The origin of Sagittal
Posted: Tue Mar 24, 2020 12:15 am
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:
> *BURIED TREASURE*
> "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]
>
> http://groups.yahoo.com/group/tuning/files/secor/notation/figures.bmp
>
> 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 ...
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:
> *BURIED TREASURE*
> "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]
>
> http://groups.yahoo.com/group/tuning/files/secor/notation/figures.bmp
>
> 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 ...