Simplified "Stoic" Sagittal for 13-limit tonality diamond use

ndentonprotsack
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Real Name: Nicholas Denton Protsack

Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by ndentonprotsack »

Hi All,

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:

Image

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
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Re: Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by Dave Keenan »

Hi Nicholas. Welcome to the Sagittal forum. Thanks for posting. This is fascinating stuff.

I took the liberty of making your image an attachment so it could be displayed directly. I suspect you found that the forum wouldn't let you attach any files or display any images in your first post. But you should be fine to do so now. I also added a link to the facebook discussion.

I didn't realise until now, that what you have here (if you were to add the obvious 2 and 3 dent symbols) is an alternative set of symbols for the medium-precision JI notation. That's the one for which we currently use the Athenian set minus :|\: . So the dent is equivalent to the "athina", which has also been called the "quarter-comma".

It doesn't divide the Pythagorean whole-tone exactly. It divides the Pythagorean chromatic semitone or sharp exactly (also called the apotome, pronounced a-POT-ə-me). It divides the apotome into 21 equal parts, hence 21-EDA.

As such, it can be extended to the 17-limit diamond, albeit with greater errors. The 17th harmonic is notated using a 3 dent symbol (as 4131/4096), with a 1.5 cent error.

Since it has the same resolution as Athenian, it is not simpler than Spartan — at least not in the same way that Spartan is simpler than Athenian, i.e. in having fewer symbols or approximating a smaller EDA. But it is simpler than Athenian in using far fewer flag types.

So both Spartan and Stoic are simpler than Athenian, but in different ways.

I don't like the new slash (for -1 dent). Graphically adding something in order to subtract, seems like it should be a last resort. I prefer having 3 right scrolls. Yes, it's a bit cluttered if you actually try to draw them as scrolls. But I think they can be drawn as 3 parallel straight lines, sloping upward to the right at about 30° to the horizontal (on the upward symbol), at about half the length of the 5-comma barbs.

I designed the Sagittal glyphs that are in Bravura — working from 8-pixel-per-staff-space (ppss) bitmaps that George and I reached consensus on. We had previously agreed on 6 ppss bitmaps of them. Those are what you see as the sagittal smilies in this forum. We found that unless you can make the symbols distinct from all other symbols when using only 6 pixels per staff space, black-and-white (no anti-aliasing), both when the symbols are aligned on a staff space, and on a staff line (1-pixel-thick), then they wont be distinct enough in any format, for any purpose.

So if you want to try your hand at making 6 ppss bitmaps for your symbols, that would be a good test and a good start towards possible future font glyphs. I just use the pencil tool to set or clear individual pixels in Microsoft Paint while zoomed way in.

Here are the 6 ppss bitmaps for the existing symbols:

download/file.php?id=39

ndentonprotsack
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Re: Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by ndentonprotsack »

Hello Dave,

Thank you for your message and information! You are completely right, both in the regard that it is an equal division of the apotome (not the Pythagorean tone), and that stoic can be seen as an alternate "minimalist" mapping of Athenian, not a simplified version of Spartan. For what it is worth, I realized both of these things right after posting :lol:

I decided to take you up on your challenge to write 6-PPSS bitmaps of the symbols, because I, too, was curious about the results. I took your advice and reverted back to the triple scroll symbol, though I had to modify the design of the original 5:7 kleisma scroll to make it fit properly. I don't know about you, but personally I felt comfortable with the idea of modifying this symbol, since the tuning of a "Dent" (or whatever we decide to call it) is slightly different than the equivalent Spartan symbol. I also took the liberty of creating a 11-L and 13-L symbol, since I realized they could logically arise from the basic design of the symbols (why not include them, in that case?)

I am quite happy with the results: (Please let me know what you think!)

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Best,
Nick

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Re: Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by Dave Keenan »

I note that you can make your image attachment display in your post by right-clicking its filename and choosing Copy link location, then pasting it and surrounding it with Image tags, which can be done by selecting it and clicking the image button (sun and mountains icon).

I think it is fine to modify the design of the right scroll. We really have no choice if we're going to have 3 of them. But an option is not to modify it for the 1 dent or 5 dent symbols, and to progressively modify it for the 6 and 7 dent symbols, in the same way that the left barb is changed in length, angle and thickness in going from the 5C symbol to the 25S symbol (4 dent to 8 dent).

Those symbols look OK. But now you need to make two copies of them and draw in staff lines for the two cases (on a line, on a space). Some features may be obliterated when you do that, forcing a redesign.

I found this old file from April 2006 — the final result of many such attempts over about 3 years.

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Re: Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by Dave Keenan »

Here's an earlier attempt, from August 2003. You can see a version of your 5 dent symbol, in a column just to the left of centre. And some horrible round-shouldered arcs. We were limiting ourselves to a maximum of 2 flags per symbol at this stage. Later relaxed to 3.

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Re: Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by ndentonprotsack »

Thanks for all of these suggestions, and materials, Dave; this is super helpful! I incorporated your suggestions, and after a couple hours of trial and error, I came up with a set of symbols that feels pretty comprehensive and easy to read. Looking forward to seeing what you think!

Image

BTW, Paul ran a few more numbers with some other people on the https://www.facebook.com/groups/xenharm ... Math Page, and came up with a rank-two temperament of the system (that still divides the apotome into 21 equal parts), and has managed to reduce the max error to just below 1 cent (0.99265 to be exact) via a 15-limit minimax! This is realized using 7-EDO as one generator, and a Dent value of 5.43169 cents as the other. Getting the accuracy within 1 cent across the board was something I had my fingers crossed for!

EDIT: Just realized, I accidentally forgot to include the 25 comma in this design, but it is unaltered from the base sagittal symbol, so it doesn't really matter in the long run :?

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Re: Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by Dave Keenan »

Hi Nicholas. That's definitely an improvement. But I think we can do better. I'm in the middle of repainting my kitchen ceiling today, or I would have tried my hand at the bitmaps. :)

Some areas are a bit too busy when combined with the staff lines. I think the double and triple scrolls need to be spaced apart more (maybe the same spacings as the double and triple barbs), and I think that the triple barbs and the double and triple scrolls need to be centred on the staff position (or more nearly so).

The rank-2 temperament stuff is interesting in its own right. But I don't think it's very relevant to notating JI in this way, because we can define each symbol to have an exact 17-limit comma value, independent of how it is composed from 7-limit elements. e.g. We can define //|(( as exactly 33/32. And further. We can define each symbol to have a different (exact) comma value depending on what nominal (and any sharp or flat) it is altering. For example, when 1/1 = C, :!(: could be defined as 7/5k when applied to Gb (thereby notating 7/5 exactly), but defined as 13/11k when applied to Eb (thereby notating 13/11 exactly). The changeover from one comma to another would occur at a specified distance from 1/1 along the chain of fifths. That way, many more rational pitches can be notated exactly, and very few will have to rely on approximation.

In that way, both Athenian and Stoic might be extended to notating 31-limit ratios. This is an area that has had very little attention since this article I wrote in 2004: http://sagittal.org/whatpitch.txt . It needs to be brought up to date using things we have learned since, including https://en.xen.wiki/w/N2D3P9 . And possibly limiting the nominals-plus-sharps-or-flats to within ±6 fifths of 1/1.

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Re: Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by ndentonprotsack »

Interesting! All good points regarding the appearance of the notation! I am curious to see what you come up with once kitchen ceiling painting has been completed 8-) . I might take another stab at it, too.

Regarding the variable accidental vs. rank-2 temperament thing, I think both could/should be an option!
If there is anything I have learned in being introduced to sagittal, it is that a flexible notation system that can be used in variety of applications (so long as the "language" is consistent) is a very good thing. I had this system in mind to notate the 15-limit extremely accurately, but also in a very uncomplicated/linear way, and I feel that the rank-2 temperament (yet to be named) I worked on with Paul achieves exactly that. For starters, I really like the idea of comma pumps and modulations always resulting in single symbols rather than combinations of them---even if they leave the tonality diamond; since the "apotome" is divided exactly, there's always a single symbol that can be reached two or three or four steps away, to accurately define whatever it is one needs to define (so long as it is a 15-limit-derived interval). I also find myself drawing the line at 15-odd limit in my music for the time being, because 17 is the first limit doesn't have a distinct enough "flavor" to my ear (perhaps my taste will change, but that's where it is now). With those points in consideration, I would like for this system to continue to be able to work in this context and I don't see any reason why it wouldn't continue to do so where we are currently heading, but I thought I would make that statement anyway.

...But, if we want to use this system where the symbols work for 17-limit exact ratios, and have variable meaning, as you described, and the syntax of the system supports it, then we should absolutely use it for that, too! I totally believe in the "both/and" approach, essentially.

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Re: Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by ndentonprotsack »

Hey Dave,

I have tried further modifying the bitmaps as you asked, but I am having a hard time coming up with a way to consolidate both the further spacing of the multiple "Dent" symbols, and not having them disappear behind staff lines, or not be spaced so far apart that they look "awkward".

I am very curious to see if you have any edits/suggestions about how to work around this, because I am feeling a bit out of my depth on this :?

I also wanted to let you know that I am currently sketching up a research paper in which I discuss this notation and the tuning system I developed with Paul, in the context of creating music that imitates (or is otherwise inspired by) elements of the natural world. It seems like a pretty specific topic for a research paper, but this area of musical study is going to be what my PhD is in, and I would love to incorporate these findings. I would also like to make sure that you and Paul get credited for the work you have done to help out, so we should definitely chat about that in the coming weeks.

Best,
Nick

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Re: Simplified "Stoic" Sagittal for 13-limit tonality diamond use

Post by Dave Keenan »

Yes, it's quite a punishing discipline, forcing yourself to make them distinct on a 6-ppss black & white bitmap. It's a way of simulating poor reading conditions. Here's my attempt:

Image

The scrolls aren't really scrolls any more. They are small inverted barbs, which is the way I suggested they could be hand-written.
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