*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:

I have emailed the scan to Paul, and I await his approval to upload it here.Dave Keenan wrote: Hi Paul Rapoport. Great to hear from you. What is the article about?

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.Paul Rapoport wrote: Notation of equal temperaments, implying all the edos.

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?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.)

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.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.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.

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.

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."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.

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?

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.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.

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

How would you feel about me scanning your paper and posting it to the Sagittal forum? I can write in the correction.Okay. It'll be a little while.

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.