it is applied, or by the musical context. Alternatively, accent marks (from the Herculean and subsequent extensions) may be added to distinguish these commas.

This Desmos graph shows the theoretical minimum Sagittal set required to notate an EDO. However the standard Sagittal EDO notations don't always adhere to this because there are many other considerations that go into choosing a good EDO notation from among the possible ones.

Sharps/Flats

Using and (or for Revo flavor) is still technically Sagittal notation, however, it's just a reskin of the usual chain-of-fifths notation. Ditto for Stein-Zimmermann half-sharps and half-flats in hemipythagorean.

Relationships between sagittal symbol subsets (excluding accents)

Spartan

It is the simplest and coarsest of the Sagittal sets. The Spartan set has a maximum resolution of 13EDA[3], which is sufficient to notate 13*-limit just intonation (if used for JI), and all EDOs which are at most sharp-13, including all EDOs from 1 up to and including 111, the zeta peaks 130, 142, among many others. If used with tempered systems, it can be used to write music in the 23-limit, such as with 94edo.

*In this set, ratios of 13 are represented by reusing the accidentals for ratios of 35 (7*5). This is because the resulting interval, 512/315 (the ~13/8 interval) is only 0.4 ¢ (4096/4095) from just. The prime 13 will not have a distinct accidental up until the Olympian set.

The eight pairs of single-shaft accidentals shown below are sufficient to provide these capabilities when used alone, and to the left of the standard sharp, flat and their doubles (the Evo flavor).

As an alternative, the multi-shaft Spartans provides a complete set of stand-alone accidentals to replace each of the above combinations of a single-shaft Sagittal with a standard accidental (the Revo flavor). The standard natural is used alone in both Evo and Revo variants, but only to cancel a previous accidental when a barline will not suffice.

Sagittal extensions following Spartan allow notation of JI ratios with primes beyond 13 (and 13 proper), and more combinations of lower primes, as well as finer tone-fractions, degrees of larger EDOs, and more complex temperaments, all with single Sagittal accidentals. The same choice of Evo versus Revo is available with each extension.

Athenian

It is a handy symbol set, adding 10 symbol pairs to Spartan, with a total of 23 symbol pairs*, allowing for a maximum resolution of 21EDA. Early in the design of the Sagittal notation system, Secor and Keenan found that by extending the Spartan set with a further five pairs of single-shaft accidentals shown below an economical universal JI notation system could be defined, by dividing the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This set of thirteen pairs is called the Athenian set. If the divisions were made exactly equal (5.4136 ¢), this would be an example of Brahmagupta temperament, of which the two most salient EDOs are 217 and 224.

When used for JI, it defines the Standard Medium Precision JI, capable of writing in the 17-limit. In tempered systems, it can be used to write music in the 31-limit, such as with 217edo.

*There are two symbol pairs that are interchangeable in this level of precision[4], these being / and / . They will not become distinct until the next level of precision.

Trojan (12-EDO relative)

This is a special set that combines some Spartans with some from the Athenian extension and adds 3 more symbol pairs, which can be used to notate any tuning relative to 12-EDO with medium precision (typical error ±2 cents). It also provides the exact standard notations for the compton or 12N EDOs up to 192, in which the apotome is 100 cents.

Promethean

It adds 20 more symbols to Athenian symbol set, allowing for a maximum resolution of 47EDA. When used for JI, it defines the Standard High Precision JI capable of writing in the 23-limit or 41-limit, however, this JI notation is not recommended. It instead can be used to write in EDOs such as the zeta edos 270 and 311, the latter to write music in the 41-limit.

The accent sets

Fine-grained Sagittal notations can use accents, also called diacritics, to the left of a symbol or a bare shaft, to indicate very subtle distinctions in pitch.

Relationships between sagittal symbol subsets including accents

Herculean

It adds the schisma accent (a diacritic) to the Promethean symbol set, which can be stacked with the remaining alterations, allowing for a maximum resolution of 58EDA, of which a great edo is 612. When used for JI, it defines the Standard Ultra Precision JI capable of writing in the 23-limit with higher precision.

Olympian

It adds the mina accent to the Herculean symbol set, able to be stacked up to twice with the schisma and the remaining alterations, allowing for a maximum resolution of 233EDA, The zeta peak 2460edo has been used as a base to define the mina as an interval measure, and the Olympian set of intervals generally, due to its extremely precise 27-odd-limit palette. When used for JI, it defines the Standard Extreme Precision JI capable of writing in the 47-limit with great precision. It also is the smallest precision level that has an "exact" mapping for prime 13, thanks to the mina's appearance. 13/8 is now written as a major sixth minus the 35 large diesis and a mina. From C, this would be C - A .

Magrathean

It adds the tina accent to the Olympian symbol set, able to be stacked up to thrice with any of the symbols (three tinas make a ~mina), allowing for a whopping maximum resolution of 809EDA. The strict zeta peak 8539edo has been used to define the tina as an interval measure, due its insanely precise 27-odd-limit (and beyond) interval palette. When used for JI, it defines the Standard Insane Precision JI capable of writing in the 127-limit with astonishing precision. There is no level of precision higher than this, and it is unlikely that one will ever exist. Unless you want some hot sauce.[5]

Gallery of symbols

Spartan single-shaft

nai
7/5 kleisma up
nao
7/5 kleisma down
pai
5 comma up
pao
5 comma down
tai
7 comma up
tao
7 comma down
phai
25 small diesis up
phao
25 small diesis down
patai
35 medium diesis up
patao
35 medium diesis down
pakai
11 medium diesis up
pakao
11 medium diesis down
jatai
11 large diesis up
jatao
11 large diesis down
jakai
35 large diesis up
jakao
35 large diesis down

Phai/phao, patai/patao, pakai/pakao, jatai/jatao, and jakai/jakao can be spelled (respectively) fai/fao, gai/gao, vai/vao, wai/wao, and dai/dao. Pronunciation is intentionally loosely defined to accommodate various languages.

Spartan multi-shaft

Multi-shaft sagittals are only used in the Revo flavor of Sagittal.

sharp phao
sharp 25S down
flat phai
flat 25S up
sharp tao
sharp 7C down
flat tai
flat 7C up
sharp pao
sharp 5C down
flat pai
flat 5C up
sharp nao
sharp 7/5k down
flat nai
flat 7/5k up
sagi­sharp
sharp
sagi­flat
flat
sharp nai
sharp 7/5k up
flat nao
flat 7/5k down
sharp pai
sharp 5C up
flat pao
flat 5C down
sharp tai
sharp 7C up
flat tao
flat 7C down
sharp phai
sharp 25S up
flat phao
flat 25S down
sharp patai
sharp 35M up
flat patao
flat 35M down
sharp pakai
sharp 11M up
flat pakao
flat 11M down
sharp jatai
sharp 11L up
flat jatao
flat 11L down
sharp jakai
sharp 35L up
flat jakao
flat 35L down
double sharp phao
double sharp 25S down
double flat phai
double flat 25S up
double sharp tao
double sharp 7C down
double flat tai
double flat 7C up
double sharp pao
double sharp 5C down
double flat pai
double flat 5C up
double sharp nao
double sharp 7/5k down
double flat nai
double flat 7/5k up
sagi­double sharp
double sharp
sagi­double flat
double flat

Athenian extension single-shaft

ranai
11/7 kleisma up
ranao
11/7 kleisma down
sanai
17 comma up
sanao
17 comma down
kai
55 comma up
kao
55 comma down
jao
11/7 comma down
janao
11/5 small diesis down

Athenian extension multi-shaft

Multi-shaft sagittals are only used in the Revo flavor of Sagittal.

sharp janao
sharp 11/5S down
flat janai
flat 11/5S up
sharp jao
sharp 11/7C down
flat jai
flat 11/7C up
sharp kao
sharp 55C down
flat kai
flat 55C up
sharp sanao
sharp 17C down
flat sanai
flat 17C up
sharp ranao
sharp 11/7k down
flat ranai
flat 11/7k up
sharp ranai
sharp 11/7k up
flat ranao
flat 11/7k down
sharp sanai
sharp 17C up
flat sanao
flat 17C down
sharp kai
sharp 55C up
flat kao
flat 55C down
sharp jai
sharp 11/7C up
flat jao
flat 11/7C down
sharp janai
sharp 11/5S up
flat janao
flat 11/5S down
double sharp janao
double sharp 11/5S down
double flat janai
double flat 11/5S up
double sharp jao
double sharp 11/7C down
double flat jai
double flat 11/7C up
double sharp kao
double sharp 55C down
double flat kai
double flat 55C up
double sharp sanao
double sharp 17C down
double flat sanai
double flat 17C up
double sharp ranao
double sharp 11/7k down
double flat ranai
double flat 11/7k up

Trojan extension

zai
23C up
zao
23C down
prai
19/5C up
prao
19/5C down
pazai
23/5S up
pazao
23/5S down
sharp pazao
sharp 23S down
flat pazai
flat 23S up
sharp prao
sharp 19/5C down
flat prai
flat 19/5C up
sharp zao
sharp 23C down
flat zai
flat 23C up
sharp zai
sharp 23C up
flat zao
flat 23C down
sharp prai
sharp 19/5C up
flat prao
flat 19/5C down
sharp pazai
sharp 23/5S up
flat pazao
flat 23/5S down
double sharp pazao
double sharp 23S down
double flat pazai
double flat 23S up
double sharp prao
double sharp 19/5C down
double flat prai
double flat 19/5C up
double sharp zao
double sharp 23C down
double flat zai
double flat 23C up

Promethean extension single-shaft

Promethean symbols are rarely used, but they allow Sagittal to notate some EDOs with more than 400 pitches per octave.

rai
19s up
rao
19s down
sai
17k up
sao
17k down
srai
143C up
srao
143C down
shai
49/11C up
shao
49/11C down
razai
19C up
razao
19C down
ratai
19/7C up
ratao
19/7C down
satai
49S up
satao
49S down
sakai
23S up
sakao
23S down
phrai
13/5M up
phrao
13/5M down
jazai
19/11M up
jazao
19/11M down
jpai
49M up
jpao
49M down
prakai
49/5M up
prakao
49/5M down
ktai
49L up
ktao
49L down
khai
19/11L up
khao
19/11L down
rakhai
13/5L up
rakhao
13/5L down

Srai/srao, phrai/phrao, prakai/prakao, khai/khao, and rakhai/rakhao can be spelled (respectively) slai/slao, frai/frao, vrai/vrao, chai/chao, and rachai/rachao.

Promethean extension multi-shaft

Multi-shaft sagittals are only used in the Revo flavor of Sagittal.

sharp sakao
sharp 23S down
flat sakai
flat 23S up
sharp satao
sharp 49S down
flat satai
flat 49S up
sharp ratao
sharp 19/7C down
flat ratai
flat 19/7C up
sharp razao
sharp 19C down
flat razai
flat 19C up
sharp shao
sharp 49/11C down
flat shai
flat 49/11C up
sharp srao
sharp 143C down
flat srai
flat 143C up
sharp sao
sharp 17k down
flat sai
flat 17k up
sharp rao
sharp 19s down
flat rai
flat 19s up
sharp rai
sharp 19s up
flat rao
flat 19s down
sharp sai
sharp 17k up
flat sao
flat 17k down
sharp srai
sharp 143C up
flat srao
flat 143C down
sharp shai
sharp 49/11C up
flat shao
flat 49/11C down
sharp razai
sharp 19C up
flat razao
flat 19C down
sharp ratai
sharp 19/7C up
flat ratao
flat 19/7C down
sharp satai
sharp 49S up
flat satao
flat 49S down
sharp sakai
sharp 23S up
flat sakao
flat 23S down
sharp phrai
sharp 13/5M up
flat phrao
flat 13/5M down
sharp jazai
sharp 19/11M up
flat jazao
flat 19/11M down
sharp jpai
sharp 49M up
flat jpao
flat 49M down
sharp prakai
sharp 49/5M up
flat prakao
flat 49/5M down
sharp ktai
sharp 49L up
flat ktao
flat 49L down
sharp khai
sharp 19/11L up
flat khao
flat 19/11L down
sharp rakhai
sharp 13/5L up
flat rakhao
flat 13/5L down
double sharp sakao
double sharp 23S down
double flat sakai
double flat 23S up
double sharp satao
double sharp 49S down
double flat satai
double flat 49S up
double sharp ratao
double sharp 19/7C down
double flat ratai
double flat 19/7C up
double sharp razao
double sharp 19C down
double flat razai
double flat 19C up
double sharp shao
double sharp 49/11C down
double flat shai
double flat 49/11C up
double sharp srao
double sharp 143C down
double flat srai
double flat 143C up
double sharp sao
double sharp 17k down
double flat sai
double flat 17k up
double sharp rao
double sharp 19s down
double flat rai
double flat 19s up

Gallery of accents

Fine-grained Sagittal notations can use accents, also called diacritics, to the left of a sagittal or a bare shaft, to indicate very subtle distinctions in pitch.

Herculean extension accents

⁠ai
shaft up
⁠ao
shaft down

Ai and ao are not accents but bare shafts which do not alter the pitch. They are used when there would otherwise be bare accents.

bi bo
5s up 5s down

The interval is called a "schisma". The accent's shape is called "tick" (up or down).

Olympian extension accents

mi mo mimi momo
5⋅7⋅13n up 5⋅7⋅13n down 77/65n up 77/65n down

The average unit interval is called a "mina" (rhymes with ballerina) and is approximately 0.42 of a cent. The shapes are called "wing" and "bird".

Magrathean extension accents

qui quo quiqui quoquo mi mo
7²⋅11⋅19/5n up 7²⋅11⋅19/5n down 7³⋅17n up 7³⋅17n down 5⋅7⋅13n up 5⋅7⋅13n down
quimi quomo quiquimi quoquomo mimi momo
5²⋅11²/7n up 5²⋅11²/7n down 7⁴/25n up 7⁴/25n down 77/65n up 77/65n down
quimimi quomomo quiquimimi quoquomomo mimimi momomo
5²⋅17/7n up 5²⋅17/7n down 11⋅17/(5²⋅7)n up 11⋅17/(5²⋅7)n down 7²⋅11n up 7²⋅11n down
i o
77/(5⋅37)n up 77/(5⋅37)n down

The average unit interval is called a "tina" (rhymes with ballerina) and is approximately 0.14 of a cent. The new shapes are called "horn" and "wedge". Notice how 3 tinas is approximately equal to one mina, so the system just equates the 3. Either way, this is an insane level of pitch precision. The "i/o" accent, whose shape is called "dot", represents some unit fraction of a tina, often a half as shown, but it is intentionally left to be defined by the user.[5]

Prime approximations

Here are some approximations to primes from D, using the several precision sets available in JI. Values in parentheses are absolute error in cents from just; if none is shown, the notation is exact.

5 7 11 13 17 19 23 29 31
Spartan F C G B (0.42) D (2.971) F (2.380) A (3.008) C (6.223) D (1.691)
Athenian E A (1.009) C (0.339)
Promethean F A D (0.436)
Olympian B C (0.130) D
Magrathean C

See also

External links


View • Talk • EditMusical notation
Universal Sagittal notation
Just intonation Functional Just System • Ben Johnston's notation (Johnston–Copper notation) • Helmholtz–Ellis notation • Color notation
MOS scales Diamond-mos notation • KISS notation (Quasi-diatonic MOS notation)
Temperaments Chain-of-fifths notation • Stein–Zimmermann–Gould notation • Ups and downs notation • Syntonic–rastmic subchroma notation • Extended meantone notation • Fractional sharp notation
See musical notation for a longer list of systems by category. See Category:Notation for the most complete, comprehensive list, but not sorted by category.
  1. ↑ https://sagittal.org/sagittal.pdf p. 7 Figure 4
  2. ↑ Most of these are symbols are rarely used. is its own complement. https://sagittal.org/sagittal.pdf p. 24 Figure 13
  3. ↑ https://forum.sagittal.org/viewtopic.php?t=516
  4. ↑ https://sagittal.org/sagittal.pdf p.10
  5. ↑ 5.0 5.1 https://forum.sagittal.org/viewtopic.php?p=2714&hilit=bomb#p2714 "A tina is approximately 1/809th of an apotome, 1/8539th of an octave (a zeta-peak EDO), or 0.14 cents. The fractional-tina is generally half a tina but is intentionally arbitrary, because if you need any more precision than that, I have a bottle of Da' Bomb Beyond Insanity Hot Sauce with your name on it"
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