Sagittal notation

Sagittal notation is a system for notating microtonal music developed by George Secor and David Keenan, in cooperation with many other microtonalists, with the aim of providing a unifying approach to notating microtonal music, for many different tuning systems. It can be used for exact notation of just intonation music based on pure rations as well as music based on equal divisions of the octave.

For pure ratios, the system is based on notating everything relative to notes constructed through a cycle of fifths in the pythogorean twelve tone system together with notation symbols for various commas, and for equal divisions of the octave it uses the closest approximations to intervals in those tunings, as well as commas built up from those approximations (examples of papers that cite Sagital:  and some that use it: ).

To start with the exact notation for pure ratios, you can get to the just intonation E as a 5/4 above C using the pythagorean E (81/64) flattened by a syntonic comma (81/80). With a symbol for the syntonic comma it's easy to notate not just 5/4, but, as it turns out, 15/8, 8/5, 5/3, etc. By adding notation symbols for several other commas you can notate any of the small number ratios like 7/6, 11/8 etc exactly. This example from their paper shows many of them in use. You can see the syntonic comma used to notate a 5/4 on the first chord, and to notate 5/3 on the second chord. The same example is notated in two different ways, the mixed version which combines new symbols with traditional sharps and flats, and the pure version, which uses a single symbol wherever possible.



They comment on the "mixed" version:

"While this version  requires  fewer total symbols for a music font, it results in a greater number of symbols on a  manuscript,  which  tends  to  give  it a  more  cluttered  appearance  when  chords  are  notated.    However,  this  version  would  have  an  easier  learning  curve,  which  would enable wind and string players to master sight-reading more quickly."

Sagittal notation follows similar lines to the Sabat-Schweinitz system. It was developed around the same time, starting in 2001 and first published in Xenharmonikôn in 2004. . The design principle for the notation is to use arrows pointing up or down for alterations of pitch, away from the circle of pure fifths. Following Tartini, multiple vertical strokes are used to distinguish the accidentals as well as half arrows to left or right and both. Curved barbs are used to help with reading at speed. It includes symbols sufficient to notate not just the fifth, and seventh harmonics, but also the eleventh, thirteenth and seventeenth harmonics. The Sagittal accidentals are included in the Standard Music Font Layout (SMuFL), intiated by Steinberg (who publish the Cubase) and is now maintained by the W3C Music Notation Group.

This system is designed for the practical musician with a system of increasing precision levels. For instance, the difference between the 5-comma 81/80 and the 41-comma 82/81 is only 0.3 cents. In most circumstances they re-use the same symbol for higher commas. However sometimes composers need to be able to notate just intonation pitches more precisely, and so they have designed symbols if necessary all the way up to the 61st harmonic.

Most musicians will find the Athenian level sufficient, and this has notations for just intonation ratios up to the seventeenth harmonic. At that level the symbols are reasonably distinct from each other, and where they are similar in pitch, they are similar in shape. The next higher levels of precision are the Promethean and Herculean. The highest precision level in the published system is the Olympian level with a precision of half a cent. The idea behind this name is that it requires god-like powers to hear the difference between adjacent symbols and to read them.

When the system is used for equal divisions of the octave, the idea is to use the tempered harmonic series as a basis for notation of equal temperaments. For instance instead of a 3/1, the equal tempered notation uses the best approximation to a tempered 3/1 in the equal temperament to be notated (19 steps in twelve equal for instance). Instead of a 5/1 it will use the nearest tempered version (28 steps in twelve equal) and so on. The various commas are replaced by tempered commas which are constructed from the tempered intervals.

Sometimes the resulting comma vanishes. For instance in twelve equal, then the 3/1 when repeated 12 times reaches an exact number of octaves, 19 octaves (since there are 12 notes to an octave) and so that shows that the pythagorean comma, the difference between 19 octaves and 12 3/1s, vanishes in this tuning, if you use the tempered intervals.

Similarly, the tempered 81/80 or 34/(24*5) becomes 4*19 steps upwards (using the tempered version of the 3/1 at 19 steps) and then 4*12 and 1*28 downwards (using the tempered 5/1 at 28 steps), which gets you to 4*19-4*12-28 = 0, so the syntonic comma vanishes in this tuning. This should be no surprise, as it just shows that it makes no distinction between its equivalent of the Pythagoran major third, 81/64 and it's equivalent of 5/4, and indeed they are used interchangeably in twelve equal harmonies.

In this way the authors Dave Keenan and George Secor developed exact notations for all the equal tempered tunings in common use and many rarely used ones as well, all based on the same principles.

It also includes a system called "Trojan" based on the same principles, which can be used to notate approximations to just intonation pitches relative to twelve equal as fractions of an equal tempered semitone. The idea of this notation is to guide the performer to within a few cents of the desired pitch, after which they can find the just intonation note by ear.