Sagittal notation

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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

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 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 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

Spartan single-shaft

⁠ ⁠
nai
7/5 kleisma up
⁠ ⁠
nao
7/5 kleisma down
⁠ ⁠
pai
5 comma up
⁠ ⁠
pao
5 comma down
⁠ ⁠
tai
7 comma up
⁠ ⁠
tao
7 comma down
⁠ ⁠
phai
25 small diesis up
⁠ ⁠
phao
25 small diesis down
⁠ ⁠
patai
35 medium diesis up
⁠ ⁠
patao
35 medium diesis down
⁠ ⁠
pakai
11 medium diesis up
⁠ ⁠
pakao
11 medium diesis down
⁠ ⁠
jatai
11 large diesis up
⁠ ⁠
jatao
11 large diesis down
⁠ ⁠
jakai
35 large diesis up
⁠ ⁠
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
⁠ ⁠
jai
11/7 comma up
⁠ ⁠
jao
11/7 comma down
⁠ ⁠
janai
11/5 small diesis up
⁠ ⁠
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

See also

External links