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.
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
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| nao
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| 7/5 kleisma down
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| phao
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| 25 small diesis down
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| patao
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| 35 medium diesis down
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| pakao
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| 11 medium diesis down
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| jatao
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| 11 large diesis down
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| jakao
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| 35 large diesis down
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Spartan multi-shaft
Multi-shaft sagittals are only used in the Revo flavor of Sagittal.
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| sharp phao
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| sharp 25S down
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| sharp tao
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| sharp 7C down
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| sharp pao
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| sharp 5C down
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| sharp nao
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| sharp 7/5k down
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| flat nao
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| flat 7/5k down
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| sharp pai
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| sharp 5C up
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| flat pao
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| flat 5C down
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| sharp tai
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| sharp 7C up
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| flat tao
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| flat 7C down
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| flat phao
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| flat 25S down
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| flat patao
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| flat 35M down
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| flat pakao
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| flat 11M down
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| flat jatao
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| flat 11L down
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| flat jakao
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| flat 35L down
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| double sharp phao
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| double sharp 25S down
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| double flat phai
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| double flat 25S up
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| double sharp tao
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| double sharp 7C down
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| double flat tai
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| double flat 7C up
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| double sharp pao
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| double sharp 5C down
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| double flat pai
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| double flat 5C up
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| double sharp nao
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| double sharp 7/5k down
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| double flat nai
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| double flat 7/5k up
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| sagidouble sharp
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| double sharp
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| sagidouble flat
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| double flat
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Athenian extension single-shaft
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| ranao
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| 11/7 kleisma down
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| janao
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| 11/5 small diesis down
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Athenian extension multi-shaft
Multi-shaft sagittals are only used in the Revo flavor of Sagittal.
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| sharp janao
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| sharp 11/5S down
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| sharp jao
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| sharp 11/7C down
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| sharp kao
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| sharp 55C down
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| sharp sanao
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| sharp 17C down
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| sharp ranao
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| sharp 11/7k down
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| flat ranao
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| flat 11/7k down
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| flat sanao
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| flat 17C down
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| flat kao
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| flat 55C down
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| flat jao
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| flat 11/7C down
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| flat janao
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| flat 11/5S down
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| double sharp janao
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| double sharp 11/5S down
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| double flat janai
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| double flat 11/5S up
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| double sharp jao
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| double sharp 11/7C down
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| double flat jai
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| double flat 11/7C up
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| double sharp kao
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| double sharp 55C down
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| double flat kai
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| double flat 55C up
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| double sharp sanao
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| double sharp 17C down
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| double flat sanai
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| double flat 17C up
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| double sharp ranao
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| double sharp 11/7k down
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| double flat ranai
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| double flat 11/7k up
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See also
External links
