Sagittal notation: Difference between revisions

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* [[: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 Athenian ~~symbol set~~ ==

== The Symbol sets ==

Early in the design of the Sagittal notation system, Secor and Keenan found that an economical JI notation system could be defined by dividing the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This ~~required only five~~ pairs of up and down microtonal accidentals, although a few others were added for convenience in alternative spellings. This 10-symbol set ([[Sagittal notation#Athenian extension single-shaft|shown below]]), in combination with the Spartan symbol set which it extends, is called the Athenian ~~symbol~~ set~~. It exactly notates many common 11-limit ratios and the 17th harmonic, while also approximating many common 13-limit ratios within ±0.4{{c}}~~. If the divisions were made exactly equal, this would be an example of [[Brahmagupta]] temperament.

=== 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.

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. 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.

=== 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.

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.

== Gallery ==

@@ Line 56: / Line 56: @@
 * [[: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 Athenian symbol set ==
+== The Symbol sets ==
-Early in the design of the Sagittal notation system, Secor and Keenan found that an economical JI notation system could be defined by dividing the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only five pairs of up and down microtonal accidentals, although a few others were added for convenience in alternative spellings. This 10-symbol set ([[Sagittal notation#Athenian extension single-shaft|shown below]]), in combination with the Spartan symbol set which it extends, is called the Athenian symbol set. It exactly notates many common 11-limit ratios and the 17th harmonic, while also approximating many common 13-limit ratios within ±0.4{{c}}. If the divisions were made exactly equal, this would be an example of [[Brahmagupta]] temperament.
+=== 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.
+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. 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.
+=== 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.
+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.
 == Gallery ==