Sagittal notation: Difference between revisions

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== 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. With '''Spartan''' and '''Athenian''', it 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.

=== Spartan ===

It is ~~not necessary to learn~~ the ~~complete~~ Sagittal ~~microtonal notation system~~. The Spartan set is sufficient to notate 13-limit just intonation, 1~~/12-tones~~, ~~50 common equal divisions of~~ the ~~octave (EDOs)~~, and ~~their related higher-rank temperaments~~.

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 135, and 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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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. ~~The error~~ is only ±0.4 ¢. Sagittal extensions following Spartan allow notation of JI ratios with primes beyond 13, 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.

In this set, ratios of 13 are represented by reusing the accidentals for ratios of 35. 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 ===

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.

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.

~~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~~.

=== 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 notation is not recommended. It instead can be used to write in EDOs such as [[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>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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* [https://www.panix.com/~mj/tuning/EDO-Charts.html Interactive EDO Charts] by [[Mark Johnson]]

== References ==

[[Category:Notation]]

[[Category:Sagittal notation| ]]

@@ Line 45: / Line 45: @@
 == 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. With '''Spartan''' and '''Athenian''', it 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.
 === Spartan ===
-It is not necessary to learn the complete Sagittal microtonal notation system. The Spartan set is sufficient to notate 13-limit just intonation, 1/12-tones, 50 common equal divisions of the octave (EDOs), and their related higher-rank temperaments.
+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 135, and 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 52: / Line 54: @@
 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. The error is only ±0.4 ¢. Sagittal extensions following Spartan allow notation of JI ratios with primes beyond 13, 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.
+In this set, ratios of 13 are represented by reusing the accidentals for ratios of 35. 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 ===
-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.
+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.
-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.
+=== 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 notation is not recommended. It instead can be used to write in EDOs such as [[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>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 140: / Line 156: @@
 * [https://www.panix.com/~mj/tuning/EDO-Charts.html Interactive EDO Charts] by [[Mark Johnson]]
+== References ==
 {{Navbox notation}}
 [[Category:Notation]]
 [[Category:Sagittal notation| ]] <!-- Main article -->