Gift of the Gods

Episode One

The Arrow and the Dibbler

The music that plays on loading this page is Harry Partch's Study on Olympos' Pentatonic,
transcribed from the original score into MIDI format by George Secor.

On one particular April day in 2001 one of the Alternate Tunings group members, composer Joseph Pehrson, asked a question (in message #21671) regarding what subsets of the 72-tone equal temperament (72-ET) might be most useful as musical scales. An answer was found jointly by Paul Erlich and David Keenan in the form of a temperament capable of generating several different structurally consistent (technically described as moment-of-symmetry, or MOS) scales, so remarkable in its efficiency of harmonic resources that it was named with the acronym MIRACLE: multitudes of integer ratios approximated consistently, linearly, and evenly. However, as later reported by Joe Monzo (in the "MIRACLE" article in the Tonalsoft Encyclopaedia of Tuning,, ), "in late June 2001 it became evident to tuning list subscribers that George Secor actually has precedence as the originator of the MIRACLE temperament ... [as] originally published in Xenharmonikôn 3" some 26 years earlier.

It was not until the beginning of the following year that I joined the Alternate Tunings group and was able to congratulate Paul and Dave for their accomplishment. I also mentioned that, in case anyone was interested, I had recently come up with a new "Sagittal" notation for 72-ET that could also be used for several other divisions of the octave (including 17, 19, 22, 24, 29, 31, 34, 36, and 41) in such a way that the symbols retained the same harmonic meanings across all of those tunings. There was plenty of interest, but one thing that I had not anticipated was that others might want to notate all sorts of alternate tunings that I had never seriously considered, some of these having literally hundreds of tones in the octave – a seemingly herculean task.

In the ensuing discussion Dave Keenan, following a suggestion by Gene Ward Smith, observed with great enthusiasm that a comprehensive microtonal performance notation that could handle virtually any tuning might well be within the realm of possibility, and I responded by expressing my desire to expand my notation to meet that challenge. For the next year and a half Dave and I worked together to develop and repeatedly refine the details of the Sagittal notation, and it is only now that the complete story can be told – and what a story to tell!

* * * * * *

Before continuing, I have a confession to make – I’m not really George Secor, the original discoverer of the MIRACLE temperament. My true identity will be revealed momentarily, along with my reasons for assuming an alias. You will probably find the things I’m about to disclose extremely difficult to believe, but I ask you to hear me out.

In the ancient world people were in some ways more open-minded and accommodating than those in modern times. If a traveling poet or musician sang and played in a strange tuning, there would always be one or two in the audience curious to learn more about that tuning. Or if a particular song might perchance (on some rare occasion) pay homage to a god with whom the inhabitants of that locality were not very familiar, they would, more likely than not, desire to include that deity among those they worshipped. Thus, for many centuries did the inhabitants of the lands surrounding the waters of the Mediterranean enjoy the blessings of the gods, who favored them with a "golden age" of civilization. But this was not to last indefinitely.

The trouble began in a remote corner of the Roman Empire in the land of Judea. Its inhabitants would worship only their own god, stubbornly refusing to worship the Greco-Roman (Olympian) gods of the Pantheon. Even when the Roman authorities generously offered to provide a prominent place for their god in the Pantheon, the Jews insisted that their god would be satisfied with nothing less than a monopoly.

Things only got worse once a certain Jewish sect (called Christians) started swarming all over the Empire, seeking to convert anyone and everyone else to their way of worship, in exchange for divine forgiveness and the promise of eternal life – an offer that, for many, was too good to refuse. As devotion to the Pantheon waned and eventually ceased, so did the world cease to benefit from its blessing, and much of it entered what would later be called the "dark ages". Since then, only the Muses have been able to have any influence on or interaction with the mortal world, and then only when a poet or musician might be persuaded to honor them with an occasional phrase in acknowledgement of their help and inspiration.

The dark ages are still going on for Zeus and company, but lately there has been a glimmer of hope. Not very long ago Zeus summoned us all to Mount Olympus with the news that support for that monopolistic god (the one that so rudely put us out of business) has been steadily declining. He said, "Now that mortals are increasingly being encouraged to choose their own beliefs and values, there is practically no limit to what some of them have been (and could be) persuaded to believe. And if that’s not incredible enough, some have even been foolish enough to have chosen not to believe in any gods – now that’s really frightening!"

He then paused for a moment to ensure that he had our full attention, at which point his face took on a mischievous smile as he prepared to make his conclusion. "This is our chance to restore ourselves to a position as one of the world’s great religions – to regain a respectable share of the market. But in order to accomplish this, we would have to set aside our disputes and differences and agree to work with one another."

No sooner had he finished, than Mount Olympus suddenly became filled with joyful shouting, cheering, and merrymaking that lasted through the night and continued for several days thereafter (as only the gods know how to celebrate).

* * * * * *

Once the festivities had run their course and the participants had sufficient time to recover, Zeus appointed my half-brother Apollo to form a Worship Restoration Committee. As Messenger of the Gods, I, Hermes (you may be more familiar with my Roman name, Mercury), was the logical choice to assume the role of public relations manager, with responsibility for implementing a plan to accomplish our objective.

We faced a great dilemma in that we were not able to appear to any mortals unless they already believed in us, and those few mortals who did happen to believe in us were invariably judged by others to be highly eccentric or insane, thereby nullifying any testimony they might give for our existence. What to do?

I roamed the earth for a time, unseen by its inhabitants, but merely observing them to see if any ideas might occur to me. I could not help noticing their increasing preoccupation with computers and the Internet, through which it was possible for them to take on new identities in the alternate reality of cyberspace, where individuals, although separated by great distances in the physical realm, are able to discuss all sorts of interests ranging (as the saying goes) from the sublime to the ridiculous. Hmmm, I thought: This just might have possibilities.

Taking advantage of the fact that many universities have computer facilities available to their students around the clock, I entered into one of these places and waited till everyone had left. After a bit of experimentation I soon discovered a way to generate electrical impulses in a computer that correspond to the pressing of keys and movement of a mouse. In short order I started "surfing the net" to see what I could find.

After extensive searching on various individuals and topics associated with the ancient world, I happened upon the Yahoo! Alternate Tunings group and quickly realized that they had at least a couple of things in common with us – an appreciation for the artistic and mathematical heritage of the ancient world, and alienation from an established order that does not take their interests seriously. While we were marginalized by a monopolistic deity, they are ignored by a musical establishment that insists that one tuning should have a monopoly in both theory and practice. Surely there would be some here who would be able to identify with our plight, with whom we would be able to begin rebuilding our following.

I returned immediately to Mount Olympus and reported what I had found to the Restoration Committee. As one of the principal gods of music (specializing in rational harmony), Apollo was at first so delighted that he could hardly contain himself. But once we had gotten past the initial excitement, we began to consider a few hard realities. In the first place, as gods worthy of worship, what did we have to offer these Yahoo! Alternate Tunings enthusiasts in return?

Artemis noted that, because of their noble efforts to advance theory and practice in the arts, most of them would probably be judged worthy for admission to Elysium, where they would join the ranks of the blessed in the Underworld and thereby enjoy the company of the ancient philosophers and musicians that had preceded them.

Dionysus (pronounced di-o-NY-sus) responded that rewards in the afterlife are well and good, but these mortals would be more interested in what they might receive in the "here and now."

As if on cue, Calliope cleared her throat and proceeded to remind us that it was she and the other Muses that had provided the inspiration for countless artists, poets, philosophers, and musicians through the ages. She eloquently drove home her point by reciting some lines from the poet Hesiod (The Theogony, English translation by Apostolos N. Athanassakis):

Here are the words the daughters of aegis-bearing Zeus,
The Muses of Olympos, first spoke to me.
"Listen, you country bumpkins, you swag-bellied yahoos,
We know how to tell many lies that pass for truth,
And we know, when we wish, to tell the truth itself."
So spoke Zeus’s daughters, masters of word-craft,
And from a laurel in full bloom they plucked a branch,
And gave it to me as a staff, and then breathed into me
Divine song ...

Once she finished we all sat in silence, stunned by the word "yahoos," as it seemed to be a veiled prophetic allusion to those in the Alternate Tunings group – as if she somehow knew that this meeting would one day take place.

But once more Dionysus brought us back down to earth by insisting that these mortals would need to be presented with something tangible and delightful to the senses – something enjoyable that they could see or touch.

Apollo countered that he thought it should also be useful or practical – preferably something that would appeal to the intellect.

I responded that they now have computers and all sorts of other gadgets. What could we possibly give them that they don’t already have? After further discussion, we concluded that my next assignment would be to consult the Alternate Tunings group archives to find the answer to that question.

As I read through numerous messages I found that there were three things about which there seemed to be a lack of consensus for this, that, and the other tunings: notation, notation, and notation. After studying a few of the notations that had been proposed or used for the various tunings, I reached the conclusion that the best solution would be to devise a new comprehensive notation that would handle all of the tunings.

Once again I returned to the Mount Olympus Restoration Committee with my recommendation. We quickly agreed that, for a project of this sort, we were going to need all the help we could get. This was going to involve some mathematics, so we decided that I should seek out the best talent in the Elysian Underworld to help us with a few things, including some of the grunt work, such as number-crunching. For this purpose I enlisted Pythagoras of Samos, Plato, and Didymus the Musician as the core of the Elysian notation committee. They were only too happy to oblige, for inasmuch as only the worthiest of believers in the Olympian gods are admitted there, newcomers are now very few and far between. (Those who believe in other gods go to other destinations in the afterlife, depending on a combination of their beliefs and conduct, but beyond this, I can tell you no more.)

Until recently these luminaries of the ancient world would have been ill equipped for this task, for they knew next to nothing about the important developments in music and mathematics over the past millennium. (As I recall, the last sane mathematician I had the privilege of escorting to Elysium was Hypatia of Alexandria, and brilliant though she is, she arrived nearly 1600 years ago.)

That all changed with the passing of a hydraulics engineer who, believing himself to be Archimedes, had spent his last years in a mental institution, most of it in the bath. He was well educated in both physics and mathematics (besides being an amateur musician), and upon his arrival in the Underworld he passed along his knowledge of these subjects to a sizable group, many of whose names you would immediately recognize (including the real Archimedes, who wisely remained incognito so as not to jeopardize this arrangement). In return, his students rewarded him with such attention and respect as he had never before enjoyed, despite the fact that he insisted on lecturing them stark naked from a steaming tub.

* * * * * *

It was now the time of year when Persephone, daughter of Zeus and Demeter, made her annual journey from Olympus to rejoin her husband Hades, God of the Underworld, for the coming of winter, so it seemed fitting that I should accompany her there to inform him of our plan. I also obtained permission for Apollo and Artemis to visit Elysium from time to time to observe the progress of the committee, which he granted on the condition that they not allow themselves to be seen by its inhabitants. Therefore, as the only Olympian god having direct contact with the Elysian Committee, it fell to me to coordinate their activities with those of the Olympian Committee.

At our very first meeting in Elysium we agreed that the new notation should be backward-compatible with existing notational practice, so that we would retain the conventional staff with 7 nominals representing pitches in a chain of fifths from F through B. As in conventional notation, this chain of fifths may be extended in either direction by the employment of sharp and flat symbols (and doubles thereof) used in conjunction with these 7 nominal notes. For each division of the octave the notational "fifth" would be defined as that interval most closely approximating the Pythagorean fifth (with ratio 2:3).

We observed that certain tunings such as the Pythagorean tuning, the meantone temperament, 12-ET, and 19-ET are capable of being notated entirely with conventional symbols, even though some intervals may vary considerably in size from one tuning to another. Any of these tunings having more than 12 pitches per octave will have enharmonic pairs of tones with sharps and flats that differ in pitch, with the sharps being higher in pitch than the flats for tunings having fifths wider than those in 12-ET (such as the Pythagorean tuning) and lower in pitch for tunings having narrower fifths (such as the meantone temperament and 19-ET). In 19-ET there is a single circle of 19 fifths, with E-sharp and F-flat (and also B-sharp and C-flat) being equivalent in pitch and the remaining sharps being equivalent to double-flats (and vice versa). Naturally, it would be necessary for a composer to indicate the intended tuning in a musical score, since the symbols themselves would not be sufficient for that purpose.

Didymus observed that tunings in which sharps and flats differ significantly in pitch will contain scales of the ancient enharmonic genus, and he expressed the opinion that the abandonment of the meantone temperament for a 12-tone octave was not progress at all, but a giant step backward.

Plato laughingly agreed that western civilization would have some catching up to do if it ever wanted to hear the music of the ancients played on its most popular instruments.

Before leaving, I briefed the committee on several matters, including the divisions of the octave that were the most popular with the Yahoo! Alternate Tunings group, those features of existing notations with which I was most impressed, and the logarithmic interval-measuring system of cents (hundredths of a 12-ET semitone) devised by Alexander Ellis in the 19th century. As they began studying these things in more detail, I returned to Mount Olympus.

Once I showed the members of the Olympian committee the various symbols used in notations that I had seen, they agreed that these were the best ideas:

1) Arrows pointing up or down that have been used to indicate alterations in pitch in each direction, most often (but not always) for quartertones;

2) Multiple vertical strokes (used by Richard Stein in 1909) to indicate multiples of a semi-sharp;

3) Sloping lines (used by Robert H. M. Bosanquet around 1875) to indicate commatic alterations in pitch (smaller than a semi-sharp).

After suggesting that we begin by deciding how to notate half-sharps and half-flats, I explained that most composers writing for 31-ET have followed Adriaan Fokker in using the van Blankenburg / Tartini fractional flats with the Stein fractional sharps, whereas quartertone composers have over the years used a great variety of symbols, including arrows and the Stein symbols. (See the figure depicting quartertone arrows vs. Stein's symbols; the latter are shown with Mildred Couper’s sesquiflat symbol.)

Apollo thought that it would be desirable that our new notation combine the best features of existing notations, and without offering any further explanation he decreed that this could best be accomplished if arrows were used for the half-sharps and half-flats. When I questioned him as to whether he failed to appreciate the logic of the Stein fractional sharps, he responded that the examples I showed him of attempts to extend those symbols in logical fashion to finer divisions of a semitone were too cumbersome. (See the figure showing Wyschnegradsky's symbols for 72-ET, which includes yet another semiflat symbol.)

Ivan Wyschnegradsky's symbols for 72-tone equal temperament

* * * * * *

Apollo accompanied me on my next trip to the Elysian Underworld, along with Artemis his twin sister (known to the Romans by the name Diana and, like Apollo and myself, highly skilled in archery). I situated them on a rock ledge high enough to provide sufficient darkness in which they could easily avoid being seen by the notation committee, yet from which they could comfortably see and hear everything that took place. As I made my way to where the members of the committee were gathered, I could hear that they were in the process of discussing whether the arrows or the Stein symbols would be better to indicate the semisharp and semiflat. Since I was curious to see what conclusion they might reach, I decided not to interrupt them with the news that the issue had already been settled on Mount Olympus.

Pythagoras favored the arrows for their clear indication of the direction of pitch alteration, whereas Plato liked the Stein semisharp symbol, since its meaning was so obvious. Didymus liked both of these, yet wished that there were another alternative that combined the invertibility of the arrow with the intuitive appearance of the Stein symbol.

Pythagoras and Plato pressed Didymus to make up his mind, but Didymus insisted that he would not be rushed, for this decision was extremely important in that it would affect the notation of many octave divisions (as a single degree of 17, 24, and 31; as 2 degrees of 34, 41, and 48; and as 3 degrees of 72; or simply as a generic quartertone), as well as 11-limit just intonation (for altering tones by an 11-diesis, 32:33, an interval of approximately 53 cents).

By this time Apollo had lost his patience and decided to settle the matter his own way. He placed one of his distinctive golden arrows into position against his bow and shot it directly into the center of the table around which the committee was seated. As startled as they were at this sudden interruption, they were even more startled when they realized that I was not the one responsible for it.

Their eyes grew wide as I explained that Apollo himself had made a special visit from Mount Olympus to cast the deciding vote: that the arrow was to be the 11-diesis symbol. Filled with awe, Pythagoras slowly uttered the phrase that would henceforth be remembered in honor of this momentous occasion: "The Apollo 11-diesis – one quarter-step for a tone, one giant leap for tonality!"

Now that we had gotten the committee started on the new symbols, I decided that I would go up to where Apollo was concealed to commend him for his resourcefulness. Meanwhile the committee had gone back to work to decide how to notate multiples of a semisharp and semiflat. To Pythagoras this seemed quite simple: Since mortals already had symbols for single and double sharps and flats, all that remained was the sesquisharp and sesquiflat (i.e., one-and-a-half times a sharp or flat), and he suggested using an arrow in combination with a sharp or flat symbol for that purpose (as some quartertone composers have already done).

Neither Plato nor Didymus was pleased with that idea, for it meant not only that two symbols would be required to modify a single note, but also that this would not incorporate Stein’s idea that vertical strokes could symbolize in a single symbol the number of half-sharps the symbol represented. Pythagoras immediately pointed out that the Stein feature was present only in the fractional sharps and not the flats. Didymus did not respond, but only stared in silence at Apollo’s golden arrow, which was still embedded in the tabletop. As the minutes passed and Didymus appeared lost in thought, Plato could only shrug his shoulders about what to do next.

"Does this sort of thing happen very often down here?", Apollo asked me, in a whisper. I explained to him that these folks were now accustomed to enjoying their "retirement" from the responsibilities of their earthly lives. Although their minds are no less keen than before, their creative abilities are greatly diminished now that they no longer have contact with the Muses as a source of artistic inspiration. While they are quite capable of, and even relish opportunities for learning new things, their days of glory are long past, and much of their time is now spent sharing memories with one another.

Suddenly, as if to change the subject, Apollo pointed to his golden arrow embedded in the tabletop below and challenged me to shoot one of mine as close beside it as I possibly could. Taking careful aim, I dispatched my bronze arrow, which landed so close to Apollo's that the arrowheads were as one, with the arrow shafts parallel, but not touching one another. Once again Pythagoras and Plato were startled, but Didymus did not react until several seconds afterward, and then only slowly did his eyes light up and a broad smile appear on his face as he turned toward Pythagoras to answer his question: "There is your new flat symbol – a downward pointing arrow with two shafts." Without hesitation, Apollo carefully drew his bow and shot another of his golden arrows, which landed close beside mine, directly opposite his first one. Without missing a beat, Didymus continued, "And there is your new sesqui-flat symbol – a downward pointing arrow with three shafts. If we invert these, we then have our new sharp and sesqui-sharp symbols, so that both the fractional sharps and flats now have the Stein feature."

Satisfied with the resolution of the arrow-Stein issue, I quickly rejoined the committee and arrived just as Pythagoras was posing his next question for Didymus: "Would you then use an arrow symbol with four shafts for the double flat, and if so, would that not be liable to be mistaken for the sesqui-flat symbol?"

As Plato was in the process of voicing his agreement with Pythagoras’ concern, Apollo directed Artemis to go some distance along the ledge so that she could shoot one of her silver arrows so that it would penetrate the table top at an angle. Once she had done that, he then shot another of his golden arrows from another part of the ledge (in the opposite direction) that impacted the table top so as to form an "X" with her arrow. The crossed golden and silver arrows were immediately recognized by all as belonging to Apollo and Artemis, respectively.

"I believe the gods have spoken!", Didymus declared with great delight, as he observed that a single arrowhead with crossed shafts would be easy to distinguish from the other three symbols. When inverted, this would become the double-sharp symbol, with the crossed arrow-shafts serving as a reminder that this symbol was replacing the small "x" that, contrary to logic, was presently being used for the largest upward alteration of pitch.

Now that they had been presented with a sequence of symbols that was both logical and elegant, both Plato and Pythagoras enthusiastically agreed that the conventional sharp and flat symbols (those "haphazard relics of the dark ages", as Plato fittingly described them) should be replaced with new ones better suited to the music of a new millennium. (See the figure showing Apollo’s logical and elegant "pure Sagittal" solution.)

With these four pairs of symbols we could now notate the 17, 24, and 31 divisions of the octave, all of which have the apotome (a concise name for a chromatic semitone, the interval of alteration indicated by a sharp or flat) as two degrees and Apollo’s 11-diesis (single-shaft arrow) symbol as a single degree (a half-apotome in these three tunings).

Inasmuch as we had a symbol associated with Apollo, Pythagoras suggested that the new apotome symbol should be dedicated to King Zeus, since, with a little stretch of the imagination, one could see in it a resemblance to the multiple branches of one of his lightning bolts.

Zeus with lightning bolt

Plato thought it was good that we were keeping the existing natural sign, which resembles a pair of interlocking half-arrows, one pointing up and the other down, thereby cancelling each other out. Since this symbol has some similarity to the lowercase Roman letter "h" and somewhat resembles a throne, he proposed that it be dedicated to Zeus’s wife, Queen Hera, who spends much of her time avenging her husband’s infidelities, just as a natural sign cancels the effectiveness of sharp and flat (apotome) symbols.

Hera on her throne

Once these things had been agreed upon, the committee decided that it was almost time to take a lunch break, so I excused myself to rejoin Apollo and Artemis. As we began leaving, Apollo informed me that he was getting hungry, but I had previously cautioned him not to eat anything from the Underworld. He therefore asked me for one of the honey cakes that I had brought with me from above. I advised him that we would need them to feed the dog, Cerberus, that guards the gate to the Underworld – a distraction that would enable us to make a quick exit.

Meanwhile, Artemis had something else on her mind. As I later learned, she was miffed because Apollo hadn’t given her more opportunity to show off her archery skills, which might have persuaded the committee to dedicate a symbol to her. So she lingered behind for a moment to provide a demonstration. Drawing her truest and sharpest arrow, she loosed it, striking Apollo's 11-diesis arrow just behind the head, cleaving both its head and shaft in two. The smaller half was knocked sideways and landed on the ground some distance away, where it was not discovered until later. This all happened so quickly that Apollo and I were completely unaware of it, but the committee was a bit frightened, not knowing what to make of it.


When we reached the bank of the Acheron, Charon was already waiting with his boat to ferry us back across its waters, and before long we made our way back to the surface, where we had lunch. As we dined, Apollo explained that our next task would be to find a way to extend the arrow symbols in a simple and logical fashion to notate the intervals required for 72-ET.

* * * * * *

Once we finished eating, Apollo and Artemis decided that they wanted more time in the fresh air and would wait for me while I gave the committee their next instructions. Upon my arrival back in Elysium I announced that we would be working on the notation for 72-ET, beginning with a single degree of 1/6 apotome.

Pythagoras expressed his concern that, before we decided on a symbol for this interval, perhaps we should clarify how that interval should be defined. He had brought a small harmonic canon (a multi-stringed instrument with movable bridges) to the meeting to enable the committee to try out some of the tunings used by the Yahoo! group, but he had become dismayed to learn that most of these would be temperaments – tunings involving irrational ratios that would be difficult to measure as string lengths – rather than justly intoned scales involving rational intervals. Ellis’s cents, while useful in comparing intervals in various tunings with one other, were of little use in determining string lengths for tempered intervals.

I explained that much of the sophistication of the harmonic system developed over the past several centuries depended on the ability to modulate freely within the 12-ET tuning. Now that the resources of that harmonic system had become virtually exhausted, the Alternate Tunings group was exploiting higher divisions of the octave for new harmonic resources.

Plato then responded that, although equal divisions of the octave allow all of the tones to serve in all possible roles, they don't serve any of those roles nearly as well as they do in just intonation (where they are permitted to be more specialized) – as if a carpenter, stonemason, blacksmith, baker, and tailor were routinely called upon to perform each other's tasks in a game of musical chairs. Perhaps if western civilization had stuck to rational harmony, it would not now have to consider tunings incompatible with the present one (12-ET) to find new harmonies to its liking.

Picking up on this thought, Pythagoras suggested that all of the new symbols in the notation should be defined as rational ratios rather than degrees of an octave division. We could then follow the very same principle that we adopted in notating a chain of just (Pythagorean) fifths by defining the new symbols as commas – small rational intervals that would modify the tones in a single chain of Pythagorean fifths to arrive at pitches that produce consonant intervals with the tones in the original chain. The symbols for these commas could then be used to designate appropriate numbers of steps or degrees in various octave divisions, according to the harmonic properties of each division. From one tuning to another the actual (melodic) size of the interval represented by a given symbol might vary somewhat, but the harmonic meanings of the symbols would remain constant across all tunings.

He then presented an example: Four just fifths (of ratio 2:3) exceed two octaves and a just major third (4:5) by the comma of Didymus (80:81). If one starts from C and proceeds by just fifths to a Pythagorean E, an E on a manuscript would then be modified by the particular symbol that we would choose to represent a downward alteration of Didymus’ comma in order to notate the slightly lower E a just major third above C. This symbol could then be used to represent a single degree of 22, 41, or 72-ET, since this is the amount in each of those divisions by which the major third in a chain of fifths must be narrowed to arrive at the best approximation of a just major third. Therefore, in each of these tunings the E modified by our Didymus-comma-down symbol would have the same harmonic meaning in those tunings that it has in just intonation. In divisions with narrow fifths (such as 19 and 31-ET), Didymus’ comma is tempered out (and thereby vanishes), so that particular symbol would not be used for those divisions.

Plato noted that the practice of using the same notational semantics to indicate intervals that differ somewhat in size from one tuning to another has a parallel in written languages and dialects that share the same alphabet. For a given letter the sound may vary somewhat from one language to another (or even within a language), but within certain boundaries there is a commonality of sound across those languages that enables a person to transfer the knowledge already possessed in reading one of them to read another with little difficulty, once the meanings of the words are understood.

With that issue settled, I then explained that a popular way of notating Didymus’ comma is with the sloping lines used by Bosanquet in the 19th century. Didymus volunteered to illustrate this by making some sketches in the ground. Looking around for a convenient writing stick, he happened to pick up the half of Apollo’s golden arrow that Artemis had cleaved. With its sharp point Didymus drew a natural sign and downward and upward-sloping lines to the left and right of it, respectively. These resemble the backslash and slash characters on a computer keyboard: \ and /.

Plato was concerned that sloping lines do not differ sufficiently in appearance to avoid the potential for confusion as to the intended direction of alteration, and the others nodded in agreement. While we were discussing this, Inanates (pronounced in-AN-uh-teez), the half-wit that had been observing our activities all day, dropped some Asphodel seeds into the depressions that Didymus had created in the ground and brushed the dirt back into place to cover them. The others teased Didymus about taking committee time to do his gardening with his makeshift dibbler, which gave us all a good laugh.

With his artwork obliterated, Didymus placed what remained of Apollo’s golden 11-diesis arrow on the table so that the half-arrowhead happened to be at the left side of the shaft and pointing away from him. Not noticing until now that the arrow had been split, I was about to ask what had caused this, when Didymus suddenly exclaimed that here was a better symbol for his comma. He picked up the half-arrow and held it so that it pointed upward with the half-arrowhead sloping upward (from left to right), giving a clear indication of the direction of pitch alteration for his comma. "Yes!" I responded, slapping my palm against his raised hand in the manner of a "high-5" salute – an appropriate gesture, I thought, to reinforce the point that his 5-comma symbol would be used to notate ratios of 5 (except that, in my exuberance, I forgot that this custom was not practiced in the ancient world, and Didymus was somewhat taken aback.).

Continuing with his illustration, he inverted the arrow while keeping the half arrowhead at the left side of the shaft, so that we saw it point downward at the same time that the half-arrowhead sloped downward (from left to right). With this example we established the principle that any new symbol would be inverted not by rotating it by 180 degrees, but rather by mirroring it vertically.

Once again I was about to ask what had happened to Apollo’s arrow, when Inanates handed me a ragged piece of parchment on which he had written the following rhyme that summarized the day’s events:

Apollo's a fellow whose arrow is yellow,
eleven for heaven and quartertone mellow.
Zeus gets a double and Hera a nullo
and takes her revenge on many a fellow.
Artemis' cleavage makes Didymus' dibble
when Apollo's arrow is split down the middle.

This puzzling reference to Artemis suddenly brought to mind the misfortune of the hunter Actaeon in his unexpected fatal encounter with the virgin goddess, and I fervently hoped that its meaning was not what I thought. Deciding that I should immediately get to the bottom of this, I excused myself and, taking both the split arrow and parchment with me, made a quick exit to rejoin Apollo and Artemis above.

Once I had confronted Artemis with the evidence I had gathered, she confessed to her impulsive act as Apollo slowly shook his head in disapproval. But when I explained how this had resulted in Didymus’ modification of Bosanquet’s slashes to arrive at a new 5-comma symbol, Apollo was all smiles. Using the "dibbler" to make a few sketches in the ground, he quickly devised a method for extending these arrow symbols to notate all of the intervals required for 72-ET, from 1/6-apotome to a double apotome (the complete sequence in Wyschnegradsky's table) – all with single symbols, simple enough that none would require more than five straight strokes of a musician’s pen!

I then went back to the Elysian committee, where I quickly explained Apollo’s fait accompli for 72-ET. At one point Plato seemed to express a look of concern, but didn’t say anything.

Since we realized that subsets of these symbols could also notate several other lesser divisions of the octave, it seemed that our task had been completed. After thanking the members of the Elysian committee for their help and returning Didymus’ dibbler to him as a keepsake, I departed.

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Copyright © 2004, 2007 George D. Secor and David C. Keenan