source file: m1420.txt Date: Mon, 18 May 1998 18:59:39 -0400 Subject: Reply to Graham Breed's Miscellany From: "Paul H. Erlich" >LATTICES >-------- >It would be more correct to say "Paul Erlich's Triangular >Lattice Diagram, which is of the type used by Breed". Paul >was using them before me, definitely extended them to the >7-limit first, and drew them in ASCII first. He also named >them "triangular". As I say on my website (and have said >before on this list) _I_ would have called them hexagonal, >like I was taught in Condensed Matter Physics. >I don't know how much, if anything, is original to Paul. >He can speak for himself on that. My 7-limit lattice is of course a face-centered cubic lattice, distinct from the hexagonal lattice which packs space just as efficiently but is less symmetrical. I came up with the idea on my own, and while at first it perplexed John Chalmers, he later found it to be quite similar to what Erv Wilson had done (exactly the same, in fact, except that Wilson may (or may not) have simply seen the lattice as existing in two dimensions, while I definitely see the two-dimensional representation as being a projection of the three-dimensional one). In fact, my triangular lattices show Wilson's CPS scales as the most symmetrical arrangements of tones, while cubic lattices show Euler genera as more symmetrical. >PRIME VS ODD LIMIT >------------------ >The most paradoxical chord from an odd limit perspective I >find is 4:6:9. I think of it as extended 3-limit, because >it's the fifths that give it it's character. It is also >highly octave specific. It sounds more consonant >(concordant) than a major triad in root position, but >transpose it to 8:9:12 and it's clearly more dissonant. Well, I don't know if it really beats a major triad, but otherwise, I quite agree. Here's the interesting thing. When I worked out a model for harmonic entropy, which should also describe critical band roughness if the partials decrease in amplitude in some specific fashion, I derived that to a good approximation, the complexity of a just ratio is directly related to its DENOMINATOR. Later, imposing octave equivalence made me change this to ODD LIMIT, but I admit that it's possible that octave equivalence does not really come in to the "objective" dissonance of an interval. Now the chord 4:6:9 has three intervals: 3:2, 3:2, and 9:4. The highest denominator is 4. The chord 4:5:6 has a 3:2, a 5:4, and a 6:5. The highest denominator is 5. So judging from the intervals alone, you appear to be right that the 4:6:9 is more consonant. However, I think the chord as a whole has something that the intervals alone don't explain, as evidenced by the differing levels of consonance in otonal and utonal chords. On the whole-chord basis, which has proved impenetrable to the type of analysis that led me to the denominator rule for intervals, I think the major triad would win. The chord 8:9:12 has the intervals 3:2, 4:3, and 9:8. The highest denominator is 8 -- pretty dissonant! >There may be ways the ear relates to prime factors. Different >overtones will be reinforced, and the difference tone pattern >may change. Can you give a concrete example? >The good intervals in equal temperaments don't usually >constitute anything like an odd limit. It's generally more >efficient to say how well different primes are approximated. This may not work if consistency doesn't hold. >For exactness, state the signed errors, and you can work >other intervals out from that. Isn't that what Carl Lumma said? But no, you guys are wrong, and that's the whole reason for the consistency concept. Wendy Carlos, Yunik and Swift, and others also seem to have missed out on the importance of consistency. >Incidentally, it seems to me that the concept of level-n >consistency is closely related to the recently defined >radius of the scale. If you want to use all the notes in >a radius 2 scale, it helps if it's level-2 consistent as well. I don't think it helps all that much. I think Paul Hahn would agree with you, though. >LUCYTUNING >---------- >Gary Morrison wrote: >> The semitruth: LucyTuning definitely can stack up more fifths above a tonic >> before approximately closing the circle than either quarter-comma or third- >> comma meantone. Third-comma meantone comes "close enough" at 19 fifths, >> and quarter-comma at 31 fifths. LucyTuning doesn't get there until about >> 88 fifths. But this is only a semitruth, because taken in absolutes, the >> circle of fifths never closes in ANY typical meantone tuning. >The most irrational meantone in this respect is Kornerup's >phi based tuning. Is that right? I think so. There are more "irrational" tunings with one generator, but they cannot be considered meantones. >I think of it as the >standard melodic meantone. I've thought about that too, but I don't know. It is awfully close to the harmonically optimal meantones in my paper. >GUITAR TUNING >------------- >Incidentally, having different notes on different strings is >a _good_ thing. It means you get more chords than you would >otherwise. Huh? It definitely means fewer positions in which to play a given chord.