User:Robertinventor/Sagittal

I proposed this as a new article here: Talk:Just_intonation#Staff_notations

Sagittal is a similar system for notating just intonation music and the harmonic series (examples of papers that cite Sagital: and some that use it:) following 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.

It 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. So 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.

Sagittal has been extended to a system of notations for many 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 pure fifth the equal tempered notation uses the best approximation to a tempered fifth in the equal temperament to be notated. The various commas are replaced by tempered commas. In this way the authors Dave Keenan and George Secor developed notations for all the equal tempered tunings in common use and many rarely used ones as well.

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.