Hi Mike and others.

I'll tell you my view on this prime limits and temperaments and other stuff, although some of you might (as I've seen earlier) find it a "limited" or "narrow" approach. Anyway, at least it works for me. To the people who are very familiar with the topic of 2D tunings, I apologize for too much introductory material. But I think it can help in explaining the whole point.

Let's begin with the prime of 2. We get octaves -- i.e. some sort of equidistant "pitch frames" to which we attribute identical musical "meanings".
Now let's add 3. Combining this with 2, we get fifths and fourths and their octave equivalents. This makes it possible to fill a single octave with loads of other tones, one of which (3/2) represents the new prime, and another (4/3) represents its inversion. It allows us to view the 2/1 a bit differently than "just a distance between two pitches" and to consider one "trine" some kind of "anchor" or "default state" (from which other trines are derived), which can be either 2:3:4 or 3:4:6 (i.e. subharmonics of 4:3:2). So out of the 1-dimmensional system of octaves, we've got a 2D system of octaves and fifths (or fourths). To make a few things easier in practice, we can temper something out to get a 1D system. One of many possibilities is, for example, to temper out the Pyth. comma, which results in 12-EDO. Although we can also temper out other intervals and thus get tunings like 5-EDO or 17-EDO, 12-EDO seems to do "the least harm" to the original Pythagorean tuning -- if I'm not mistaken, the next closest is 53-EDO.

Now, let's get back to JI and add the prime 5. This makes it possible to fill in a single fifth with other tones than its octave equivalents or inversions, which means it offers a completely different view on a fifth. Similarly as I used "trines" of 2:3:4 or 3:4:6 as a kind of "anchor state" for Pythagorean tuning, I can view either 4:5:6 or 10:12:15 (i.e. subharmonics of 6:5:4) as some sort of "anchor" triads for "Didymic" tuning (I've often heard 5-limit JI called like that). So now we've got a 3D system. This is the key point because it's the basis of Baroque/classical harmony. Octaves serve as equivalence intervals, fifths determine the position of the "fundamental bass", and thirds determine what's called "tonal gender" in Czech (i.e. whether a triad is major or minor -- is there really no English translation for that?). Again, to make some things easier, we can temper something out to change our 3D system into a 2D tuning. So we have the 1st dimmension of 2/1 and the 2nd dimmension changes according to the interval tempered out. Of course, there is a huge number of possibilities what can be tempered out and therefore what tunings eventually we can get out of this (some of which may sound pretty weird for the first time), but the famous example is tempering out 81/80, which results in a generator of a fifth (or fourth if we assume octave equivalence). This is the way standard notation works (i.e. two tones which would be a syntonic comma apart are represented by the same note). So in fact, in the Renaissance and early Baroque times, there was the notational experience which was 2-dimmensional and the auditory experience which, even when represented by 2D meantone, was based on the 5-limit 3D system and suggested treating meantone accordingly (i.e. a kind of imitating a 3D tuning using a 2D temperament). This might well suggest another possible clue why Baroque/Classical harmony developped in the way it did -- i.e. the idea of triads having its origin in the auditory experience, and the idea of some particular harmonic progressions having its origin in the notational experience (similarly as many Renaissance composers never viewed C# and Db as "possibly being the same" because of the meantone distinction, I think many composers would never, if they hadn't known about anything else than 5-limit JI in their lives, have claimed progressions like "A-E-A-C, F-D-A-D, G-D-G-B, C-E-G-C" as "possibly having the same top note in both the initial and the final chord" because the final C is a syntonic comma lower than the initial C if the common tones of consecutive chords are preserved).

Now here comes something which is a bit of a problem for me (although for many people it's completely okay). Because it was "necessary" to make even more things easier in practice, the 1D system of 12-EDO was later used as a "pitch model" for almost all European music, which lead to a slight "distortion" of meanings of enharmonically related pitches in some cases. And I'm also "speculating" that this may be a possible answer why, when alternative tunings started being "mildly" examined during the 20th century, equal tunings were thoroughly described from various views, while 2D tunings were noted much later. But if you show another EDO than 12 to someone who's just beginning to get interest in microtonal music, he/she will, most probably, be first interested in getting some meantone (in one case) or getting as close to Pythagorean as possible (in another) or getting as close to JI intervals in general as possible (in another). When I first discovered 31-EDO and 34-EDO, I kept saying they were "just excellent". Actually, it was just because I liked 31 as a very nice meantone and 34 as a nice approximation to 5-limit JI (using still less notes than 53 which seemed like a lot to me). But ONLY as late as in 2006, I finally discovered something more important. If you temper out two intervals in a 3D system, you get a 1D system, which means (lo and behold) that not only is 34-EDO good at approximating JI BUT (and there's the big but) it offers completely new possibilities of temperaments and chord progressions which I had never thought of before. Why? Because I was unable to think of them -- because I was unable to get out of the "meantone or JI" restriction. So not only can 34 do, for example, hanson (9-step generator, tempering out 15625/15552) but it can also do tetracot (5-step generator, tempering out 20000/19683). And not only can 19-EDO do meantone (81/80) but it can also do hanson like 34 (15625/15552) or it can do magic (3125/3072) or semisixths (78732/78125) or negri (16875/16384). However, no sooner was I able to discover these things than after improvising some music in real 2D tunings, not their EDO simulations. That was an unforgettable eye-opener. One of my first discoveries at that time was the "rediscovery" of the semisixths temperament whose generator I finally decided to tune as the 16th root of 60, or ~443 cents. In my oppinion, this "proper" version of semisixth is not easy to imitate in a 1D tuning. Although 19-EDO or 46-EDO can temper the "comma" out, none gets so close to JI as the proper semisixths does. And I can tell you with confidence that about 5 years ago, I wouldn't at all think about anything else than simply approximating JI in 46-EDO or meantone in 19-EDO. And I would not be able to discover the new chord progressions which work only in one particular 2D tuning and are impossible to convert to standard notation -- like those in my "Run Down The Whistle" or "Among Other Things" or "Dinepome's Adventure". What this all means is that even if we keep using the 5-limit "model" as our starting point, we can get many possible "harmonic systems", only one of which is reproducible in standard notation (i.e. the one based on meantone).

And now (drum roll), another change arrives. The prime 7. This gives us, in total, 4 primes to deal with -- i.e. a 4D system. Because 7 is surrounded by 6 and 8 in the integer series, one choice might be to think of chords like 6:7:8 or the subharmonic 21:24:28 (if these can be called chords) as possible "anchors" in the 7-limit field, or we could, if we wish, do some sort of "octave inversion" and get 4:6:7 and 14:21:24, if we want to have a fifth there. But now there's still something else to say. Because classical harmony is more or less based on a 3D system with some 2D ambiguities, something like a 4D system can never be exactly converted to the chord relations which we know from there. If I tried to do that, it's a bit like if I took the 2D Pythagorean tuning and said that "5/4 is a Didymic ditone because it's close to the Pyth. ditone of 81/64" or that "5/4 is a Didymic diminished fourth because it's close to the Pyth. diminished fourth of 8192/6561". Although C-E can represent both a ditone and a major third, major thirds don't have the same meanings as ditones do. In fact, I was doing just the same thing back in 2002 when I kept saying that 7/4 was a 7-limit augmented sixth (just because of the silly reason that it was close to the 5-limit augmented sixth of 225/128). Many people were doing this much sooner than I was. And many people were defending it, including me, by saying things like "Look how close it is in quarter-comma meantone or in 31-EDO, you can't deny that." But do you know why I was doing it? Because I only knew about equal temperaments or JI, nothing else. I was thinking of Pythagorean tuning simply as of "something which you can nicely approximate in 29-EDO, 41-EDO, or 53-EDO, to name a few" and meantone was just "something which you can approximate in 19-EDO, 31-EDO, 50-EDO or others". But then in 2006, things changed radically. Again and again, within my own calculations, I was realizing that you can get good 5-limit approximations even in other 2D tunings than those based on fifths -- or that you can temper out any interval in a given 3D system, which gives you the chance to approximate the 3D system using many different 2D temperaments. And a few months later, I eventually found those papers about regular temperaments written by Paul or Graham or Herman or George, which was, I think, perhaps an even better experience than if it had happened in reverse order. Well, and what do I say about 7-limit intervals today? They're simply another harmonic dimmension which has nothing to do with the three other dimmensions that we know from classical harmony. And if I really want to know more about that, one of the possible ways is to try some 3D temperaments. Of course, you could, for example, approximate temperaments like marvel or starling using meantone. But this is like approximating meantone or miracle using 31-EDO, which may not tell you the whole story (which was my case). Within a few months, actually, I'm planning to make a short piece of music in the "breed" temperament (2401/2400) and another in the one which I've been privately calling "norwell" (6144/6125).

Okay, that's it, for the time being. I think I've said all I wanted to say. How much of it seems useful or useless to others, that's just upon the readers themselves.

Petr