Sagittal notation: Difference between revisions

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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.

Using {{sagittal| # }} and {{sagittal| b }} (or {{sagittal|/||\}} {{sagittal|\!!/}} for Revo flavor) is still ''technically'' Sagittal notation, however, it's just a reskin.

=== 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]] [[~~135edo~~|~~135~~]], [[142edo|142]], among many others.

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).

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<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.

=== Trojan (12-EDO relative) ===

This is a special extension to Spartan which adds 3 more symbol sets, 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 your needs, maybe Athenian proves to be more useful.

=== Promethean ===

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=== 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>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'' 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 ==

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=== Athenian extension single-shaft ===

<div style="margin-bottom: 20px;">{{sge| )¦( | )!( | ran | 896/891 | 11/7 kleisma }}

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</div>

=== Trojan extension ===

=== Promethean extension single-shaft ===

=== Promethean extension multi-shaft ===

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

{{sge| )¦¦¦\\ | )!!!// | rakh | 885735/851968 | 13/5L | mid=48px }}

=== Herculean extension diacritics ===

{| class="wikitable"

!<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

|5832/5831 up

|5832/5831 down

|[https://en.xen.wiki/w/4096/4095 35/13sx] up

|35/13sx down

|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 ==

@@ Line 47: / Line 47: @@
 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.
+Using {{sagittal| # }} and {{sagittal| b }} (or {{sagittal|/||\}} {{sagittal|\!!/}} for Revo flavor) is still ''technically'' Sagittal notation, however, it's just a reskin.
 === 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]] [[135edo|135]], [[142edo|142]], among many others.
+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).
@@ Line 66: / Line 68: @@
 <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.
+=== Trojan (12-EDO relative) ===
+This is a special extension to Spartan which adds 3 more symbol sets, 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 your needs, maybe Athenian proves to be more useful.
 === Promethean ===
@@ Line 77: / Line 82: @@
 === 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>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'' 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 ==
@@ Line 116: / Line 121: @@
 === Athenian extension single-shaft ===
-<div style="margin-bottom: 20px;">
+<div style="margin-bottom: 20px;">{{sge| )¦( | )!( | ran |  896/891  | 11/7 kleisma }}
-{{sge| )¦( | )!( | ran |  896/891  | 11/7 kleisma }}
 {{sge| ~¦( | ~!( | san | 4131/4096 | 17 comma }}
 {{sge|  ¦\ |  !/ | k   |   55/54   | 55 comma }}
@@ Line 143: / Line 147: @@
 {{sge|//X  |\\Y  | ran |  896/891  | 11/7k | mid=48px }}
 </div>
+=== Trojan extension ===
+{{sge|¦~|!~|z|736/729|23CT| mid=48px }}
+{{sge|)/¦|)\!|pr|41553/40960 |19/5CT| mid=48px }}
+{{sge|/¦~|\!~|paz|46/45|23/5ST| mid=48px }}
+{{sge|¦¦~|!!~|paz|46/45|23ST| mid=48px }}
+{{sge|)¦¦)|)!!)|pr|41553/40960 |19/5CT| mid=48px }}
+{{sge|/¦¦~|\!!~|z|736/729|23CT| mid=48px }}
+{{sge|¦¦¦~|!!!~|z|736/729|23CT| mid=48px }}
+{{sge|)/¦¦¦|)\!!!|pr|41553/40960 |19/5CT| mid=48px }}
+{{sge|/¦¦¦~|\!!!~|paz|46/45|23/5ST| mid=48px }}
+{{sge|X~|Y~|paz|46/45|23ST| mid=48px }}
+{{sge|)X)|)Y)|pr|41553/40960 |19/5CT| mid=48px }}
+{{sge|/X~|\Y~|z|736/729|23CT| mid=48px }}
+=== 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.
+{{sge| )~¦¦ | )~!!    | sak |  24/23  | 23S | mid=48px }}
+{{sge| ~~¦¦ | ~~!!    | sat |  6561/6272  | 49S | mid=48px }}
+{{sge| )/¦¦ | )\!!    | rat |  5103/4864  | 19/7C | mid=48px }}
+{{sge| (¦¦  | (!!     | raz |  19/18  | 19C | mid=48px }}
+{{sge| ~¦¦) | ~!!)    | sh  |  11907/11264  | 49/11C | mid=48px }}
+{{sge| ~¦¦\  | ~!!/   | sr  |  34749/32768  | 143C | mid=48px }}
+{{sge| )//¦¦ | )\\!!  | s  |  17/16  | 17k | mid=48px }}
+{{sge| (¦¦~ | (!!~    | r  |  81/76  | 19s | mid=48px }}
+{{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 }}
+{{sge| )~X | )~Y    | sak |  24/23  | 23S | mid=48px }}
+{{sge| ~~X | ~~Y    | sat |  6561/6272  | 49S | mid=48px }}
+{{sge| )/X | )\Y    | rat |  5103/4864  | 19/7C | mid=48px }}
+{{sge| (X  | (Y     | raz |  19/18  | 19C | mid=48px }}
+{{sge| ~X) | ~Y)    | sh  |  11907/11264  | 49/11C | mid=48px }}
+{{sge| ~X\  | ~Y/   | sr  |  34749/32768  | 143C | mid=48px }}
+{{sge| )//X | )\\Y | s  |  17/16  | 17k | mid=48px }}
+{{sge| (X~ | (Y~    | r  |  81/76  | 19s | mid=48px }}
+=== 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 ==