My name is Nicholas Denton Protsack. Since this is my first time posting here, I would like to introduce myself: I am a composer/cellist (and part-time music theorist) currently residing in Kelowna, British Columbia. I am currently in the first year of a PhD at Victoria University of Wellington (and am studying via distance until the NZ borders open and I am able to move there).

I am posting here on the request of Dave Keenan in order to carry over an extended conversation we (and others) have been engaging in on the 'Microtonal Music and Tuning Theory' Facebook Page.

In short, I would like to propose an even simpler version of Sagittal than the Spartan set, that allows one to map all the intervals of the 13-limit tonality diamond (and, by extension, the 15-limit tonality diamond) within 1.033 cents of error, while only introducing 8 new accidentals. This can be achieved via a spine of (essentially) perfect 5ths tempered by -0.00011 cents, and a small generator of 5.414 cents; (in other words, this can be achieved with a subset of 665-ET). A prototype of this mapping was initially discovered by me, and subsequently optimized via the tempering of several intervals that Paul Erlich and Steve Martin calculated. Paul began referring to the 5.414 cent generator (much to my amusement) as a "Dent", and, for the sake of brevity, I will refer to this interval by that name as well.

With these generators accepted, the following commas vanish (courtesy of Paul Erlich):

[0, 2, 2, 1, -2, -1⟩ (1575:1573)

[5, -3, 1, -1, -1, 1⟩ (2080:2079)

[5, -1, 3, 0, -3, 0⟩ (4000:3993)

[12, -2, -1, -1, 0, -1⟩ (4096:4095)

[-7, -1, 2, 0, -1, 2⟩ (4225:4224)

[-1, -7, 4, 1, 0, 0⟩ (4375:4374)

These commas are so small (and a number of them are already tempered out in Spartan Sagittal) that the system can essentially be equated with 13-limit Just Intonation, and function as well as JI for just about all of its basic applications.

The version of Sagittal notation I have "synthesized" for this new tuning system (in consultation with Dave Keenan) aims to reduce the information needed to communicate pitch-meaning, to the lowest extreme possible: Hence the name, "Stoic Sagittal" (with an alternate name candidate being "Diogenic Sagittal"). All accidentals used in this system derive their component parts from only two symbols: and Where equals exactly one "Dent" and equals exactly four Dents. When this basic premise is accepted, various combinations of these symbols are all that are needed to notate every element of the 13-limit tonality diamond. The one exception to this rule is the symbol for "7 Dents", which uses a slash to indicate the subtraction of 1 Dent. (This prevented an alternate "cluttered" representation of the accidental from being used; sort of like how the Roman Numeral for '4' is written as IV instead of IIII).

I have written up and attached to this post a rough, handwritten document containing all the proposed symbols, and their applications in all JI intervals up to the 15-limit. I have also included their respective errors to 0.001 of a cent:

**What are the advantages of this system?**

-There are only 3 unique symbols (and 8 accidentals) necessary to map every interval in the 13-limit Diamond using Stoic Sagittal. Spartan Sagittal cannot map the entire 13-limit tonality diamond, and Athenian Sagittal recommends the use of 7 unique symbols (and 11 accidentals).

-Since only two(ish) elements are used in the construction of these accidentals, they can be seen as perhaps more comprehensive than the Spartan Sagittal set. As long as you know the meaning of the two "root" symbols, the the meaning of all symbols can be understood with a linear and unambiguous logic.

-It is significantly easier to write legibly by hand using these symbols, as there is no potential for "look-alikes" when drawing them. (Even if one is drawing them sloppily).

-Comma pumps can occur indefinitely without introducing new symbols: Since the symbols are all evenly-spaced apart from one another, and (I believe) the Pythagorean whole tone is divided exactly, there is no need for extra symbols when modulating outside the 13-limit diamond. One need only modulate by one (or two, or three, etc.) Dents.

**What are the disadvantages of this system?**

-It is less accurate than Spartan Sagittal, and all (rather than some) intervals are approximated (albeit by VERY small amounts). Spartan Sagittal has (I believe) a max error of 0.828 cents through the 15-odd limit. Stoic Sagittal has a max error of 1.033 cents through the 15-odd limit (the same as 665-ET).

-There is no logically consistent way to extend the system any higher than the 13-limit without introducing new symbols.

-Creating a Pure Sagittal version of this system may prove to be difficult. I created it with the intention of it being used for Mixed Sagittal, so I am not sure how it will translate.

Please let me know what you think of this system, and if you would have any of its symbols/semantics changed in some way, I would like to hear your thoughts! I would love if whoever designed the glyphs for the Bravura font of Sagittal would consider making a version of these, since my skills with Inkscape (and the like) are practically non-existent. Ultimately, it would be a dream (and honor) to see this included in the Bravura font, even!

All the best,

Nicholas Denton Protsack