Gift of the Gods

Episode Two

Archytas Takes a Bow

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

Previous Episode

Once we had returned to Mount Olympus, Apollo, Artemis, and I presented the results of our work with the Elysian committee. Having completed the notation for 72-ET (as well as several other popular divisions of the octave), we decided that it was time for me to present our gift to the Alternate Tunings group.

Dionysus was beginning to delight in his talent for identifying problems (or creating them if he couldn’t find any), and he now pointed out that we were facing a dilemma: If I were to join the group as "Hermes, famed Messenger of the Gods" (as he put it), either nobody would take me seriously, or they would be so absorbed with trying to persuade me to reveal my "true" identity that it would be difficult for me to get them to concentrate on the notation. We agreed that I would need to assume an alias.

But even that would present problems. If I were to use a fictitious name, I would be regarded as a newcomer whose proposals for microtonal notation would probably not be considered too seriously by very many in the group. In order to be treated as a credible microtonal authority, it might be better to impersonate someone with an existing reputation working with alternate tunings who would be respected by the members of the group – a non-participant who was also not in frequent contact with anyone else in the group. But who?

The answer to that question could likely be found in the archives of the Alternate Tunings group, so Apollo and Artemis volunteered to help me scan the thousands of messages posted since the beginning of that year (2001). We were not engaged in this for very long when Apollo made the comment that, although the active participants in the group shared a common goal, he observed occasional differences of opinion that resulted in lively and provocative discussions reminiscent of the ancient philosophers of the Areopagus (Mars Hill forum) in Athens. These moments made our work much less tedious.

Artemis, Apollo and Hermes

As Artemis reached the end of the messages for April she could not help noticing a subject thread proclaiming "DAVE KEENAN’S MIRACLE SCALE" initiated by Paul Erlich, nor for that matter, one (in mock retaliation) titled "PAUL ERLICH’S AMAZING 11-LIMIT GENERATOR" by Dave Keenan, nor a third one titled "MIRACLE OF MIRACLES" by Paul Erlich, all expressing excitement about the remarkable characteristics of a "MIRACLE" tuning that had just been discovered jointly by Dave and Paul. As we read through these messages, I could not help but lament, "What a name for a tuning! If only we had come up with that."

But as we read through additional subject threads that went on for weeks discussing the possibilities of the new MIRACLE tuning, we learned that it was originally discovered over a quarter-century earlier by George Secor, whom they honored by agreeing to name the Miracle generating interval the "secor". Unfortunately (for them, but fortunately for us), no one seemed to have been able to contact the honoree to inform him of this. "By Jove!", Apollo exclaimed. "Hermes, I believe we’ve found an alias for you! This is our chance to give these folks a real miracle – a notation by divine revelation!"

After we had searched the archives to learn as much as we possibly could about Secor and his work with alternate tunings, I obtained a Yahoo email address in Secor's name and joined the Alternate Tunings group. In my introductory message, I explained that I was re-establishing contact with the microtonal community after nearly 20 years of inactivity, thus providing a convenient explanation for my lack of knowledge regarding recent developments that might be brought up in the course of discussion.

After congratulating Paul Erlich and Dave Keenan on their rediscovery of the Miracle tuning, I began my presentation of the Sagittal notation as a new performance notation for 72-ET that was downward-compatible with a number of other divisions of the octave (including 17, 19, 22, 24, 29, 31, 34, 36, and 41), while incorporating the best features of previous notations. The 5-comma and 11-diesis symbols were generally well received, but three major issues were raised as obstacles to the acceptance of the notation as I had presented it.

There was strong opposition to the replacement of the conventional sharp and flat symbols with double-shaft Sagittal symbols by a few members (most notably Joseph Pehrson and Dave Keenan), even though there were a few others who seemed to favor the idea. It appeared that most of the members preferred to withhold comment or to defer judgment on this issue until they had a chance to see how the notation would work in some actual musical examples.

Lacking any clear consensus, we eventually decided that the notation would need to be offered in two forms: 1) with pure Sagittal symbols (both single- and multi-shaft), and 2) as a mixed-symbol option that combines single-shaft Sagittal symbols with conventional sharp and flat symbols, as Pythagoras first suggested. Although it would lack the Stein feature, the mixed-symbol version would have the advantage of a gentler learning curve (since fewer new symbols would need to be learned), while the pure version would have the advantage of greater clarity (i.e., a less cluttered appearance and reduced ambiguity) on a manuscript containing chords.

None of us in either the Olympian or Elysian committees anticipated the sheer multitude and complexity of tunings that some would want to notate, as demonstrated in the following conversation with Dave Keenan and mathematician Gene Ward Smith (who chimed in to confirm that Dave’s viewpoint was shared by others in the group):

Dave: I'm curious as to what George Secor proposes to do about these. So far George, you've been notating the easy ones. How does your notational semantics handle 20, 25, 27, 28, 30, 32, 37, 42, 44-tET?

My reply: So I'm notating the easy ones, huh? I thought I was notating the good ones. I threw systems like these into my microtonal garbage can during the first week or so that I spent investigating microtonality (yes, rejected without a hearing, late in 1963), and since that time I have never even entertained the thought of making a trip to the microtonal junkyard to reclaim them.

Gene: If you really threw 27 into the garbage heap as well as 58, I think you'd better take a trip to the junkyard and reclaim them. :)

My reply: And I think not. As I observed in my very first posting, life is short, and we must establish our priorities for ourselves. Considering all of the music that has come out of the resources of a 12-tone octave, I believe that those tonal systems that I have judged to be excellent would occupy me for many lifetimes, so that I have no need to concern myself with others that I perceive as less worthy of my time and effort.

You and your fellow "Scavengers" (which warm appellation I intend not to be taken in any derogatory sense) are more than welcome to these "treasures" that have been thoughtlessly discarded by myself and others of like mind, even as Margo Schulter offered some very beautiful words (re: microtonal "recycling"):

<< Since, as Henry James has said, life is "all inclusion and confusion," and art "all discrimination and selection," we all make choices, explicit or implicit; one musical style's "garbage" may be another's cornerstone; Therefore, we might respond most nobly by seeking to understand each other's choices -- and, if so inclined, taking them as a creative incitement to make beautiful music in "rejected" tunings, thus recycling the "garbage" in most fertile fashion. >>

And so the very essence of discovery, innovation, and creativity is to see beauty or utility in those places where the rest of us pass by, unaware of the possibilities.

Gene’s response: I'm simply doing the mundane math here.

Paul Erlich also insisted that it would be essential that a comprehensive microtonal notation handle divisions of the octave with significantly more than 72 tones, and he expressed a particular interest in 152-ET.

True, it was possible for me to argue that certain divisions of the octave would be most popular on the grounds that they more accurately represent the small-number ratios of just intonation, or that they would be most practical on the grounds that they contain fewer tones than larger-numbered divisions. But I could not get around the argument that, if the Sagittal notation were intended to be used by all microtonal composers, then it must not only be able to notate just about anything that anyone might want to do, but also just about anything that anyone thinks that they might want to do. This would include not only tunings "scavenged" from the microtonal junkyard, but also tunings with hundreds of tones in the octave that might be of interest to the theorist or composer of electronic music. At the same time it would have to be simple enough to be practical as a performance notation, capable of being read rapidly in real time.

Dave Keenan (who was beginning to rival Dionysus with his persistent criticism) also objected to the problem of "left-right confusability" in a notation that contains symbols that are lateral mirror images of one another, as was the case with a few of the symbols for 72-ET that Apollo had devised. After it became apparent that I could not convince him that the problem was not as serious as he imagined, I decided that we could remedy this at the same time as we considered ideas for expanding the notation in response to the previous issue, which was by far the most significant shortcoming.

With these three issues to address, it was clear that our Olympian and Elysian notation committees had their work cut out for them.

* * * * * *

Before returning to Mount Olympus with my report, I decided that I would first visit Elysium to reconvene the notation committee, which was now joined by tuning theorist Claudius Ptolemy of Alexandria, whom Didymus invited.

I began by explaining that the first problem with acceptance of the notation by the Alternate Tunings Group was resistance to the abandonment of conventional sharp and flat symbols, so that we would need to develop the notation in both pure and mixed-symbol versions. The latter would not employ any new multi-shaft symbols, but would instead combine single-shaft symbols with conventional single- and double-sharp and flat symbols. The mixed version might then serve as a transition to the eventual exclusive adoption of the pure version of the notation.

The next problem that we addressed was that of lateral confusability in 72-ET. Whereas Bosanquet used multiple diagonal strokes to indicate multiples of the 5-comma: /, //, and ///, or \, \\, and \\\, Apollo believed that he had a better idea. Since a left half-arrow (or dibbler), /| up and \| down, symbolized a single degree of 72-ET, his 11-diesis symbol with its full arrowhead (used for 3 degrees) could be interpreted as consisting of an arrow shaft with two half-arrowheads: /|\ up or \|/ down. He arrived at a symbol for 2 degrees by removing the left half-arrowhead from his 11-diesis symbol to leave only the arrow shaft with right half-arrowhead, |\ up and |/ down, as the difference between 3 degrees and 1 degree.

One of the half-arrowheads (or "flags", as we would call them from this point on) could also be removed from either side of each of the new Sagittal apotome, sesqui-apotome, and double-apotome symbols to arrive at a complete sequence of symbols for 72-ET from 1 to 12 degrees. Subsets of these symbols would also serve to notate the 22, 29, 34, 36, and 41 divisions.

The accompanying figure illustrates Apollo’s sequence of symbols for 72-ET (with red marks to indicate those that would be replaced). Each of the new symbols in the sequence is arrived at by subtracting either a left or right flag to reduce the amount of alteration produced by a full-arrow symbol (a multiple of 3 degrees) by either 1 or 2 degrees, respectively. The figure also shows how these symbols can be used to notate just major and minor triads, as well as tones representing harmonics 8 through 12 of a lower octave of C, exactly as they would be notated in 72-ET. (It is important to observe that, in the 3rd and 4th chords of the JI example, the slope of the right barb of each double-shaft symbol indicates the direction that the JI ratio of 5 differs from a Pythagorean sharp or flat, even though the symbol itself is pointing in the opposite direction. It should also be noted that the symbols that were about to be replaced are not ones that are used for 5-limit JI.)

I went on to explain that the specific problem that Dave Keenan and others had with this sequence of symbols consists in the fact that, since the 1 and 2-degree symbols are lateral mirror images of one another, they might be confused too easily when read rapidly (and likewise with the other three adjacent pairs of symbols that are not multiples of 3 degrees.)

At this point Plato volunteered the information that he had also noticed a potential for confusion between these symbols when I was hurriedly explaining Apollo’s complete 72-ET symbol sequence at our last meeting. But since Apollo seemed to be calling all of the shots (except for that last one that Artemis made), he felt reluctant to express his concern, even though the same problem had already been observed with the Bosanquet slashes.

Pythagoras was also concerned that the 2-degree symbol pair had not been defined as a rational ratio, but rather in terms of an octave division, giving us an additional reason to consider replacing this symbol.

I decided that it would not be necessary to get Apollo’s permission to work on this problem, since we all seemed to be in agreement that something needed to be done. While Apollo’s symbols had a logical elegance fit for a god, they would need to be adapted to the shortcomings of mere mortals, who are less proficient in distinguishing left from right.

After reviewing the simple-number ratios that were represented by various degrees of 72, we found that our problem could be remedied simply by creating a new symbol to represent pitches relating to the 7th harmonic. Most of these ratios of 7 could be arrived at by altering tones in a chain of fifths by a septimal comma (or 7-comma) of 63:64 (approximately 27.3 cents) in just intonation, which corresponds to 2 degrees of 72-ET (approximately 33.3 cents). So what should a new 7-comma symbol look like?

We agreed that all new symbols should be invertible so as to indicate clearly the direction that pitch is to be altered, which is most easily accomplished by using symbols that resemble arrows. Ptolemy suggested that, since we had already used additional arrow shafts to indicate an increase in size (by multiples of a half-apotome), perhaps we could make some sort of difference in the arrowhead to indicate this new comma, if not in size, then perhaps in shape – but what shape?

Plato suddenly remembered that his good friend Archytas of Tarentum, whom he had just seen the day before, had devised a diatonic scale in which this particular comma occurred. But unlike Pythagoras and Didymus, whom the mortals had subsequently honored by naming commas in their behalf, Archytas and his 7-limit scale had remained in relative obscurity, since traditional musical harmony does not recognize any consonances above the 5 limit. Plato recommended that we rename the 63:64 comma the "comma of Archytas", and the others quickly agreed.

I asked Plato to find Archytas, in order to brief him on the activities of the notation committee and to invite him to join us (but not to mention the new comma name that we had just agreed upon). Plato immediately did this, returning a short time later with his good friend, whom he warmly introduced to us.

I then explained to Archytas how we were attempting to solve the problem of lateral confusability in 72-ET with a new symbol for the 7-comma, 63:64, and that we had just agreed on a new name for it, but that we had yet to agree on a shape for the symbol arrowhead. After suggesting that the appearance of the symbol should somehow remind one of the new comma name, I had Archytas stand facing the others and then asked him to hold my arc-shaped bow with his left hand near one end, while he touched the other end of the bow’s arch to the top of his head.

The others made a great effort to suppress any hint of laughter as Archytas, stood there, looking much bewildered and a bit apprehensive. Suddenly his face lit up as he ventured a guess: Might the new 7-comma symbol have an arrowhead in the shape of a rainbow because of its seven colors? While everyone agreed that this would be a good memory aid, we also thought that we should immediately put an end to the suspense. No sooner did I finish my announcement, "Members of the committee, may I proudly present Archytas’ comma!", that the others broke into applause, not only to celebrate the creation of the modified half-arrow symbol with an arc on the (viewer’s) right-hand side, but also to congratulate the individual after whom it was named.

Archytas then gave a brief speech thanking his good friend Plato for remembering him in the committee’s time of need, and thanking the committee for bestowing this honor upon him, and also thanking me for preparing him with a proper attitude to receive such an honor. This was followed by even greater applause than before – a welcome contrast with the difficulties that I had encountered as a newcomer to the Alternate Tunings group, and even more so with the difficulties that I was about to encounter on Mount Olympus.

Shortly after the Elysian committee had gotten back to work, Pythagoras made the observation that, since the comma of Archytas is almost exactly half the size of Apollo’s 11-diesis, then a double-shaft symbol with a right-hand arc would serve as the apotome-complement of its single-shaft counterpart, which would give us a complete revised set of symbols for 72-ET, as shown in the accompanying figure. As before, the symbols used for lesser divisions of the octave contain only straight-line flags, which we now called barbs, in order to distinguish them from convex curved flags, or arcs, that are used to notate ratios of 7.

As I had used the slash "/" and backslash "\" characters on the computer to represent barbs, from this point on I would need to use a right parenthesis ")" to represent Archytas’ arc for the 7-comma-up symbol, |), and ||) for its double-shaft apotome complement. In order to represent the downward symbols, the vertical bar or "pipe" characters would now be replaced with exclamation marks, !) and !!), and the upper-case "X" character for the upward direction in crossed-shaft symbols would now be replaced with an upper-case "Y" character to indicate downward alteration.

Now that the lateral confusability problem had been solved and Archytas would be contributing his mathematical abilities to the project, I instructed the committee to begin working on the task of devising a symbol that could be used for 3 degrees of the 39, 46, and 53 divisions of the octave, which we had overlooked earlier. (Note that the above figure shows divisions having 1, 2, 3, 4 and 6 steps to the apotome, but not 5 as required for the 39, 46 and 53-ETs.) They could also consider what symbols might be used for the commas that had been tentatively identified by Gene Ward Smith and Dave Keenan to notate harmonics beyond the 11-limit. Meanwhile, I would return to Mount Olympus to report on the problems posed by the Alternate Tunings Group and the considerable progress that we had just made. With the symbols for 72-ET, we were now able to notate equal divisions of the octave with the following numbers of tones: 2, 3, 4, 6, 8, 9, 11, 12, 17, 18, 19, 22, 24, 26, 29, 31, 34, 36, 38, 41, 65, 72 and 79; by taking 2, 3, 4, and 6 as subsets of 12; 8 as every third step of 24; 9, 18 and 36 as subsets of 72; 11 as every second step of 22; and 65 and 79 using the 72-ET symbol sequence.

I previously mentioned that Inanates was regarded as a half-wit, and indeed the others had given him his name in jest at the quality of the poems he was forever scribbling. But looking back over events, I suspect that in certain ways he was really quite intelligent, maybe even prescient, for just as I was about to leave, he again handed me a piece of parchment containing a rhyme. It made little sense to me at the time, so I tucked it into my robe as I departed:

Of the arc of Archytas and Didymus' dibbler
What can one say but that Zeus is a quibbler
Dionysus or Hermes, which is the fibbler?
Tippler tattler, fiddler riddler?
But enough from this adlibbing dribbling scribbler.
Do Didymus' thirds make him a tribbler?
Or Archytas' sevens a septimal nibbler?
Let Zeus consult the all-seeing Sybil re
Archytas' arcing and Didymus' dibblry.

* * * * * *

Upon my arrival at Olympus, Zeus eagerly summoned all of the Olympian gods to hear my report. First I informed them that I had successfully established a new identity in the Alternate Tunings group as George Secor, the original discoverer of the MIRACLE tuning, who was now making his comeback by presenting a new comprehensive "Sagittal" notation. While there was indeed a great need for, and consequently a great deal of interest in a notation that held out the promise of handling just about any tuning that anyone might want to use, the members of the group found some serious problems with our notation, not the least of which was that we had considerably underestimated the number of tunings that the notation would be required to handle.

I went on to explain that not only was the Elysian committee already working on these problems, but that we had already solved one of them, having enlisted the help of Archytas of Tarentum, the renowned musical theorist and mathematician. As I proudly showed Zeus and the others the modified set of symbols for 72-ET that eliminated the problem of lateral confusability that existed with our original set of symbols, there was suddenly a loud cry of protest from Dionysus. Although I couldn’t imagine what he could possibly find fault with in the new symbols, it seemed as if he were compelled to do his utmost to uphold his reputation as chief critic.

Everyone was stunned as Dionysus shouted at me, accusing me of subterfuge and calling me a traitor because I was using symbols pertaining to a foreign god – the flail and crook of Osiris, the ancient god of the Nile River, of corn and wine, of the rising and setting sun, and of the Egyptian Underworld. What an outrage to use the trademark of a god who had been in direct competition with certain esteemed members of our own Pantheon – Apollo, Demeter, and Hades! He was particularly incensed that Osiris (in his role as Egyptian god of wine) had on numerous occasions been confused with Dionysus himself (most often by misguided worshippers who were too drunk to tell a tulip from a turnip).


Osiris, whose flail and crook bear an uncanny resemblance to the symbols for the 5 and 7 commas

I tried to explain that the 5 and 7-comma symbols originate from Bosanquet, Didymus, and Archytas, and that the 11-diesis honors Apollo (at which point Apollo spoke up to confirm what I was saying was true), but Dionysus was not the least bit impressed. A division erupted, with Apollo and Artemis siding with me, while Ares (who, as God of War, rarely passed up a chance to promote a good conflict) and Athene (who is noted for her strong sense of loyalty to the Olympian cause) both sided with Dionysus. When Demeter turned her head to look at me, she did not have to say a word, her face expressing both hurtfulness and sadness.

Zeus was extremely upset, and after making the magnitude of his displeasure known with a sudden outburst of lightning and thunder, he sternly ordered me to fix the problem – the sooner the better, lest I incur the wrath of Hades!

Fortunately, Hades’ wife Persephone was with him in the Underworld and would not be returning to Mount Olympus for several weeks, so there was a good chance that he would not hear about this for a while. On the other hand, if I didn’t take care of this matter soon, the faux Archimedes wouldn't be the only Elysian in hot water. As I made my way back to the Underworld, I reflected on the thought that the objections of Dave Keenan and his fellow yahoos had suddenly become laughable in comparison to this latest turn of events.


To be continued ...

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