The origin of Sagittal

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Dave Keenan
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The origin of Sagittal

Post by Dave Keenan »

Here is the first of the two posts in which George Secor presented the first version of Sagittal to the world, on the Yahoo Group "Tuning", in January 2002. These are the posts that caused me to become involved in Sagittal development.

I note that, at the time, it was widely assumed that the common double-dagger-like semi-sharp symbol had first been used by Guiseppe Tartini. Later research found no evidence for this, and instead found the earliest recorded use was by Richard Stein.

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> "Notation - Part 1"
> From: George Secor
> January 22, 2002
> Patience comes to those who wait for it, and I thank you all for your
> patience. Here at last is Part One of my saggital notation
> presentation, and I hope you agree that it was worth waiting for.
> In this first part of the presentation I will illustrate the process
> by which I arrived at the 72-EDO form of the notation. Subsequent
> installments will address its multi-system application, both in
> native (or EDO-specific) and transcendental (or trans-system generic)
> forms, leaving most of the questions and comments that have been made
> regarding the more controversial aspect of the subject for the final
> installment.
> It is perhaps a bit of a stretch to call this buried treasure,
> inasmuch as this is so new that there has barely been enough time to
> get any "dust" on the paperwork (most of which is virtual, in the
> form of computer files; hmmm, I do seem to notice some dust on the
> monitor screen). I reasoned that the presentation would be more
> widely read, especially by future members of the Tuning List, if I
> put it in my Buried Treasure column.
> At the beginning of the year I made a new year's resolution to
> complete the development and testing of my notation, and I am sharing
> it with you to elicit your comments and suggestions to make this the
> very best notation possible, one that will come closest to "doing it
> all" and doing it well.
> So as not to keep you in further suspension, let the resolution begin!
> *A Challenge I Couldn't Resist!*
> Please note: The figures for this presentation are in: [At the end of this post]
> I have always believed that the best notation is that which is
> simplest. A good example of this is the Tartini fractional sharps
> (shown in the right half of the top row of Figure 1), which are so
> clear that they require virtually no explanation. Although these
> were employed by both Ivan Vyshnegradsky (for 24-EDO) and Adriaan
> Fokker (for 31-EDO), it is rather surprising that quartertone
> composers never adopted these as a standard notation. Instead, they
> often preferred to place arrows in front of notes, which, in
> combination with sharps or flats, tend to clutter a musical
> manuscript, especially when chords are notated on a single staff.
> Existing methods of notating 72-EDO have also used this approach, and
> the diversity of symbols used somewhat arbitrarily (and not always
> logically) to designate three different amounts of alteration in
> pitch strikes me as a conglomeration of add-ons or do-dads intended
> to supplement traditional notational practice. However, I did not
> see these 72-EDO notations (including the one devised by Ezra Sims)
> until several months after I had produced the initial (expanded)
> version of my saggital notation, so they had absolutely no influence
> in its development. To be completely honest, once I did see them, I
> was appalled. I later learned that the symbols that were proposed by
> those on the Alternate Tuning List were ASCII versions for
> theoretical use only, not practical notation intended for use on an
> actual musical manuscript, and the goal was largely to emulate the
> Sims notation. Inasmuch as my goal was to arrive at the very best
> notation possible, it is understandable that, immediately upon seeing
> it, I found that I had absolutely no desire to emulate the Sims
> notation, and it should be evident by the end of this part of the
> presentation that any similarity between the Sims and the saggital
> symbols is purely coincidental.
> It is not an easy matter to arrive at a simple notation that would
> require only a single symbol to modify the pitch of the seven
> naturals notes on the staff for 72-EDO. In the first place, 24
> symbols would be needed for a complete range of alteration by a whole
> tone, both upward and downward. In order for this approach to be
> successful, the new symbols would need to have an intuitiveness that
> would enable them to be quickly and easily understood. They would
> also need to be similar enough that they could be easily remembered,
> yet different enough that there would be no difficulty in
> distinguishing them from one another. This was a challenge that I
> couldn't resist!
> The solution did not come quickly, however, as it soon became evident
> that this is one situation where the desired result would not be
> achieved without investing a considerable amount of time and effort.
> I spent hours putting all sorts of symbols, both old and new, on a
> piece of paper, seeking as many ideas as possible from which to
> choose. In the end I found that the best ideas were ones that had
> already been successfully used in the past, and my saggital notation
> integrates three of these into a unified set of symbols. These three
> ideas are: 1) the use of arrows to indicate alterations in pitch up
> and down, 2) the intuitiveness of the Tartini fractional sharps, and
> 3) the slanted lines used by Bosanquet to indicate commatic
> alterations.
> *Tartini Plus Arrows*
> Up and down arrows can be employed to indicate clearly the direction
> in which the pitch is to be altered, and it was immediately obvious
> that it would be necessary to have only 12 different symbols if each
> symbol of the new notation could be inverted or mirrored vertically
> to symbolize equal-but-opposite amounts of alteration. This would
> require discarding the traditional single and double sharp symbols
> (as well as excluding the Tartini fractional sharps from
> consideration), inasmuch as they look virtually the same when
> inverted. A traditional flat symbol can be inverted and does
> resemble a hand with a finger pointing; the problem is that it points
> in the wrong direction, so I concluded that it would also need to be
> discarded. Of the conventional symbols, only the "natural" symbol
> would be retained.
> In my first version of the sagittal notation of August 2001 (which I
> now call the expanded saggital symbols), I used arrows as semisharp
> and semiflat symbols, with multiple arrowheads for single, sesqui,
> and double sharps and flats. These are shown in the second row of
> Figure 1. The use of arrows to represent semisharps and semiflats
> may seem somewhat arbitrary, inasmuch as they have been used in
> different instances to represent various amounts of pitch alteration,
> but I felt that their frequent use for notating quartertones was
> adequate justification.
> In December I realized that these symbols could be simplified by
> replacing the multiple arrowheads with single arrows that are
> combined with one to three vertical strokes, as in the Tartini
> fractional sharps, with an "X" for the double sharp and flat, as
> shown in the third row of Figure 1. The single arrowheads not only
> make the symbols more compact, but they also permit a bolder print
> (or font) style to be employed, which improves legibility.
> If the abandonment of the conventional sharp and flat symbols seems a
> bit shocking, we need to realize that, although they have served us
> well since they were devised in the Middle Ages, 21st-century
> microtonality will be better served by something new and better, and
> I think that it is safe to say it is about time for an upgrade. We
> can continue to call these sharps and flats with semi, sesqui, and
> double prefixes added as appropriate, inasmuch as it is only the
> symbols that are changing, not their names or meanings.
> This set of 9 symbols is sufficient to notate 17, 24, and 31-EDO.
> However, more symbols would be needed for 72-EDO.
> *Plus Bosanquet*
> The third idea to find its way into my saggital notation was the
> symbol for commatic alterations in 53-EDO that Bosanquet used around
> 1875. These are shown in the top row of Figure 2, which illustrates
> a lateral grouping for multi-comma alterations. The single degree of
> 72-EDO is similar in size to that of 53-EDO, with the intervals
> representing just (5:4) and Pythagorean (81:64) major thirds
> differing in size by this amount in each system, so the use of this
> sort of symbol would not be inappropriate to indicate an alteration
> of a single degree in 72-EDO. I first added a stem to the Bosanquet
> symbol to form a sort of half-arrow or flag. I then stacked several
> of these flags to indicate multiple-degree alterations, as in the
> second row of Figure 2.
> I quickly realized that the symbol that I was already using to alter
> by 3 degrees differed from the 1-degree symbol by only a right half-
> arrow or flag, and that it would be quite logical to represent a 2-
> degree alteration with a backward 1-degree symbol. The resulting
> expanded saggital symbols are shown in the third row of Figure 2.
> These were subsequently simplified into the compact saggital notation
> shown in the fourth row of Figure 2. Observe that each new half-
> arrow (or Bosanquet flag) symbol is adjacent to a full-arrow symbol,
> with the slant of the Bosanquet flag corresponding to the direction
> in which the pitch symbolized by the adjacent (full-arrow) symbol
> must be altered to arrive at the pitch symbolized by the Bosanquet
> flag symbol: upward slope signifies alteration one degree (or comma)
> up, while downward slope signifies one degree (or comma) down.
> The full range of symbols is shown in Figure 3, along with some
> examples on a musical staff comparing other notations with the new
> saggital notation.
> Both the compact and expanded versions of the saggital symbols may be
> simulated with ASCII characters for e-mail messages, etc., using a
> combination of the slash, backslash, pipe, and capital X characters.
> One comma down is \|, semisharp is /|\, and doubleflat is \X/
> (compact) or \\\\|//// (expanded). While this generally involves
> more characters than with other proposed ASCII notation, it is more
> intuitive, and it inconveniences the theorist rather than the
> musician. (Please note that the combination of ASCII symbols has a
> better appearance when a proportionally spaced font is used; my
> choice is Ariel.)
> The next part of this presentation will discuss how the notation may
> be applied logically and consistently to other EDO's, beginning with
> 31 and 41, as well as the use of the 72-EDO symbols as a
> transcendental notation for sets of just (or near-just) tones mapped
> onto a lesser division of the octave.
> Until next time, please stay tuned!
> --George
> Love / joy / peace / patience ...

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Dave Keenan
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Re: The origin of Sagittal

Post by Dave Keenan »

Here is the second of the two posts in which George Secor presented the first version of Sagittal to the world, on the Yahoo Group "Tuning", in January 2002.

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> "Notation - Part 2"
> From: George Secor
> January 28, 2002
> (Prerequisite: "Notation - Part 1", #32971)
> *Background Note*
> It had not been more than a week or two after devising the expanded
> version of the saggital symbols that Margo Schulter contacted me
> about my 17-tone well temperament, and in the course of our
> correspondence we frequently mentioned the occurrence of commas in
> various tonal systems that are realizations of her neo-medieval
> approach to microtonality. In the course of our discussion, I made a
> suggestion concerning the nomenclature of commas that we adopted.
> This will be included in the 17-tone article that will be appearing
> in the next issue of Xenharmonikon (#18), and is as follows:
> << The two different sizes of whole tone [9:8 and 8:7 in Archytas'
> diatonic scale] differ in size by 63:64 (27 cents), which Alexander
> Ellis referred to as the septimal comma. Ptolemy listed Archytas'
> diatonic tuning as the diatonic toniaion, from which we might be a
> little hesitant to coin the term toniaic comma. Instead, I believe
> it would be fitting to honor the originator of this scale by calling
> this Archytas' comma. (For many years I have felt that the use of
> the names Pythagoras and Didymus in association with their respective
> commas is a clearer and more memorable way of identifying them than
> the adjectives ditonic and syntonic, which only a scholar could
> love. Confusion between these two terms can happen to the best of
> us: Even as knowledgeable an authority as J. Murray Barbour slipped
> up in this regard in the beginning of the first chapter of his book,
> Tuning and Temperament. >>
> With our discussion of commas fresh in my mind, I happened to come
> across a couple of papers on which I had sketched out the expanded
> saggital symbols, and I began to see a connection between the two
> that led to the idea of using the saggital symbols, not only for
> systems other than 72-EDO, but also as a transcendental notation.
> *Didymus and Archytas*
> In the first part of the presentation I mentioned a distinction
> between *native* (or EDO-specific) and *transcendental* (or EDO-trans-
> generic) applications of the saggital notation. As a general
> principle, a composition of even moderate complexity written for a
> specific EDO could not be expected to be directly transferable into
> another EDO, and it would be appropriate to employ a native (or
> system-specific) notation for this purpose. However, it would be
> highly desirable for the symbols used in a system-specific notation
> to be selected from a master superset of symbols. An analogy from
> written language would help to illustrate this point.
> It is more difficult for those whose native language is English to
> learn to read a language that uses a different alphabet or set of
> symbols different from the Roman alphabet (e.g., Hebrew, Arabic,
> Chinese, or Japanese) than one that uses essentially the same
> alphabet (e.g., Spanish, French, or German). For the same reason it
> is better, if at all possible, to use a single unified set of symbols
> for many EDO's, rather than different symbols for different EDO's –
> provided that each symbol were to have the same meaning across all
> (or at least most) of those EDO's, just as specific vowels and
> consonants in the Roman alphabet represent similar (but not
> identical) sounds in the various European languages that employ them.
> The fundamental principle that sets the saggital notation apart from
> other systems is as follows:
> Whereas other systems of notation use symbols to represent
> alteration in pitch in terms of specific numbers of system degrees or
> specific fractions of a sharp or flat, the saggital symbols
> represents alterations in pitch by using symbols to represent
> alterations *by approximations of certain superparticular ratios*,
> namely Didymus' comma (81:80), Archytas' comma (64:63), and the
> unidecimal diesis (33:32).
> As before, the figures for this presentation are in: [At the end of the previous post]
> Reference to the expanded saggital symbols in the third row of Figure
> 3 will serve to illustrate this fundamental principle. Immediately
> to the left of the natural sign is a symbol with a stem with a left
> flag, \|. In 72-EDO this would represent one degree down, but in the
> saggital notation this actually represents a Didymus-comma-down
> (80:81), which just happens to be one degree of 72-EDO. The next
> symbol to the left is a stem with a right flag, |/. Specific to 72-
> EDO this would represent two degrees down, but in the saggital
> notation this represents an Archytus-comma-down (63:64). The next
> symbol to the left combines these in a single-headed arrow with both
> left and right flags, \|/, symbolizing a unidecimal-diesis-down
> (32:33). For those interested in the numbers, observe the following
> interval arithmetic (in which it should be understood that plus and
> minus symbols actually indicate multiplication and division of the
> numerical ratios):
> Unidecimal diesis – (Archytus' comma + Didymus' comma) = 4.503 cents
> 33/32 – (64/63 + 81/80) = 385/384, or 4.503 cents
> Those in the tuning-math group will immediately recognize this as a
> zero vector encountered in their exploration of the Miracle tuning,
> perhaps more recognizable as the difference between just intervals
> represented by certain basic intervals in that tuning:
> (3 secors) – (2 secors + 1 secor) = a zero vector
> 11/9 – {8/7 + 16/15) = 385/384, or 4.503 cents
> This serves to demonstrate not only the close relationship of the
> saggital notation to the Miracle tuning geometry, but also to point
> out that the power and versatility of the saggital symbols is drawn
> from the very same geometry that gives the Miracle tuning its
> versatility and efficiency, so it would not be inappropriate to call
> this a Miracle notation. But first let's see if it accomplishes any
> miracles.
> As we noted, in 72-EDO Didymus' comma is 1 degree and Archytus' comma
> is two degrees. In 41-EDO, these are each one degree, while in 31-
> EDO they are zero degrees and one degree, respectively. Now look at
> any saggital symbol in the third of figure 3; the number of degrees
> of alteration accomplished by that symbol in a given EDO is found by
> totaling the number of degrees represented by Didymus (left) and
> Archytas (right) flags in the symbol for that EDO. That's how it's
> done, plain and simple! And this principle can also be used to
> notate many other EDO's as well.
> Just as a certain vowel symbolized by a letter of the alphabet is
> pronounced somewhat differently from one spoken language to another,
> so does a Didymus-comma-down symbol, for example, lower the pitch by
> different amounts in different EDO's. In the case of systems that
> disperse Didymus' comma, such as 24 or 31-EDO, the Didymus-comma-down
> symbol will indicate an alteration of zero degrees, so that the left
> flags in the expanded saggital symbols are disregarded in these
> systems.
> It is obvious that nobody is going to take the time or trouble to
> count flags, which is why the compact saggital symbols were
> developed. These contain all of the information conveyed by the
> expanded symbols in a simpler form. Each of the compact symbols can
> be readily identified as having a Didymus (left-handed), Archytas
> (right-handed), or diesis (laterally symmetrical) appearance and
> function.
> *Native vs. Transcendental*
> It was noted above that compositions written for one EDO are not
> generally transferable into another EDO. However, there are
> circumstances under which such transference is not only possible, but
> highly desirable and practical. For example, music written in the
> Blackjack or Canasta scales (~72-EDO) could easily be played on wind
> instruments (with relatively small amounts of pitch-bending)
> specifically built for 31 or 41-EDO, and the result would be far
> better than using 12-EDO instruments with extended (or extraordinary)
> techniques. (I believe that 72-EDO wind instruments are a bit out of
> the question for the foreseeable future, if ever.) Another example
> would be just intonation (e.g., the 43-tone Partch set) mapped onto
> 72 and played on 31 or (for Partch, preferably) 41-EDO instruments.
> (I have to assume that, should microtonality go mainstream, we would
> no longer want to suffer the limitations of 12-EDO instruments, and
> whatever notation we might use should take this into account. It
> never hurts to plan ahead!) In these instances the 72-tone notation
> could be employed as a transcendental notation that could be read
> directly into 31 or 41, making it unnecessary to have separate parts
> for instruments in each of those systems. This would remove a
> considerable burden from the composer, who might not know in advance
> what instruments would eventually be used to perform a given
> composition.
> The saggital notation accomplishes this by a principle similar to the
> way decimal numbers are rounded off to whole numbers, as illustrated
> in Figure 4. In native 31-EDO notation, all symbols are multiples of
> a single diesis. (As it happens, in all three EDO's – 31, 41, and
> 72 – three 5:4's fall short of an octave by the number of degrees
> representing 33:32 in each of those systems, so the term "diesis" can
> be used here in a broader sense to refer to either the unidecimal or
> meantone diesis.) Each left or right-handed symbol in the
> transcendental notation may be regarded as an alteration of the
> neighboring symmetical symbol by a Didymus flag that represents
> 81:80, which is a zero vector in 31-EDO. (The left-handed symbols
> are thus interpreted as a symmetrical symbol plus Didymus flag, while
> right-handed symbols are interpreted as a symmetrical symbol minus a
> Didymus flag.) The small arrows in the figure show how
> mental "rounding" to the nearest diesis can be accomplished.
> In the 41-EDO native notation some new symbols are required to
> indicate odd-number multiples of a half-diesis (i.e., quarter-sharp
> and quarter-flat) alteration. It should be evident that
> mental "rounding" (to the nearest half-diesis) is even simpler than
> what is required in 31-EDO.
> This process of mental rounding works in this fashion with almost all
> simple 11-limit ratios, and the exceptions are easy to remember: the
> few ratios having a 7 factor in the numerator and a 5 factor in the
> denominator (or vice versa) – 7/5, 10/7, 21/20, 40/21 – must be duly
> noted and memorized. As far as I have been able to tell, most
> everything else works like a charm!
> For your reference, there is a one-octave diagram of 72-EDO (with
> numbered system degrees and saggital symbols) compared with 19-limit
> just intonation at: [At the end of this post]
> The next part of this presentation will continue with application of
> the saggital notation to some other EDO's.
> And your questions and comments will be appreciated.
> Until next time, please stay tuned!
> --George
> Love / joy / peace / patience …

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Dave Keenan
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Re: The origin of Sagittal

Post by Dave Keenan »

I found the following draft of an unfinished paper, dated 2002-Feb-25, while looking for something else.

A Common Notation System for Extended Just Intonation and Equal Temperaments

Gene Ward Smith, George Secor, David C Keenan

In 1991 ('The Notation of Equal Temperaments', XH16 1995) Paul Rapoport proposed a system to notate the various equal divisions of the octave (EDOs or ETs) using the familiar seven note names A to G, and sharps and flats, with the addition of accidental symbols corresponding to various commas, and fractions of these commas. The commas used most extensively for this purpose were the syntonic comma [4, -1], the diesis [0, -3] and the diaschisma.[-4,-2]. These, and the others commas he mentioned, involve (octave-equivalent) ratios of only the primes 3 and 5. Rapoport concluded that "The use of any of the above signs may not completely mitigate the complications inherent in this manner of notating close approximations to just tunings ... More concise signs are needed which represent the higher harmonics better ..."

[ Box: A comma is ... Explain the vector notation]

It has occurred to many people that rational tunings (meaning tunings based on frequency ratios involving only whole numbers), whether just or not, can be notated by extending conventional notation with pairs of new accidentals, each pair corresponding to a comma for each prime number higher than 3.

In both cases the conventional names are applied to a Pythagorean chain of perfect fifths (or the best approximation thereof), centered on D (or D-A if an even number is required), i.e. F C G D A E B. The flat "b" and sharp "#" can be considered to be a pair of accidentals that correspond to a comma for the prime number 3. This comma is [7, 0] and is called an apotome. It simply allows the chain of fifths to be extended in both directions.

It occurred to us to see whether a one-comma-per-prime system might also form the basis of a system for notating ETs in the manner of Rapoport. We hoped to avoid the need for symbols representing fractions of a comma, since these are not required for rational tunings. We found that a system suitable for notating most ETs of interest, well into the hundreds, could be obtained using commas for primes only up to 19. This requires only 6 new pairs of accidentals. This is less than were used by Rapoport in his 5-limit system.

The chosen commas are these

5	[4, -1]			\ /
7	[-2, 0, -1]		LP
11	[1, 0, 0, 1]		[ ]
13	[4, 0, 0, 0, 1]		{ }
17	[7, 0, 0, 0, 0, -1]	yh
19	[3, 0, 0, 0, 0, 0, 1]	o*

All these commas vanish in 12-tET. Well almost. 12-tET treats ratios of 11 inconsistently. We consider it best to use 32:33 rather than 704:729 for 11s because the former comes out smaller in cents on a true Pythagorean chain and spans fewer fifths.

Using those commas a
1:3:5:7:9:11:13:15:17:19 chord on G would be spelled
G:D:B:F:A:C :Eb:F#:G#:Bb
with the addition of the appropriate comma symbols.
Note that the above all fall within a range of fifths from Eb to G#.

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