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<span style="display: block; text-align: right;">[[de:Sagittalnotation]]</span>
[[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. 

== 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. {{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.

<span id="Evo-SZ"></span> 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.

=== 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| !!!) }}. When the Sagittal part alters in the opposite direction to the sharp or flat part, the rules are not so simple, e.g. {{sagittal| \! }}{{sagittal| # }} becomes {{sagittal| ||\ }} and {{sagittal| |) }}{{sagittal| b }} becomes {{sagittal| !!) }}; one must simply learn these ''[[2187/2048#Notation|apotome]] complements''.

== Notation software support ==
=== Sibelius ===
Sagibelius 2.0 – plugins for using Sagittal notation in Sibelius 4 and up. By [http://www.jacobbarton.net/2011/10/sagibelius-2-0-released/ Jacob Barton]. Hosted on this wiki. Donationware.

[[:File:Sagibelius_2.0.zip|Sagibelius_2.0.zip]]

=== Lilypond ===
[http://x31eq.com/lilypond/ Plugin for Sagittal notation in Lilypond] by Graham Breed

=== MuseScore ===
Sagittal accidentals are available in MuseScore via the [https://www.smufl.org/fonts/ 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 Sagittal Songbook]]
** [http://oneforall.ytmnd.com One for All]
** [[land_urchin|Land Urchin]]
** [[Clouds_(Andrew_Heathwaite)|Clouds]]
** [[Prayer of Thanks]]

* [[:File:sunday3.pdf|Sunday Pipes]] in [[22edo]] by [[Mats Öljare]]
* Tibia in [[22edo]] by [[Paul Erlich]] ([http://www.youtube.com/watch?v=d44Lfp9lAG8 Listen]). Sagittal score [[:File:TIBIA.pdf|in F||\]] or [[:File:tibia_in_g.pdf|in G]] (contains errors in measures 9, 19 and 20)
* [[:File:Ivor_Darreg,_Suite.pdf|On the Enharmonic Tetrachord (from Suite, Op. 62)]], in [[22edo]], by [[Ivor Darreg]]. Originally printed in the Spring 1975 issue of Xenharmonikon in quarter-tone notation. Transcribed to Sagittal by [[Juhani Nuorvala]].([https://www.youtube.com/watch?v=DvHvza1vtfo Listen])

== 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 the complete Sagittal microtonal notation system. The '''Spartan''' and '''Athenian''' sets will be more than enough for most purposes. 
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.

[https://www.desmos.com/calculator/iehdworjko You can look up what the minimum precision is required to write an EDO in this Desmos graph.]

=== Sharps/Flats ===
Using {{sagittal| # }} and {{sagittal| b }} (or {{sagittal|/||\}} {{sagittal|\!!/}} for Revo flavor) is still technically Sagittal notation.

=== Spartan ===
It is the simplest and coarsest of the Sagittal sets. The Spartan set has a maximum resolution of 13EDA<ref>https://forum.sagittal.org/viewtopic.php?t=516</ref>, 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 111, the [[Zeta peak edo|zeta peak]] [[130edo|130]], [[142edo|142]], among many others.

The eight pairs of single-shaft accidentals [[Sagittal notation#Spartan single-shaft|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.

In this set, ratios of 13 are represented by reusing the accidentals for ratios of 35 (7*5). Thusly, [[8505/8192]] (the ~13/8 interval) is only 0.4 ¢ ([[4096/4095]]) from just. 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 [[Sagittal notation#Athenian extension single-shaft|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.

<nowiki>*</nowiki>There are two symbol pairs that are interchangeable in this level of precision<ref>https://sagittal.org/sagittal.pdf p.10</ref>, these being  {{sagittal|(|}} {{sagittal|(!}} / {{sagittal||\}} {{sagittal|!/}}   and   {{sagittal|)||~}} {{sagittal|)!!~}} / {{sagittal|/||}} {{sagittal|\!!}} . They will not become distinct until the next level of precision.

=== 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, 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]].

=== 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. 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 [[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. When used for JI, it defines the ''Standard Extreme Precision JI'' capable of writing in the 47-limit with great precision.

=== Magrathean ===
It adds the ''tina'' diacritic (1 step of [[8539edo]]) 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. 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>

== Gallery ==
=== Spartan single-shaft ===
<div style="margin-bottom: 20px;">
{{sge|  ¦( |  !( | n   | 5120/5103 | 7/5 kleisma }}
{{sge| /¦  | \!  | p   |   81/80   | 5 comma }}
{{sge|  ¦) |  !) | t   | 64/63| 7 comma }}
{{sge|//¦  |\\!  | ph  | 6561/6400 | 25 small diesis }}
{{sge| /¦) | \!) | pat |   36/35   | 35 medium diesis }}
{{sge| /¦\ | \!/ | pak |   33/32   | 11 medium diesis }}
{{sge| (¦) | (!) | jat |  729/704  | 11 large diesis }}
{{sge| (¦\ | (!/ | jak | 8505/8192 | 35 large diesis }}
</div>

=== Spartan multi-shaft ===
Multi-shaft sagittals are only used in the [[#Revo|Revo]] flavor of Sagittal.
<div style="margin-bottom: 20px;">
{{sge| )¦¦( | )!!( | ph  | 6561/6400 | 25S }}
{{sge|  ¦¦) |  !!) | t   |   64/63   | 7C }}
{{sge|  ¦¦\ |  !!/ | p   |   81/80   | 5C }}
{{sge| /¦¦) | \!!) | n   | 5120/5103 | 7/5k }}
{{sge| /¦¦\ | \!!/ }}
{{sge|  ¦¦¦( |  !!!( | n   | 5120/5103 | 7/5k }}
{{sge| /¦¦¦  | \!!!  | p   |   81/80   | 5C }}
{{sge|  ¦¦¦) |  !!!) | t   |   64/63   | 7C }}
{{sge|//¦¦¦  |\\!!!  | ph  | 6561/6400 | 25S }}
{{sge| /¦¦¦) | \!!!) | pat |   36/35   | 35M }}
{{sge| /¦¦¦\ | \!!!/ | pak |   33/32   | 11M }}
{{sge| (¦¦¦) | (!!!) | jat |  729/704  | 11L }}
{{sge| (¦¦¦\ | (!!!/ | jak | 8505/8192 | 35L }}
{{sge| )X( | )Y( | ph  | 6561/6400 | 25S  | mid=48px }}
{{sge|  X) |  Y) | t   |   64/63   | 7C   | mid=48px }}
{{sge|  X\ |  Y/ | p   |   81/80   | 5C   | mid=48px }}
{{sge| /X) | \Y) | n   | 5120/5103 | 7/5k | mid=48px }}
{{sge| /X\ | \Y/ | mid=48px }}
</div>

=== Athenian extension single-shaft ===
<div style="margin-bottom: 20px;">
{{sge| )¦( | )!( | ran |  896/891  | 11/7 kleisma }}
{{sge| ~¦( | ~!( | san | 4131/4096 | 17 comma }}
{{sge|  ¦\ |  !/ | k   |   55/54   | 55 comma }}
{{sge| (¦  | (!  | j   |45927/45056| 11/7 comma }}
{{sge| (¦( | (!( | jan |   45/44   | 11/5 small diesis }}
</div>

=== Athenian extension multi-shaft ===
Multi-shaft sagittals are only used in the [[#Revo|Revo]] flavor of Sagittal.
<div style="margin-bottom: 20px;">
{{sge| ~¦¦( | ~!!( | jan |   45/44   | 11/5S }}
{{sge| )¦¦~ | )!!~ | j   |45927/45056| 11/7C }}
{{sge| /¦¦  | \!!  | k   |   55/54   | 55C }}
{{sge| (¦¦( | (!!( | san | 4131/4096 | 17C }}
{{sge|//¦¦  |\\!!  | ran |  896/891  | 11/7k }}
{{sge| )¦¦¦( | )!!!( | ran |  896/891  | 11/7k }}
{{sge| ~¦¦¦( | ~!!!( | san | 4131/4096 | 17C }}
{{sge|  ¦¦¦\ |  !!!/ | k   |   55/54   | 55C }}
{{sge| (¦¦¦  | (!!!  | j   |45927/45056| 11/7C }}
{{sge| (¦¦¦( | (!!!( | jan |   45/44   | 11/5S }}
{{sge| ~X( | ~Y( | jan |   45/44   | 11/5S | mid=48px }}
{{sge| )X~ | )Y~ | j   |45927/45056| 11/7C | mid=48px }}
{{sge| /X  | \Y  | k   |   55/54   | 55C | mid=48px }}
{{sge| (X( | (Y( | san | 4131/4096 | 17C | mid=48px }}
{{sge|//X  |\\Y  | ran |  896/891  | 11/7k | mid=48px }}
</div>

=== Promethean extension single-shaft ===
{{sge| )¦  | )!  | r |  513/512  | 19s | mid=48px }}
{{sge| ~¦  | ~!  | s |  2187/2176  | 17k | mid=48px }}
{{sge| )~¦  | )~!  | sr |  144/143  | 143C | mid=48px }}
{{sge| ~~¦  | ~~!  | sh |  99/88  | 49/11C | mid=48px }}
{{sge| )¦~  | )!~  | raz |  19683/19456  | 19C | mid=48px }}
{{sge| )¦)  | )!)  | rat |  57/56  | 19/7C | mid=48px }}
{{sge| ~¦)  | ~!)  | sat |  49/48  | 49S | mid=48px }}
{{sge| ~¦\  | ~!/  | sak |  16767/16384  | 23S | mid=48px }}
{{sge| )//¦  | )\\!  | phr |  416/405   | 13/5M | mid=48px }}
{{sge| (¦~  | (!~  | jaz |  176/171   | 19/11M | mid=48px }}
{{sge| (/¦  | (\!  | jp |  4096/3969   | 49M | mid=48px }}
{{sge| )/¦\  | )\!/  | prak |  405/392   | 49/5M | mid=48px }}
{{sge| ¦\) | !/)  | kt |  8680203/8388608   | 49L | mid=48px }}
{{sge| ¦\\|!//|kh|373977/360448|19/11L|| mid=48px }}
{{sge| )¦\\ | )!//  | rakh |  885735/851968   | 13/5L | mid=48px }}

=== Promethean extension multi-shaft ===
Multi-shaft sagittals are only used in the [[#Revo|Revo]] flavor of Sagittal.

TBA

=== Herculean extension diacritics ===
{| class="wikitable"
!<nowiki>|</nowiki>
!<nowiki>|</nowiki>
|-
|ai
|ao
|-
| colspan="2" |unison
|}
Ai and ao are used to write bare diacritics in the following extensions, like the schisma, schismina, and tina.
{| class="wikitable"
!{{sagittal|'}}
!{{sagittal|.}}
|-
|bi
|bo
|-
|[https://en.xen.wiki/w/Schisma 5s up]
|5s down
|}

=== Olympian extension diacritics ===
{| class="wikitable"
!{{sagittal|`}}
!{{sagittal|,}}
!{{sagittal|``}}
!{{sagittal|,,}}
|-
|mi
|mo
|mimi
|momo
|-
|[https://en.xen.wiki/w/4096/4095 35/13sx] up
|35/13sx down
|[https://en.xen.wiki/w/2080/2079 77/65sx] up
|77/65sx down
|}
The diacritic is called "mina".

=== Magrathean extension diacritics ===
{| class="wikitable"
!{{sagittal|@1}}
!{{sagittal|l1}}
!{{sagittal|@2}}
!{{sagittal|l2}}
!{{sagittal|`}}
!{{sagittal|,}}
!{{sagittal|@4}}
!{{sagittal|l4}}
!{{sagittal|@5}}
!{{sagittal|l5}}
!{{sagittal|``}}
!{{sagittal|,,}}
!{{sagittal|@7}}
!{{sagittal|l7}}
!{{sagittal|@8}}
!{{sagittal|l8}}
!{{sagittal|@8}}
!{{sagittal|l8}}
!{{sagittal|@9}}
!{{sagittal|l9}}
!{{sagittal|@.}}
!{{sagittal|l.}}
|-
|qui
|quo
|quiqui
|quoquo
|mi
|mo
|quimi
|quomo
|quiquimi
|quoquomo
|mimi
|momo
|quimimi
|quomomo
|quiquimimi
|quoquomomo
|quiquimimi
|quoquomomo
|mimimi
|momomo
|i
|o
|-
|10241/10240 up
|10241/10240 up
|5832/5831 up
|5832/5831 down
|[https://en.xen.wiki/w/4096/4095 35/13sx] up
|35/13sx down
|3025/3024 up
|3025/3024 up
|2401/2400 up
|2401/2400 down
|[https://en.xen.wiki/w/2080/2079 77/65sx] up
|77/65sx down
|1701/1700 up
|1701/1700 down
|382976/382725 up
|382976/382725 down
|382976/382725 up
|382976/382725 down
|131072/130977  up
|131072/130977  down
|1515591/1515520  up
|1515591/1515520  down
|}
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 fractions of a cent. The "i/o" diacritic is a fraction of a tina, left arbitrary intentionally.<ref name=":0" />

== See also ==
* [[Sagitype]]
* [[Pain free guide to Sagittal]] by [[William Lynch]]
* [[:File:24 Edo.pdf]] – Sagittal notation guide for 24edo by William Lynch (download: [{{filepath:24 Edo.pdf}} ''24_Edo.pdf''])
* [[Introductory_examples_in_Sagittal_notation|Introductory examples]] by [[Hans Straub]]

== External links ==
* [http://sagittal.org/ Official site]
* [http://forum.sagittal.org Sagittal Forum]
* [http://sagittal.org/sagittal.pdf The original Xenharmonikon article (updated)]
* [http://sagittal.org/gift/GiftOfTheGods.htm Gift of the Gods: a Mythical introduction to Sagittal notation]
* [http://sagittal.org/Sagittal%20Standard%20JI%20Notation%20Calculator%20Spreadsheet.xlsx spreadsheet-based calculator for Sagittal JI notation]
* [http://sagittal.org/Sagittal-SMuFL-Map.pdf Sagittal-SMuFL-Map, a table of every Sagittal symbol]
* [https://andrewmeronek.com/music-resources/sagittal-chord-lists/ Sagittal chord chart] by [[Andrew Meronek]]
* [https://www.panix.com/~mj/tuning/EDO-Charts.html Interactive EDO Charts] by [[Mark Johnson]]
{{Navbox notation}}

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