Sagittal notation: Difference between revisions

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Added Promethean single-shaft, Herculean, Olympian, and Magrathean extensions to the gallery, missing Promethean multi-shaft, and formatting the tables correctly
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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 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.
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 ===
=== Sharps/Flats ===
Line 77: Line 79:


=== Magrathean ===
=== 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 ==
== Gallery ==
Line 143: Line 145:
{{sge|//X  |\\Y  | ran |  896/891  | 11/7k | mid=48px }}
{{sge|//X  |\\Y  | ran |  896/891  | 11/7k | mid=48px }}
</div>
</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 ==
== See also ==

Revision as of 18:25, 20 August 2025

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. , 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 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 . When the Sagittal part alters in the opposite direction to the sharp or flat part, the rules are not so simple, e.g. becomes and becomes ; one must simply learn these apotome complements.

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 are defined in 6 "official" symbol sets, each one being defined as spliting the 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.

You can look up what the minimum precision is required to write an EDO in this Desmos graph.

Sharps/Flats

Using and (or 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[1], 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 130, 142, among many others.

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.

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 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.

*There are two symbol pairs that are interchangeable in this level of precision[2], these being / and / . 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 edos 270 and 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 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.[3]

Gallery

Spartan single-shaft

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

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

Promethean extension single-shaft

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

Promethean extension multi-shaft

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

TBA

Herculean extension diacritics

| |
ai ao
unison

Ai and ao are used to write bare diacritics in the following extensions, like the schisma, schismina, and tina.

bi bo
5s up 5s down

Olympian extension diacritics

mi mo mimi momo
35/13sx up 35/13sx down 77/65sx up 77/65sx down

The diacritic is called "mina".

Magrathean extension diacritics

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 35/13sx up 35/13sx down 3025/3024 up 3025/3024 up 2401/2400 up 2401/2400 down 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.[3]

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://forum.sagittal.org/viewtopic.php?t=516
  2. ↑ https://sagittal.org/sagittal.pdf p.10
  3. ↑ 3.0 3.1 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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