Sagittal notation is a musical notation system capable of notating almost any conceivable tuning while preserving, as much as possible, the notation of harmonies across different tunings. It uses arrow-like symbols made up of four simple components whose visual size is proportional to their alteration and whose alterations sum. It was developed by Dave Keenan and George Secor with significant contributions from numerous others.
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, in tempered systems, 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 , as expected. Apotomecomplements, that arise when the Sagital accidental alters in the opposite direction to the apotome, do not have a simple rule in Revo. One must simply memorize the complements one needs, as shown below. For example, becomes (flag swaps sides) while becomes (flag stays on same side).
Table of Spartan (most common) apotome complements[1]
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.
Sagittal symbols come in 7 sets of increasing size and resolution, each one being an approximately equal division of the apotome (EDA). It is not necessary to learn all of the Sagittal sets to be able to compose with it. It is like a natural language in that you don't need to know every word in the dictionary to speak the language. Some words are used extremely rarely, but they must be there to provide full coverage. The Spartan set covers 90% of what most people want to do, and the Athenian extension covers most of the rest.
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 pitch-class 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.
This Desmos graph shows the theoretical minimum Sagittal set required to notate an EDO. However the standard Sagittal EDO notations don't always adhere to this because there are many other considerations that go into choosing a good EDO notation from among the possible ones.
Sharps/Flats
Using and (or for Revo flavor) is still technically Sagittal notation, however, it's just a reskin of the usual chain-of-fifths notation. Ditto for Stein-Zimmermann half-sharps and half-flats in hemipythagorean.
Relationships between sagittal symbol subsets (excluding accents)
Spartan
It is the simplest and coarsest of the Sagittal sets. The Spartan set has a maximum resolution of 13EDA[3], 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 and including 111, the zeta peaks130, 142, among many others. If used with tempered systems, it can be used to write music in the 23-limit, such as with 94edo.
*In this set, ratios of 13 are represented by reusing the accidentals for ratios of 35 (7*5). This is because the resulting interval, 512/315 (the ~13/8 interval) is only 0.4 ¢ (4096/4095) from just. The prime 13 will not have a distinct accidental up until the Olympian set.
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.
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. In tempered systems, it can be used to write music in the 31-limit, such as with 217edo.
*There are two symbol pairs that are interchangeable in this level of precision[4], these being / and / . They will not become distinct until the next level of precision.
Trojan (12-EDO relative)
This is a special set that combines some Spartans with some from the Athenian extension and adds 3 more symbol pairs, which can be used to notate any tuning relative to 12-EDO with medium precision (typical error ±2 cents). It also provides the exact standard notations for the compton or 12N EDOs up to 192, in which the apotome is 100 cents.
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 or 41-limit, however, this JI notation is not recommended. It instead can be used to write in EDOs such as the zeta edos270 and 311, the latter to write music in the 41-limit.
The accent sets
Fine-grained Sagittal notations can use accents, also called diacritics, to the left of a symbol or a bare shaft, to indicate very subtle distinctions in pitch.
Relationships between sagittal symbol subsets including accents
Herculean
It adds the schisma accent (a diacritic) to the Promethean symbol set, which can be stacked with the remaining alterations, allowing for a maximum resolution of 58EDA, of which a great edo is 612. 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 accent 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, The zeta peak 2460edo has been used as a base to define the mina as an interval measure, and the Olympian set of intervals generally, due to its extremely precise 27-odd-limit palette. When used for JI, it defines the Standard Extreme Precision JI capable of writing in the 47-limit with great precision. It also is the smallest precision level that has an "exact" mapping for prime 13, thanks to the mina's appearance. 13/8 is now written as a major sixth minus the 35 large diesis and a mina. From C, this would be C - A .
Magrathean
It adds the tina accent 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. The strict zeta peak 8539edo has been used to define the tina as an interval measure, due its insanely precise 27-odd-limit (and beyond) interval palette. 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.[5]
Phai/phao, patai/patao, pakai/pakao, jatai/jatao, and jakai/jakao can be spelled (respectively) fai/fao, gai/gao, vai/vao, wai/wao, and dai/dao. Pronunciation is intentionally loosely defined to accommodate various languages.
Spartan multi-shaft
Multi-shaft sagittals are only used in the Revo flavor of Sagittal.
Srai/srao, phrai/phrao, prakai/prakao, khai/khao, and rakhai/rakhao can be spelled (respectively) slai/slao, frai/frao, vrai/vrao, chai/chao, and rachai/rachao.
Promethean extension multi-shaft
Multi-shaft sagittals are only used in the Revo flavor of Sagittal.
Fine-grained Sagittal notations can use accents, also called diacritics, to the left of a sagittal or a bare shaft, to indicate very subtle distinctions in pitch.
The average unit interval is called a "tina" (rhymes with ballerina) and is approximately 0.14 of a cent. The new shapes are called "horn" and "wedge". 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. The "i/o" accent, whose shape is called "dot", represents some unit fraction of a tina, often a half as shown, but it is intentionally left to be defined by the user.[5]
Prime approximations
Here are some approximations to primes from D, using the several precision sets available in JI. Values in parentheses are absolute error in cents from just; if none is shown, the notation is exact.
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.
↑ 5.05.1https://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"
