Sagittal notation
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.
In tempered systems, Evo-SZ accidentals may instead be distinct from Sagittals like , when the apotome can be split in two equal parts as in Hemipythagorean, as literal half-sharps and half-flats, and there is a distinction between mapped neutral thirds like ~11/9 and hemififths, in particularly fine-grained EDOs like 270 or 311. There is no need for this in coarser systems like 24, 31 or 41, where the mappings of the hemififth and rational neutral thirds coincide.
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.
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
- Sunday Pipes in 22edo by Mats Öljare
- Tibia in 22edo by Paul Erlich (Listen). Sagittal score in F||\ or in G (contains errors in measures 9, 19 and 20)
- 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.(Listen)
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.
You can look up what the minimum precision is required to write an EDO in this Desmos graph.
Sharps/Flats
Using and (or for Revo flavor) is still technically Sagittal notation, however, it's just a reskin.
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.
Trojan (12-EDO relative)
This is a special extension to Spartan which adds 3 more symbol sets, which is exclusively used to notate compton EDOs (12N EDOs up to 312) in which the apotome is 100 cents. However, all of these EDOs lower than 168EDO can be notated using just Spartan, in which cases, depending on your needs, maybe Athenian proves to be more useful.
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
| |
| 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 |
Trojan extension
| |
| zai |
| 23CT up |
| |
| zao |
| 23CT down |
| |
| prai |
| 19/5CT up |
| |
| prao |
| 19/5CT down |
| |
| pazai |
| 23/5ST up |
| |
| pazao |
| 23/5ST down |
| |
| sharp pazao |
| sharp 23ST down |
| |
| flat pazai |
| flat 23ST up |
| |
| sharp prao |
| sharp 19/5CT down |
| |
| flat prai |
| flat 19/5CT up |
| |
| sharp zao |
| sharp 23CT down |
| |
| flat zai |
| flat 23CT up |
| |
| sharp zai |
| sharp 23CT up |
| |
| flat zao |
| flat 23CT down |
| |
| sharp prai |
| sharp 19/5CT up |
| |
| flat prao |
| flat 19/5CT down |
| |
| sharp pazai |
| sharp 23/5ST up |
| |
| flat pazao |
| flat 23/5ST down |
| |
| double sharp pazao |
| double sharp 23ST down |
| |
| double flat pazai |
| double flat 23ST up |
| |
| double sharp prao |
| double sharp 19/5CT down |
| |
| double flat prai |
| double flat 19/5CT up |
| |
| double sharp zao |
| double sharp 23CT down |
| |
| double flat zai |
| double flat 23CT up |
Promethean extension single-shaft
| |
| rai |
| 19s up |
| |
| rao |
| 19s down |
| |
| sai |
| 17k up |
| |
| sao |
| 17k down |
| |
| srai |
| 143C up |
| |
| srao |
| 143C down |
| |
| shai |
| 49/11C up |
| |
| shao |
| 49/11C down |
| |
| razai |
| 19C up |
| |
| razao |
| 19C down |
| |
| ratai |
| 19/7C up |
| |
| ratao |
| 19/7C down |
| |
| satai |
| 49S up |
| |
| satao |
| 49S down |
| |
| sakai |
| 23S up |
| |
| sakao |
| 23S down |
| |
| phrai |
| 13/5M up |
| |
| phrao |
| 13/5M down |
| |
| jazai |
| 19/11M up |
| |
| jazao |
| 19/11M down |
| |
| jpai |
| 49M up |
| |
| jpao |
| 49M down |
| |
| prakai |
| 49/5M up |
| |
| prakao |
| 49/5M down |
| |
| ktai |
| 49L up |
| |
| ktao |
| 49L down |
| |
| khai |
| 19/11L up |
| |
| khao |
| 19/11L down |
| |
| rakhai |
| 13/5L up |
| |
| rakhao |
| 13/5L down |
Promethean extension multi-shaft
Multi-shaft sagittals are only used in the Revo flavor of Sagittal.
| |
| sharp sakao |
| sharp 23S down |
| |
| flat sakai |
| flat 23S up |
| |
| sharp satao |
| sharp 49S down |
| |
| flat satai |
| flat 49S up |
| |
| sharp ratao |
| sharp 19/7C down |
| |
| flat ratai |
| flat 19/7C up |
| |
| sharp razao |
| sharp 19C down |
| |
| flat razai |
| flat 19C up |
| |
| sharp shao |
| sharp 49/11C down |
| |
| flat shai |
| flat 49/11C up |
| |
| sharp srao |
| sharp 143C down |
| |
| flat srai |
| flat 143C up |
| |
| sharp sao |
| sharp 17k down |
| |
| flat sai |
| flat 17k up |
| |
| sharp rao |
| sharp 19s down |
| |
| flat rai |
| flat 19s up |
| |
| sharp rai |
| sharp 19s up |
| |
| flat rao |
| flat 19s down |
| |
| sharp sai |
| sharp 17k up |
| |
| flat sao |
| flat 17k down |
| |
| sharp srai |
| sharp 143C up |
| |
| flat srao |
| flat 143C down |
| |
| sharp shai |
| sharp 49/11C up |
| |
| flat shao |
| flat 49/11C down |
| |
| sharp razai |
| sharp 19C up |
| |
| flat razao |
| flat 19C down |
| |
| sharp ratai |
| sharp 19/7C up |
| |
| flat ratao |
| flat 19/7C down |
| |
| sharp satai |
| sharp 49S up |
| |
| flat satao |
| flat 49S down |
| |
| sharp sakai |
| sharp 23S up |
| |
| flat sakao |
| flat 23S down |
| |
| sharp phrai |
| sharp 13/5M up |
| |
| flat phrao |
| flat 13/5M down |
| |
| sharp jazai |
| sharp 19/11M up |
| |
| flat jazao |
| flat 19/11M down |
| |
| sharp jpai |
| sharp 49M up |
| |
| flat jpao |
| flat 49M down |
| |
| sharp prakai |
| sharp 49/5M up |
| |
| flat prakao |
| flat 49/5M down |
| |
| sharp ktai |
| sharp 49L up |
| |
| flat ktao |
| flat 49L down |
| |
| sharp khai |
| sharp 19/11L up |
| |
| flat khao |
| flat 19/11L down |
| |
| sharp rakhai |
| sharp 13/5L up |
| |
| flat rakhao |
| flat 13/5L down |
| |
| double sharp sakao |
| double sharp 23S down |
| |
| double flat sakai |
| double flat 23S up |
| |
| double sharp satao |
| double sharp 49S down |
| |
| double flat satai |
| double flat 49S up |
| |
| double sharp ratao |
| double sharp 19/7C down |
| |
| double flat ratai |
| double flat 19/7C up |
| |
| double sharp razao |
| double sharp 19C down |
| |
| double flat razai |
| double flat 19C up |
| |
| double sharp shao |
| double sharp 49/11C down |
| |
| double flat shai |
| double flat 49/11C up |
| |
| double sharp srao |
| double sharp 143C down |
| |
| double flat srai |
| double flat 143C up |
| |
| double sharp sao |
| double sharp 17k down |
| |
| double flat sai |
| double flat 17k up |
| |
| double sharp rao |
| double sharp 19s down |
| |
| double flat rai |
| double flat 19s up |
Herculean extension diacritics
| | | | |
|---|---|
| ai | ao |
| unison | |
Ai and ao are used to write bare diacritics in the following extensions, like the schisma, schismina, and tina.
| | |
|---|---|
| bi | bo |
| 5s up | 5s down |
Olympian extension diacritics
| | | | |
|---|---|---|---|
| mi | mo | mimi | momo |
| 35/13sx up | 35/13sx down | 77/65sx up | 77/65sx down |
The diacritic is called "mina".
Magrathean extension diacritics
| | | | | | | | | | | | | | | | | | | | | | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| qui | quo | quiqui | quoquo | mi | mo | quimi | quomo | quiquimi | quoquomo | mimi | momo | quimimi | quomomo | quiquimimi | quoquomomo | quiquimimi | quoquomomo | mimimi | momomo | i | o |
| 10241/10240 up | 10241/10240 up | 5832/5831 up | 5832/5831 down | 35/13sx up | 35/13sx down | 3025/3024 up | 3025/3024 up | 2401/2400 up | 2401/2400 down | 77/65sx up | 77/65sx down | 1701/1700 up | 1701/1700 down | 382976/382725 up | 382976/382725 down | 382976/382725 up | 382976/382725 down | 131072/130977 up | 131072/130977 down | 1515591/1515520 up | 1515591/1515520 down |
The diacritic is called "tina". 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, down to fractions of a cent. The "i/o" diacritic is a fraction of a tina, left arbitrary intentionally.[3]
See also
- Sagitype
- Pain free guide to Sagittal by William Lynch
- File:24 Edo.pdf – Sagittal notation guide for 24edo by William Lynch (download: 24_Edo.pdf)
- Introductory examples by Hans Straub
External links
- Official site
- Sagittal Forum
- The original Xenharmonikon article (updated)
- Gift of the Gods: a Mythical introduction to Sagittal notation
- spreadsheet-based calculator for Sagittal JI notation
- Sagittal-SMuFL-Map, a table of every Sagittal symbol
- Sagittal chord chart by Andrew Meronek
- Interactive EDO Charts by Mark Johnson
| View • Talk • EditMusical notation | |
|---|---|
| Universal | Sagittal notation |
| Just intonation | Functional Just System • Ben Johnston's notation (Johnston–Copper notation) • Helmholtz–Ellis notation • Color notation |
| MOS scales | Diamond-mos notation • KISS notation (Quasi-diatonic MOS notation) |
| Temperaments | Circle-of-fifths notation • Ups and downs notation (alternative symbols) • Syntonic–rastmic subchroma notation • Extended meantone notation • Fractional sharp notation |
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. | |
- ↑ https://forum.sagittal.org/viewtopic.php?t=516
- ↑ https://sagittal.org/sagittal.pdf p.10
- ↑ 3.0 3.1 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), 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"