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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>
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|<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 ==
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== 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 320: Line 306:
{{sge| )¦¦¦\\ | )!!!// | rakh |  885735/851968  | 13/5L }}
{{sge| )¦¦¦\\ | )!!!// | rakh |  885735/851968  | 13/5L }}


{{sge| )~X | )~Y    | sak |  24/23  | 23S | mid=48px }}
{{sge| )~X | )~Y    | sak |  24/23  | 23S }}
{{sge| ~~X | ~~Y    | sat |  6561/6272  | 49S | mid=48px }}
{{sge| ~~X | ~~Y    | sat |  6561/6272  | 49S }}
{{sge| )/X | )\Y    | rat |  5103/4864  | 19/7C | mid=48px }}
{{sge| )/X | )\Y    | rat |  5103/4864  | 19/7C }}
{{sge| (X  | (Y    | raz |  19/18  | 19C | mid=48px }}
{{sge| (X  | (Y    | raz |  19/18  | 19C }}
{{sge| ~X) | ~Y)    | sh  |  11907/11264  | 49/11C | mid=48px }}
{{sge| ~X) | ~Y)    | sh  |  11907/11264  | 49/11C }}
{{sge| ~X\  | ~Y/  | sr  |  34749/32768  | 143C | mid=48px }}
{{sge| ~X\  | ~Y/  | sr  |  34749/32768  | 143C }}
{{sge| )//X | )\\Y | s  |  17/16  | 17k | mid=48px }}
{{sge| )//X | )\\Y | s  |  17/16  | 17k }}
{{sge| (X~ | (Y~    | r  |  81/76  | 19s | mid=48px }}
{{sge| (X~ | (Y~    | r  |  81/76  | 19s }}


=== Herculean extension diacritics ===
== 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.
<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"
!{{sagittal||}}
|- style="text-align: center;"
!{{sagittal|!}}
| {{sagittal|'|size=500%}}
|-
| {{sagittal|.|size=500%}}
|ai
|- style="text-align: center;"
|ao
|-
| unison up
| unison down
|}
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.
{| class="wikitable"
!{{sagittal|'}}
!{{sagittal|.}}
|-
|bi
|bi
|bo
|bo
|-
|-
|[https://en.xen.wiki/w/Schisma 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 364: 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 387: 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 409: 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 431: 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 457: 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)]
Retrieved from "https://en.xen.wiki/w/Sagittal_notation"