## Paul Rapoport

Dave Keenan
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### Paul Rapoport

Paul Rapoport's 24-page paper The Notation of Equal Temperaments was written in 1991 and published in 1995 in Xenharmonikôn 16. It was a major influence in the design of Sagittal — not so much for its symbols, but for its systematic approach to the meaning behind the symbols. This was aimed at exposing the structural similarities between temperaments, and doing so with a minimum number of symbols, which represent various commas and subdivisions of those commas.

Paul recently posted to the Microtonal Music and Tuning Theory facebook group looking for a suitable website to host a scanned copy of this paper. We had the following fruitful exchange:
Dave Keenan wrote: Hi Paul Rapoport. Great to hear from you. What is the article about?
Paul Rapoport wrote: Notation of equal temperaments, implying all the edos.
I know and love that paper. I have a printed copy. It was important in the design of Sagittal. I too would be happy to host it on my website, alongside Paul Erlich's Middle Path paper.
Thanks, although it's not in the same class as Middle Path. It's also more limited than Sagittal, as you know, although I've expanded its principles higher than 5-limit and in JI.

At some point I'll fix the mistake I know of and scan it. Meanwhile, thanks (?) for elevating me into the same list as Pythagoras, Tartini, and all those others (I'd forgotten that, from 12 years ago.)
I just found my copy to see if I had spotted any mistake. Among my annotations, the only possibility seems to be that you may have regretted suggesting the syntonic comma for notating 17-edo, since that division is not 1,3,5-consistent. Using the semi-sharp and semi-flat seems preferable. Was that it?
Nah. The semi-sharp and -flat may be preferable, and the syntonic komma gets you only a neutral 3rd, but it's viable. Blackwood doesn't even bother with that; his restriction to sharps and flats and their doubles is viable too, given 17's relation to 19.

In those days (1991, actually) I was big on k (as I labeled it) for 3-positive EDOs and, working somewhat in isolation, not knowing a fraction (!) of what many of you did, basically engaged in inductive (some deductive) ad hocery to solve the 5-limit notation problem. It was better than other, less systematic ad hocery that preceded it, that's all.

At this point, that paper is mostly a historical relic; as you may realize, I didn't do much in microtuning theory after that and my JMT article, which the XH16 article is really an extension of.

The error I found is small, in one of the charts. Don't waste your time…although other comments on the XH16 paper are welcome, preferably not on Facebook, where they'll get lost and where real dialog isn't possible anyhow.
Dave Keenan As for 17 EDO and the question of 1-3-5 consistency, 3/1 = 27 units, 5/3 = 12, and 5/1 = 39. I suppose that doesn't work because 13 units are very slightly closer to JI 5/3 than 12. For me, reconciling that with what I see as the structure of 17 is the issue, if you follow the kommatic line.

In general, the notion of "closer" is a difficult one, because a note that's sharp may be preferable to one that's flat, even if the former is further from just…consider the perf. 5th, for example.
You got it. 17-edo's best approximation of 5/3 is 13 steps. The definition of this kind of consistency only considers the best approximation of each ratio. But even without being so strict, in the design of Sagittal EDO notations, George and I consider it poor form to use a comma whose error is close to half a step, even if it is consistent, assuming you have another option.

Paul Rapoport What I love about that paper of yours is not so much the specific results you came up with (given that you limited yourself to prime 5) but your approach to the problem and ways of thinking about it. I felt a kindred spirit.
Dave Keenan Kindred spirits are good! Thanks. Maybe I've found a few here.

I'm not aware of a hierarchy of komma options, although I'm interested in that. Having it depend on size raises questions.

The syntonic komma is a big deal in 5-limit because of 5:4 and 6:5 (obviously), and you can see it working similarly as 1 unit despite changes in absolute size through the series 5 + 12n. Not that it's the only possibility or even necessarily the best.
Paul Rapoport You wrote: "... although other comments on the XH16 paper are welcome, preferably not on Facebook, where they'll get lost and where real dialog isn't possible anyhow."

I agree, and would love to preserve this conversation in the Sagittal Forum. http://forum.sagittal.org/
If I was to start a new thread there, regarding your paper, and copy our above exchange there, would you be willing to continue the conversation there?
Dave Keenan Thanks. Sure, assuming I have anything to say. Other than the obvious online description of Sagittal, is there more I should look at first? Perhaps a few salient parts of the forum.
The first thing will be to register with the forum, so you can post replies. I recently upgraded the forum software and am a little nervous as to whether that's still all working correctly.

Then by discussing your paper I could probably bring you up to speed on how Sagittal has extended on and modified the ideas it contains.

And I would eventually love your comments on our long-running effort to simplify the notations for the "bad fifth EDOs".
A proposal to simplify the notation of EDOs with bad fifths - The Sagittal forum
Okay. It'll be a little while.
How would you feel about me scanning your paper and posting it to the Sagittal forum? I can write in the correction.
If you want to scan it, thanks; just send me the scan first. E-mail xxxxx@xxxxx.xxxxx. My correction is for a = 54. Πh should be 2, not 0.
I have emailed the scan to Paul, and I await his approval to upload it here.

Dave Keenan
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### Re: Paul Rapoport

Dave Keenan wrote: Wed Jan 09, 2019 8:12 pm I have emailed the scan to Paul, and I await his approval to upload it here.
And here it is. Thanks Paul.

Dave Keenan
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### Re: Paul Rapoport

Paul. In the paper, you limited your investigations to notating EDOs using symbols representing 5-limit commas, and 1/n of those commas, but you encouraged investigation of higher harmonics. I understand you later investigated these yourself, as well as notations for JI. I'd be interested to know what you came up with there.

I always found it strange that you proposed a single symbol (similar to , but not as wide) that could stand for a fraction 1/n of the syntonic comma (which we call the 5-comma), where n could be any whole number greater than 1, chosen so that 1/n of the comma was a single step of the EDO. It was good in that it reduced the number of symbols required, but my objection was that it could lead to 1/2 of the comma being notated as in one EDO, while 2/4 = 1/2 of the comma was notated as ▶▶ in another EDO. And similarly for 1/n of other commas and of the sharp/flat (which we call the apotome).

Sagittal aimed to notate both JI and EDOs right from the start, and we found that commas for higher primes came in a sufficient range of sizes that we could avoid having symbols for fractions of any comma — almost. Although the symbol is defined as 32:33, it is also a defacto half-apotome symbol. And in our recent efforts to simplify the Sagittal notations for EDOs with extreme fifth sizes, you could argue that we are essentially smuggling fractional comma symbols in through the back door — at least for fractions of the apotome and limma — by dual-purposing other symbols for higher primes.

But what tends to happen is that within a small range of fifth sizes we have consistent symbols for particular fractions of the apotome, but different ranges have different such mappings, as we preserve the primary meaning of the symbols as higher-prime commas.

I'm interested in what you think of this. If you need more information, or the same information presented in a different way, don't hesitate to ask.

Another interesting relationship between your paper and Sagittal is that we totally agree that the syntonic comma is the most important notational comma after the apotome, so much so that we did not see the need to directly symbolise any other 5-comma until we came to EDOs with hundreds of steps to the octave, or JI with a required precision better than ±2 cents. At that stage we symbolised the 5-schisma, as a small diacritical mark or accent. So the commas of your paper are symbolised as follows:

syntonic comma     k
schisma            s
pythagorean comma  p = k+s
diaschisma         q = k-s
diesis             d = 2k-s


mschulter
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### Re: Paul Rapoport

Please let me thank Paul Rapoport for making available this classic article, and also Dave Keenan, Paul Erlich, George Secor, and others who have played a role in the dialogues which have shaped Sagittal notation and various theoretical approaches which may be related to it.

For the moment, I have mainly a quick thought on fractional comma notations. Generally I find the half-apotome symbol useful, along with conventional sharps and flats, for 17-ed2, as well as various unequal temperaments with fifths wide of pure lending themselves to thirdtone divisions (which may involve three unequal thirdtone sizes). An alternative would be to use Arab or Persian notation, for example, which for the relevant styles of music can be very useful.

While I'm not sure how useful a fractional comma notation would be in practice for 17-ed2, the concept can be useful for locating 17-ed2 on a map of what I'll call Blackwood's continuum of regular diatonic tunings from 7-ed2 to 5-ed2. Basically, 17-ed2 could be viewed as a 1/7-Archytan-comma temperament, the Archytan or septimal comma being 64/63 (27.264 cents). A sign of this is that the apotome or 2/3-tone in 17-ed2 at 141.176 cents is a virtually just 243/224 (140.949 cents).

In some ways, as George Secor may have suggested in his paper on his 17-tone well-temperament of 1978 in Xenharmonikon 18, 17-ed2 is to Archytan temperaments (e.g. 39-ed2, 22-ed2) rather as 12-ed2 is to meantone temperaments. Just as 12-ed2 is the upper bound of meantone (where +4 fifths is the best mapping of 5/4, at least in a reasonably small tuning size), so 17-ed2 is the lower bound of the Archytan temperaments where -2 fifths, for example, is the best mapping for 7/4. Easley Blackwood, as I recall, notes that 17-ed2 has a rough approximation of 12:14:18:21 (1/1-7/6-3/2-7/4), but one that's not so accurate, because the fifth isn't tuned enough in the wide direction to account for a larger portion of the septimal comma using a chain of only 2, 3, or 4 fifths. Likewise, 12-ed2 cannot approximate 4:5:6 very closely because the narrow temperament of the fifth is not sufficient.

But if I had to view 17-ed2 in terms of a single comma, I would choose the Archytan comma. What I actually find most noteworthy about 17-ed2 is its very compact and perfectly symmetrical structure including minor, middle or neutral, and major intervals. The Sagittal sharp, flat, and half-apotome signs very nicely reflect that structure. Also, 17-ed2 is consistent for primes 2-3-7-11-13, so that might be one place to start in looking for relevant commas, with the Archytan comma (which is tempered out) only a starting point.

rapoport3a
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### Re: Paul Rapoport

First, I should say that this paper was written partly as a response to the plethora of what I considered problematic ET (EDO) notations. It seemed then (1991) that every composer or theorist had invented her/his own system without a broader or more general outlook, and in some cases without considering internal inconsistencies and awkward aspects of what came across as a partly thought out ad hoc system.

I should also note what the composer Vagn Holmboe said about revisiting old work. Although he meant it about compositions, it applies here. “You come to old work only as a guest.” I can’t get into the space I was in when I wrote that article, which I recognize now as fairly simple. I’m no more qualified to comment on it than others are.

1. At some point I may present what I think about JI notations. I have to read up on what’s happened lately, to avoid being redundant.

2. As a prelude, I’ll state that I don’t consider EDOs and JI to be compatible (in pitches/intervals, not in notation). That’s a strict theoretical attitude, because compatibility is sometimes possible because going smaller than a certain small limit in cents is impracticable. But I note that people still write about the 12 EDO P5 being (almost) indistinguishable from the just P5. While that’s true for one P5, when you concatenate them, that closeness disappears. Everyone involved in JI knows this, of course.

Re 1/n notation. Although I can’t comment yet on Sagittal’s solution, it seemed simply that if a komma was more than one unit in a tuning, you’d need the notation to deal with each fraction. In a tuning where a komma is 4 units, if the same notation is used for 2/4 as in other tunings for ½, then you might lose the information of how many units are involved in the (present) tuning’s komma. You’d also need a separate notation for ¼ and ¾ komma, and 1¼ and 1¾, and situations where the komma is divided into 3 or 5, etc. The decision on this may depend on where you want your consistency. I was working within individual tunings, whereas working across them may give a different result, possibly using numerals to designate fractions in a way that’s different from how I used them. Regardless, small numerals seemed a way to deal with both fractions and multiples of kommas, where needed.

Possibly composing in more tunings would have led me to different conclusions.

3. One thing I wonder about in Sagittal, although I may still have to read through its various versions to get the point. It’s about this: “These lines by themselves do not provide a foolproof way to distinguish up from down…”

That refers to what in various European languages are angled straight-line accents. They’re very clear in traditional use in music, reading from left to right. Sagittal even uses them as skhismatic “accents” in that way, demonstrating that they’re not unclear. The vertical line in that context I find unnecessary. The result also resembles what I consider more useful notations for a semi-flat and a 7-comma.

4. It will take me a while to respond to Margo Schulter, much reading being needed, because my article didn't involve 7 directly, nor unequal divisions. Abandoning 3 as a generator is of course another story, which I also wrote about in another article, as others have done. In my case, I think the result was a bit comical. I have to revisit that as well.