source file: m1417.txt Date: Sat, 16 May 98 13:37 BST-1 Subject: Miscellany From: gbreed@cix.compulink.co.uk (Graham Breed) I'll be a digest out of date by the time I send this. Anyway, there are some things I want to say and I'll say them all in one message. Apologies if I seem to be repeating or ignoring someone. LATTICES -------- Joseph L Monzo wrote: > Picking a particular set of pitches arbitrarily, and > Going back to Tuning Digest 1315, here's the 7-Limit > "Tonality Diamond" in Graham Breed's Triangular Lattice > Diagram, which is of the type used by Erlich: 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. > First comment: I've many times wished that we humans could > work with more than 3 dimensions when up against the problem > of visualizing and representing more than 3 prime dimensions > in music. Fortunately, 7-limit systems _can_ be represented > in three dimensions. We can, of course, work with as many dimensions as we like by using coordinate geometry. It is easer to "understand" scales when you can visualise them, though. You can specify ambiguous chords as a superposition of different matrices. I think this has application in a dynamic tuning program. > Graham leaves his 7-factor-ratios unconnected by > any lines to the 3- and 5-limit ratios. If I replace > the ratios with the prime-factor notation that I use, > and connect the 7-ratios in a manner similar to others, Usually, I connect all 5-limit planes in this fashion. I don't usually do that with three planes in a diagram, because it gets too cluttered with lines. If I were to, it would look something like this: B-----------F# / \ / \ / \ Ab / \ / \ / \ / \ / E#----------B# \ / \ / \ / \ / \ G-------\---D---/-------A \ / \ / \ / \ / \ Fb--\---/---Cb / \ / \ / \ / \ / G# \ / \ / \ / Bb----------F >From the way the lines cross, you can see which plane is supposed to be the highest, and so reconstruct it in three dimensions in your mind. Ideally, each plane would be a different colour. The structure is a regular dodecahedron, or something like that. I find it easier to understand the scales with conventional names. There's no need for prime factorisation, because it's implied by the lattice. 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. There may be ways the ear relates to prime factors. Different overtones will be reinforced, and the difference tone pattern may change. This is strongly linked to timbre, though, and breaks down before you get to 81/64. 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. For exactness, state the signed errors, and you can work other intervals out from that. 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. OCTAVE INVARIANCE ----------------- There are (at least) three different concepts involved here, which need to be distinguished for the discussion to progress: 1) The idea that harmony is unchanged if notes are transposed independently in octaves. 2) The practice of giving notes related by octaves the same name. 3) The practice of defining scales within an octave, and repeating them in other octaves. In all cases the choice of the octave as an invariant interval is not an arbitrary one. (1) I would argue with, if taken as a fundamental principle. I won't do so in detail today, as it is outside the brief of this list (we all now how to tune octaves!) If the ear/brain complex has a hard-wired mechanism for octave reduction, this would relate to the 10:12:15 vs 16:19:24 tuning of a minor triad. Personally, I see no way in which intervals an octave apart can be considered "the same" and my brother agrees with me. I'm sure people can convince themselves otherwise, the same way they learn to hear the "3-ness" in a sharp major third. TEMPERAMENT ----------- Joe Monzo again: > I think a large part of the reason Schoenberg stuck with the > 12-Eq scale was because he realized intuitively that within the > vastly expanded resources of his implied 13-Limit, were he to > work in just-intonation, the number-play involved could quickly > become a bottomless pit from which his musical inspiration would > never again emerge into the light of day. Then again, I also > think he wanted to make use of the ambiguities made possible > by a comparison of so many close ratios on the one hand and > their nearby 12-Eq equivalents on the other. Then again, perhaps Schoenberg had a piano. > I refuse to accept an equal-temperament because it makes > modulation "easy" or (excuse me while I laugh) "possible". Where were you when Gregg Gibson was stomping all over the list? Temperaments do make modulation easier, and this is important. If this is a personal decision, though, it's fair enough. > The only reasons I can see for accepting any equal-temperament > is that it is easier to play on most instruments, and, as I stated > in another post to this issue, the study of the interplay between > JI and ET in the same piece is becoming more and more interesting > to me. That sounds too dogmatic to me, perhaps it wasn't intended to be. Other reasons are: 1) You like the sound of it. The only reason you need. 2) It simplifies notation. If a piece of music makes sense in JI, good musicians will work this out from staff notation, if you point out the difference between G# and Ab. (If there are a lot of Pythagorean intervals, Erv Wilson's positive notation is more useful.) Bad musicians won't be helped by precise tuning indications. If the music doesn't make sense, who cares how it's played? 3) It allows ambiguous chords. I remember you (Monzo, TD1403) agreeing that the chord below only works in meantone. Did you really mean to imply (above) that this isn't a use for temperament? A---E---B \ / \ G---D---A 4) You can't be bothered to work out the ratios. As implied in the Schoenberg paragraph, most composers don't want to worry about ratios. They want a resourceful scale to compose in. Give them a meantone, and they'll find good harmonies in it. They might even give them to musicians who'll render them in JI. Do you really expect the composer to sit and work out where the commas should be? POPULAR MUSIC ------------- Drew Skyfyre wrote: > In an interview I just saw on TV,in which they discussed briefly,their > middle eastern/oriental explorations,Jimmy Page said of their new > album,Walking into Clarksdale,(I quote from memory) > "The Oriental stuff was done by a guy called Tim from Transglobal > Underground (a British dance music outfit) who plays an Oriental keyboard > that plays microtones." Cool! I've got one TGU album (Psychic Karaoke) and I've always wondered if it's strictly 12-equal. There's no mention of a Tim in the booklet, though. Alex Kasiek and Hamid Mantu are both credited, among other things, with "Keyboards". > Could this be one of those General Music "Arabian" keyboards ? Unlikely, I'd have thought. Perhaps they just mean a tunable synth. There's seem's to be more of an Indian than Arabic influence in TGU, along with the dance, rap and reggae. There are Indian Classical vocals on one track. When I was looking for a MIDI keyboard, one of the shops I went to said, after I asked about tuning capability, that there'd been some Arabs in before after the same thing. I think they use normal synths with tuning tables. Technically, it was someone in the shop who said that, rather than the shop itself. Anyway, you get the idea. Also, there's an "Eno" credited on this TGU album. Probably Brian, although could be Roger. I think Brian was mentioned before as having used microtonality, wasn't he? Maybe there's a connection. BTW, anyone familiar with Nassim Maalouf? 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 of it as the standard melodic meantone. GUITAR TUNING ------------- Drew Skyfyre: > I'm having a sort of eureka thing .Two recent/current threads got me > thinking about how it is possible to put together a microtonal guitar > (even non-tempered) and be able to modulate through quite a few keys,etc.. > The possibilities are extensive so,I'll just let you use your imagination. > All it entails is combining a guitar with any microtonal fretting system > (including JI) and a Steinberger TransTrem. This is pitch shifting, presumably? It's something I had in mind when I went for a non-equal fretting. Unfortunately, my Zoom 505 isn't good enough to be useful in this respect. Actually, modulation hasn't been a problem so far, as it's common to use keys with open strings anyway. The solution would be to tune the strings around the set of keys desired. Incidentally, having different notes on different strings is a _good_ thing. It means you get more chords than you would otherwise. John Starrett: > If someone > can give me a valid reason why the nut should not be treated as the 0th > fret, I will eat a bug (I get to choose the bug and the method of > preparation). Fretting a note will slightly lengthen the string, and so increase it's tension. I worked out that, to a first order approximation, this effect is independent of where on the fingerboard you put your finger. So, move the nut a bit nearer the bridge than if it were the zeroth fret. Independently movable bridges probably work for this reason, except they're at the wrong end of the guitar! For a guitar's tuning to be good enough to make just intonation relevant, we do have to think about this sort of thing. The tuning pegs are the most imprecise bit according to my ST900. Good enough for meantone, but no better. Also, the thickest string varies about 10 cents depending on how hard I pluck it. Your unloveble young nutcase, Graham Breed www.cix.co.uk/~gbreed/