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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 ==
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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]]
[[File:SagittalEulerDiagram1.png|500px|thumb|right|Relationships between sagittal symbol subsets (excluding accents)]]


=== Spartan ===
=== Spartan ===
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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>
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>
</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|(| }}
|}


== Gallery of symbols ==
== Gallery of symbols ==
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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" />
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>
</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|(| }}
|}


== See also ==
== See also ==
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== 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"