This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 10 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
300 - 325 -
Message: 301 - Contents - Hide Contents Date: Sat, 23 Jun 2001 08:06:45 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: Hi Dan, :-) I hope the actual point of my message isn't lost because of my ill-advised use of the word "trash". My apologies. I believe I defined what I meant by "best" in this context, and by implication what I meant by "trash". It's likely that "trash" isn't as emotionally loaded a term for Australians as it is for Americans. But hey some generators are clearly trash such as a 2 cent generator that doesn't form a MOS until 600 notes! Regards, -- Dave Keenan
Message: 302 - Contents - Hide Contents Date: Sat, 23 Jun 2001 14:04:51 Subject: Re: Hypothesis revisited From: Graham Breed Paul wrote:> Huh? The determinant of the matrix is usually the number of notes, > not the number of repetitions per octave (which is usually just 1).Yes, I was misteaking the terminology. The division of the octave and the generator will come together. It's getting the number of steps to an octave that's difficult.> Let's leave out the octave, octave-equivalence will be assumed (yes, > in a more general case it won't be, but let's not bite off more than > we can chew).No, you can't get linear temperaments without considering the octave. That doesn't stop it being an equivalence interval. My programs assume that anyway. To get a different equivalence interval, you re-define the coordinates. Anyway, here's a script I worked out that prints some choices for the number of steps of the other size. The results can be fed into my older module to get the octave and generator. It doesn't work for the pathological case, so I don't know what to do about that. You'll need Numeric Python, but it can probably be adapted to whatever package you use. Sometime I'll explain what's going on. I'm abbrevating "chromatic unison vector" to "chroma". import Numeric from LinearAlgebra import inverse, determinant from math import log mul = Numeric.matrixmultiply def integerize(matrix): return Numeric.array([ [int(round(y)) for y in x] for x in matrix]) def log2(f): return log(f)/log(2) primes = map(log2, [3, 5, 7, 11, 13]) for unisonVectors in ( [(-1, 2), (4, -1)], [(0, -3), (8, 1)], [(-1, 2), (8, 1)], [(-1, 2), (-4, -2)], [(0, -3), (-4, -2)], [(0, -3, 0), (-4, -2, 0), (-2,0,-1)], [(0, -3), (4, -1)]): octaveSpecific = [(1,)+(0,)*len(unisonVectors)] h = Numeric.array(primes[:len(unisonVectors)]) H = Numeric.array([1]+primes[:len(unisonVectors)]) for vector in unisonVectors: size = mul(vector,h) if size>0: octave = -int(size) else: octave = 1-int(size) octaveSpecific.append((octave,)+vector) matrix = Numeric.array(octaveSpecific) basisVectors = octaveSpecific[:2] + [(0,)*len(octaveSpecific)]*(len(octaveSpecific)-2) basisMatrix = Numeric.array(basisVectors) print "\n\nLeft hand defining matrix" print matrix octave = int(round(abs(determinant(matrix)))) inverted = inverse(matrix)*octave print "\nH' defined by octave and chroma" conversion = integerize([x[:2] for x in inverted]) print conversion guess = 1/mul(octaveSpecific[1],H) for m in range(1,int(guess*2)): for prime in mul(conversion,(m,1)): if prime%octave: break else: print m Here are the results: Left hand defining matrix [[ 1 0 0] [-3 -1 2] [-4 4 -1]] H' defined by octave and chroma [[ 7 0] [11 1] [16 4]] 5 12 19 26 Left hand defining matrix [[ 1 0 0] [ 7 0 -3] [-15 8 1]] H' defined by octave and chroma [[24 0] [38 1] [56 -8]] Left hand defining matrix [[ 1 0 0] [ -3 -1 2] [-15 8 1]] H' defined by octave and chroma [[17 0] [27 -1] [39 8]] 12 29 Left hand defining matrix [[ 1 0 0] [-3 -1 2] [11 -4 -2]] H' defined by octave and chroma [[10 0] [16 -2] [23 4]] 2 12 22 32 Left hand defining matrix [[ 1 0 0] [ 7 0 -3] [11 -4 -2]] H' defined by octave and chroma [[12 0] [19 2] [28 -4]] 10 22 34 46 Left hand defining matrix [[ 1 0 0 0] [ 7 0 -3 0] [11 -4 -2 0] [ 6 -2 0 -1]] H' defined by octave and chroma [[12 0] [19 2] [28 -4] [34 -4]] 10 22 34 46 Left hand defining matrix [[ 1 0 0] [ 7 0 -3] [-4 4 -1]] H' defined by octave and chroma [[12 0] [19 -1] [28 -4]] 7 19 31 43 55 Graham "I toss therefore I am" -- Sartre
Message: 303 - Contents - Hide Contents Date: Sat, 23 Jun 2001 18:27:20 Subject: Re: Hypothesis revisited From: Graham Breed I wrote:> Anyway, here's a script I worked out that prints some choices for the numberof > steps of the other size. The results can be fed into my older module to get > the octave and generator. It doesn't work for the pathological case, so I > don't know what to do about that.I've fixed that, and thrown a quick web page together at <Unison vector to MOS script * [with cont.] (Wayb.)>. I'll try and make it easier to throw lots of sets of unison vectors at it now. If anybody has some favourites they'd like me to try, send them in! Graham "I toss therefore I am" -- Sartre
Message: 305 - Contents - Hide Contents Date: Fri, 23 Jun 2000 11:02:43 Subject: Re: Hypothesis revisited From: monz> From: Graham Breed <graham@m...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, June 23, 2001 6:04 AM > Subject: [tuning-math] Re: Hypothesis revisited > > > ... I'm abbrevating "chromatic unison vector" to "chroma".Hmmm... that's really interesting. "Chroma" is a music-theory term with quite a background history. (Maybe Paul will say more.) -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 306 - Contents - Hide Contents Date: Sat, 23 Jun 2001 19:23:41 Subject: Re: Hypothesis revisited From: Graham Breed After getting the computer to chuck out this kind of thing:> H' defined by octave and chroma > [[ 7 0] > [11 1] > [16 4]] > 5 > 12 > 19 > 26I've now got some idea what it means. -- A complete set of unison vectors gives an equal temperament -- Take one of them away, you get a linear temperament. This can be thought of as a section of the scale tree. The usual fifth based scales are 7 5 12 19 17 26 31 29 22 And the list here is a subset of that branch. In particular, though, it's Erv Wilson's septimally positive set. This is true forever, even when the temperaments stop being consistent. They are the EDOs with 5+7n notes in them. -- Put it back as a chromatic UV, and you get an n-ly mth-ly positive set In general, the n in the n-ly bit is the number of notes in the equal temperament you got at the first step. This makes sense: the more notes you add in a Wilson/Bosanquet pattern, the closer you get to that temperament. So, in the example above, it gets closer and closer to 7-equal *the smaller the chromatic unison vector gets*. When that unison vector becomes a unison, and so commatic, you do have the equal temperament. The amount of positivity is less obvious. It assumes some kind of fifth generators, and so isn't that general. So really it's the "5" that's important for septimally positive scales rather than the "+1". Graham Graham "I toss therefore I am" -- Sartre
Message: 307 - Contents - Hide Contents Date: Sat, 23 Jun 2001 19:41:08 Subject: Re: Hypothesis revisited From: Graham Breed Monz wrote:> Hmmm... that's really interesting. > > "Chroma" is a music-theory term with quite a background history.Oh, well, see if you can come up with a better word for "chromatic unison vector" for when I start explaining this. Graham "I toss therefore I am" -- Sartre
Message: 308 - Contents - Hide Contents Date: Sun, 24 Jun 2001 03:04:39 Subject: Re: Hypothesis revisited From: jpehrson@r... --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: Yahoo groups: /tuning-math/message/298 * [with cont.]> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> Progress seems to have halted on the paper that was to introduce >> MIRACLE . . . >> >> I suggest the title >> >> _The Relationship Between Just Intonation and Well-Formed Scales_ >> >> and some sort of "proof" of the hypothesis > ...>> Then we can give a few examples, including the diatonic >> and MIRACLE scales. >> Wow! This is the complete opposite direction to where I was planning > to head. To get in the mood for writing it, I was working out how to > explain to my sister, a very _practical_ violinist and strings > teacher, what was significant about the MIRACLE scales, and how a > musician can use them.That would be a great article... I would love to read *that* one! _______ ______ ______ Joseph Pehrson
Message: 309 - Contents - Hide Contents Date: Sun, 24 Jun 2001 03:09:34 Subject: Re: Hypothesis revisited From: jpehrson@r... --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote: Yahoo groups: /tuning-math/message/300 * [with cont.]> I think my situation is perhaps made more difficult than some others > around here in that I actually enjoy and have a real interest in the > theory end of this subject... and while I do respect what I've learned > from that "small handful of likeminded folks", I can't help but at > times like this feeling that were I hostile or ambivalent towards it > all everything sure would be a hell of a lot easier. >It certainly would... because it would be a *lot* easier to "dismiss" your posts! Well, of all the people around here, with the exception of possibly Margo Schulter, you have shown Dan, that one can make music out of just about *any* possible scales... the "good," bad and the ugly... the so-called "ugly" sometimes becoming quite beautiful... It seems this attitude is also shared to some extent by Brian McLaren... who has obviously had a lot of microtonal listening experience... Frankly, I'm fascinated with the "special properties" of scales such as MIRACLE, but truly you've convinced me that there are "other things around..." ________ ______ _______ Joseph Pehrson
Message: 310 - Contents - Hide Contents Date: Sun, 24 Jun 2001 03:11:54 Subject: Re: Hypothesis revisited From: jpehrson@r... --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Yahoo groups: /tuning-math/message/305 * [with cont.]>>> From: Graham Breed <graham@m...> >> To: <tuning-math@y...> >> Sent: Saturday, June 23, 2001 6:04 AM >> Subject: [tuning-math] Re: Hypothesis revisited >> >> >> ... I'm abbrevating "chromatic unison vector" to "chroma". > >> Hmmm... that's really interesting. > > "Chroma" is a music-theory term with quite a background history. > > (Maybe Paul will say more.) >What is the history of that term again, Monz?? Joe P.
Message: 311 - Contents - Hide Contents Date: Sat, 23 Jun 2001 22:34:37 Subject: Re: Hypothesis revisited From: monz> ----- Original Message ----- > From: Graham Breed <graham@m...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, June 23, 2001 11:41 AM > Subject: [tuning-math] Re: Hypothesis revisited > > > Monz wrote: >>> Hmmm... that's really interesting. >> >> "Chroma" is a music-theory term with quite a background history. >> Oh, well, see if you can come up with a better word for "chromatic unison > vector" for when I start explaining this.I'm sorry, Graham... perhaps I should have been clear when I wrote that, that this might be an appropriate *new* additional definition of the term "chroma". Perhaps, based on the wide background history I mention, your new use of "chroma" fits right in. I'd have to dig out what I have about this term and can't do it right now. Maybe until then, you could use "chromuv"? -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 313 - Contents - Hide Contents Date: Sun, 24 Jun 2001 08:36:34 Subject: [celestial-tuning] common musical intervals in Babylonian math texts From: monz This post was originally for celestial-tuning, but it got so math-heavy that I decided to send a copy to tuning-math.> From: monz <joemonz@y...> > To: <celestial-tuning@xxxxxxxxxxx.xxx> > Sent: Friday, June 22, 2001 5:08 PM > Subject: Re: [celestial-tuning] Sumerian tuning > > I actually did find a paper on the web somewhere which was > devoted to a discussion of the syntonic comma in ancient > Babylonian mathematics. It was more in relation to other > uses than music (can't remember what right now, mainly > architecture and building). Try a search on "comma" and > "Babylonian", see what you find. > ----- Original Message ----- > From: <JGill99@i...> > To: <celestial-tuning@xxxxxxxxxxx.xxx> > Sent: Saturday, June 23, 2001 2:49 PM > Subject: [celestial-tuning] Sumerian Links > > I found a couple of links searching. One relates to > the Sumerians and Babylonians in relation to their > mathematics, with some materials involving their > music (cross-referenced abstracts of existing articles > and books): > > Sumerians and Babylonians - Mathematics and th... * [with cont.] (Wayb.)Thanks for this link, Jay! Looks like a lot of good reading on ancient math. I found the article I was referring to: Stereometric Texts * [with cont.] (Wayb.) It's part of _A History of Measures_ by Livio C. Stecchini. The author's native language appears to be Italian, because this article (in English) is *full* of typos, so its already difficult discussion is even harder to follow than it need be. This article, as I correctly remembered a few days ago, is about the volumes, weights, and specific gravities of various amounts of bricks. As the author says in section 7:> Considering the importance of bricks in the life of > Mesopotamia it is not surprising that the units of > volume and weight were so well adjusted the problem > of measuring and transporting bricks.Here are some extracts with the discussion of the Greek terms which appear in music-theory. Near the end of section 2:> (a cube that contains 300 double qa with a six-finger > edge, less a discrepancy komma or 81/80),(A "qa" is a measurement of area.) Stecchini explicitly equates the "komma" with 81:80. From section 4:> A massiqtu of 60 qa has a base of 24 x 24 fingers and > a height of 22½ (that is, 24 minus a diesis).(A "finger" is a measurement of length.) Thus, this "diesis" is 24 : 22&1/2 = 16:15. In the next paragraph, however:> The tablet indicates that when one came to the sheqel, > the unit used to weigh the media of exchange, one > reckoned by referring to the basic sheqel of 9 grams: > here the unit is a double sheqel of 18 grams reduced > of a diesis; it is equal to two sheqels of 8.4 grams.Thus, this is a different "diesis": 9 : 8.4 = 15:14. Then in section 6 we find:> ... we find a relation 1:3 3/5 between the two amounts > of seed, with a discrepancy diesis (3 3/8 * 16/15 = 3 3/5).So here again the "diesis" is 16:15. At the end of section 6:> The text describes a near-cube with a basis of 4 x 4 fingers > and a height of 3 3/4. If it were a perfect cube the qa > would be 64 cubic fingers with an excess of a diesis;The volume of the given shape is 60 cubic fingers, so this diesis is again 64:60 = 16:15.> 7. Neugebauer and Sachs have shown that a brick > measuring 15 x 10 x 5 fingers was considered a typical > brick, but they have not explained why such a brick > should have been considered typical. The explanation > is that the brick has the volume of a royal qa: > 750 cubic fingers by the barley cubit equal > 216 cubic fingers by the great cubit. However, > the brick is calculated with an excess of a leimma > over the volume of the qa, for the purpose of obtaining > a brick measured by be convenient figures 15 x 10 x 5 > barley fingers. Deducting a leimma the volume of a > brick becomes 720 cubic fingers. Since 720 = 216 x 3 1/3, > the relation between royal qa and normal qa is calculated > as 1:3 1/3, a relation frequently used instead of the > less convenient relation 1:3 3/8, with a resulting > discrepancy komma. The dimensions of the brick are > such that 6 occupy the area of a square cubit and > 36 have the volume of a cubic cubit. Below it will > appear that these bricks were counted by the dozen, > and that a dozen of bricks is as much as a man can > carry. Calculating by the great cubit each brick is > a royal qa with an excess of a leimma, so that 120 bricks, > or 10 dozens, make exactly a cube of great cubit.Note that what Stecchini is calling a "leimma" is *not* the same as the Pythagorean musical interval of that name (it's also the Pythagorean "chromatic semitone"), which is the ratio 256:243, which translates into ~90.2249957 cents. This "leimma" is 750:720 = 25:24, the familiar 5-limit "chromatic semitone", which translates into ~70.6724269 cents. This "discrepancy komma" is again the same ratio as the musical syntonic comma: 3&3/8 : 3&1/3 = 81:80. Note that Stecchini also mentions a "diesis" here. I'm a little confused about exactly how big this is. It seems to describe the difference between the "mina stereometric brutta" and the "regular mina", and I gather from the discussion that the number of sheqels contained in these minai are respectively 63&2/3 and 60, which gives the ratio 191:180, ~102.690878 cents. But I could be wrong here. The ratio between the two specific gravities is 2.4 : 2.25 = 16:15. The earlier mentions of diesis usually show it to be 16:15. Actually, my other calculations may also be wrong. I think a very close study of this article (as well as the others in the series by Stecchini) is necessary to be sure about what these terms really mean. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 314 - Contents - Hide Contents Date: Sun, 24 Jun 2001 08:59:28 Subject: Re: [celestial-tuning] common musical intervals in Babylonian math texts From: monz Wow... look what I found in the appendix to Stecchini's series of articles on ancient measurement: APPENDIX * [with cont.] (Wayb.)> I have given to the discrepancies names derived from > the accidentals of musical scales, because there is > a close correlation between units of measures and > ancient musical scales. This is made clear by Chinese > musical treatises. Actually I have found that the > reading of the Greek musical treatises or of the > similar Chinese ones, which must have been derived > from a common source, is the best preparation for the > understanding of the arithmetic of ancient metrology. > > The arithmetic of discrepancies is essential to the > understanding of the development of problems in > cuneiform mathematical texts. It is disputed among > musicologists whether musical scales have a physiological > or conventional origin. The evidence I have gathered > indicates that musical scales were derived from the > arrangement of the units of volume. The ancients used > to arrange the units of public reference standards in > a series, in ascending or descending order. The relation > among the contents of the basic units of volume appears > to have been adopted as determining the basic tetrachord.Stecchini then gives examples, and quite a bit of further discussion on music and tuning. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 315 - Contents - Hide Contents Date: Sun, 24 Jun 2001 09:05:48 Subject: Re: Hypothesis revisited From: monz> ----- Original Message ----- > From: <jpehrson@r...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, June 23, 2001 8:11 PM > Subject: [tuning-math] Re: Hypothesis revisited > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > > Yahoo groups: /tuning-math/message/305 * [with cont.] >>> Hmmm... that's really interesting. >> >> "Chroma" is a music-theory term with quite a background history. >> >> (Maybe Paul will say more.) >> >> What is the history of that term again, Monz??Joe, I'd have to shift gears in my mind and do some real research to answer this question in the detail it deserves. You (and others who have my book) can read a few references to "chroma" in the beginning chapters. In brief, "chroma" is a Greek term referring to color. It has been used by later music-theorists primarily to refer to the idea of a categorical pitch-class _gestalt_. Some theorists have used it very similarly to the way I have... that is, to refer to the difficult-to-describe "affect" produced by the basic prime intervals (i.e., 2:1, 3:2, 5:4, 7:4, 11:8, 13:8, etc.). But IIRC Paul has criticized me for using it this way because it had various other meanings in the past which may be more well-established. Paul, help! -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 316 - Contents - Hide Contents Date: Sun, 24 Jun 2001 11:32:15 Subject: Re: common musical intervals in Babylonian math texts From: monz> ----- Original Message ----- > From: monz <joemonz@y...> > To: <celestial-tuning@xxxxxxxxxxx.xxx>; <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, June 24, 2001 8:59 AM > Subject: [tuning-math] Re: [celestial-tuning] common musical intervals inBabylonian math texts> > > Wow... look what I found in the appendix to Stecchini's > series of articles on ancient measurement: > > APPENDIX * [with cont.] (Wayb.) > >>> I have given to the discrepancies names derived from >> the accidentals of musical scales, because there is >> a close correlation between units of measures and >> ancient musical scales.I was just talking with Brink and had this idea: If the Sumerians based their musical interval measurements on the same ones they used for building physical brick structures, they could possibly have viewed a wall (for example) as a lattice-diagram! Would some of the geometers out there take a look at Stecchini's articles Units of Length * [with cont.] (Wayb.) Stereometric Texts * [with cont.] (Wayb.) and construct some plausible examples? I'd be very interested in that, but am having a hard time understanding the proliferation of different measurement units. If I can get a better understanding, perhaps I'll give it a try. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 317 - Contents - Hide Contents Date: Sun, 24 Jun 2001 21:17:59 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:> Even if you solve the problem you have proposed, how do you then > choose sets of unison vectors to give you the the _bes_ scales (small > JI errors and low cardinality for complete otonalities). Some unsison > vectors, although small, "pull in opposite directions".One way would be to examine the geometry of the unison vectors in the triangular lattice -- if the angles between them are small, the periodicity block will not contain a lot of consonant structures . . .> > The basic requirement remains the same: One algorithm for _generating_ > linear temperaments and another to _filter_ out the trash. It's the > filter that is by far the most significant here, since one can, as I > did, simply feed it _every_ possible generator to a sufficiently fine > resolution.Who's to say what's trash?> > The way I see it, the solution of your problem would merely give us > another way of generating linear temperaments to be filtered. We > already have two different ways of doing that. >But the most significant part of the paper, I believe, would be to show how well-formed scales, which have received a great deal of attention in the music-theoretic literature of late, can be seen as flowing naturally from a fundametally JI-oriented framework, which has received virtually none.
Message: 318 - Contents - Hide Contents Date: Sun, 24 Jun 2001 21:26:51 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., Graham Breed <graham@m...> wrote:>>> Let's leave out the octave, octave-equivalence will be assumed (yes, >> in a more general case it won't be, but let's not bite off more than >> we can chew). >> No, you can't get linear temperaments without considering the octave. That > doesn't stop it being an equivalence interval. My programs assume that anyway.Then what do you mean, "No"? What I'm thinking is, let's not bother with a column for powers of 2 in the matrices . . . along the lines of what Fokker did.
Message: 319 - Contents - Hide Contents Date: Sun, 24 Jun 2001 21:30:41 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., Graham Breed <graham@m...> wrote:> After getting the computer to chuck out this kind of thing: >>> H' defined by octave and chroma >> [[ 7 0] >> [11 1] >> [16 4]] >> 5 >> 12 >> 19 >> 26 >> I've now got some idea what it means. > > -- A complete set of unison vectors gives an equal temperamentIf you temper them all out.> > -- Take one of them away, you get a linear temperament.Yes, and reasonable choices of an additional, non-tempered-out unison vector lead to MOSs of the linear temperament.
Message: 320 - Contents - Hide Contents Date: Sun, 24 Jun 2001 23:58:22 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:>> Even if you solve the problem you have proposed, how do you > then>> choose sets of unison vectors to give you the the _bes_ scales > (small>> JI errors and low cardinality for complete otonalities). Some > unsison>> vectors, although small, "pull in opposite directions". >> One way would be to examine the geometry of the unison vectors > in the triangular lattice -- if the angles between them are small, > the periodicity block will not contain a lot of consonant structures > . . .But doesn't that depend which set of unison vectors you use for a given PB, since they are not unique. I think of Canasta having many consonances because 224:225 and 385:384 have such a _small_ angle between them when projected onto the 5-limit plane. But yes, I'm sure you could do something like this, but why bother, when we have a "near-JI filter" on the end of the pipeline.>> The basic requirement remains the same: One algorithm for > _generating_>> linear temperaments and another to _filter_ out the trash. It's > the>> filter that is by far the most significant here, since one can, as I >> did, simply feed it _every_ possible generator to a sufficiently > fine >> resolution. >> Who's to say what's trash?This rhetorical question only serves to strengthen my argument that it is the _filter_ that is most significant. If the question is not quite rhetorical: 1. No one thinks that all linear temperaments are equally interesting. 2. Masses of people over centuries have effectively given us a short list of those they found useful. (Popularity of Partch's scales would in effect tell us that MIRACLE is useful) 3. There is wide acceptance (even by Dan Stearns :-) that approximation of small whole-number ratios contributes _something_ towards making a linear temperament useful. It's certainly one of _your_ key assumptions Paul. It's presumably the reason why you're interested in unison vectors in the first place.>> The way I see it, the solution of your problem would merely give > us>> another way of generating linear temperaments to be filtered. > We>> already have two different ways of doing that. >>> But the most significant part of the paper, I believe, would be to > show how well-formed scales, which have received a great deal > of attention in the music-theoretic literature of late, can be seen > as flowing naturally from a fundametally JI-oriented framework, > which has received virtually none.But they don't "flow naturally", do they? What is the definition of "well-formedness"? Is it simply MOS/Myhill's? But presumably all you want to do is show that the current definition of "well-formedness" isn't enough, and that additional criteria of near-JI-ness should be added. To do this, one can show the near-JI-ness of some historical scales. (Of course some are not). Then you can generate well formed scales that have no approximations of SWNRs and let people decide whether they find them useful. A maximally dissonant MOS, there's a fun project. :-) All this scan be done without having to mention periodicity blocks or unison vectors at all. -- Dave Keenan
Message: 321 - Contents - Hide Contents Date: Mon, 25 Jun 2001 01:15:15 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:>> >> One way would be to examine the geometry of the unison vectors >> in the triangular lattice -- if the angles between them are small, >> the periodicity block will not contain a lot of consonant structures >> . . . >> But doesn't that depend which set of unison vectors you use for a > given PB, since they are not unique.Well, the picture is not that simple when you're talking about one of the unison vectors (the chromatic one) _not_ being tempered out. Then it _does_ matter which set you choose.> > I think of Canasta having many consonances because224:225 and 385:384> have such a _small_ angle between them when projected onto the 5-limit > plane.Hmm . . . can you explain the thinking behind that? Of course, the fact that you're projecting them makes it very different from the idea I was thinking about.> > But yes, I'm sure you could do something like this, but why bother, > when we have a "near-JI filter" on the end of the pipeline.To make the process more transparent and intuitive for those who prefer to look at, and work with, JI lattices.> > > If the question is not quite rhetorical: > 1. No one thinks that all linear temperaments are equally interesting. Erv Wilson? > 2. Masses of people over centuries have effectively given us a short > list of those they found useful. (Popularity of Partch's scales would > in effect tell us that MIRACLE is useful)wha . . . wha . . . what??> 3. There is wide acceptance (even by Dan Stearns :-) that > approximation of small whole-number ratios contributes _something_ > towards making a linear temperament useful. It's certainly one of > _your_ key assumptions Paul. It's presumably the reason why you're > interested in unison vectors in the first place.Yup! I just thought this paper would be better if it were capable of unifying different fields of tuning theory, and presenting a few new interesting scales with descriptions according to this new unified theory, than being some sort of attempt to crown a few scales with the title of "best". Of course, mentioning these searches would be very relevant in the context of the paper, but I see it as more of a footnote than as the main subject of the paper. Every scale has its unique properties, so ruling out any just because others are "better" means blocking off many potentially interesting musical effects.>>> But the most significant part of the paper, I believe, would be to >> show how well-formed scales, which have received a great deal >> of attention in the music-theoretic literature of late, can be seen >> as flowing naturally from a fundametally JI-oriented framework, >> which has received virtually none. >> But they don't "flow naturally", do they? What is the definition of > "well-formedness"? Is it simply MOS/Myhill's?Yes. So why don't they "flow naturally"?> > But presumably all you want to do is show that the current definition > of "well-formedness" isn't enough, and that additional criteria of > near-JI-ness should be added.More than that -- I want to show that well-formedness should not be an "axiom" at all but could instead be derived from more "fundamental considerations". A JI-friendly underpinning to much modern scale theory. One might even include a case where _two_ of the unison vectors are not tempered out, and related this to a second-order ME scale, such as the Indian 7-out-of-22.
Message: 322 - Contents - Hide Contents Date: Mon, 25 Jun 2001 02:43:09 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>>> One way would be to examine the geometry of the unison > vectors>>> in the triangular lattice -- if the angles between them are > small,>>> the periodicity block will not contain a lot of consonant > structures>>> . . . >>>> But doesn't that depend which set of unison vectors you use for > a>> given PB, since they are not unique. >> Well, the picture is not that simple when you're talking about one > of the unison vectors (the chromatic one) _not_ being tempered > out. Then it _does_ matter which set you choose.Yes. That's what I thought I said. It _does_ matter. But choosing one to be chromatic, still doesn't uniquely determine the others does it? (except in 5-limit). So how do you know which vectors to check angles between?>> I think of Canasta having many consonances because> 224:225 and 385:384>> have such a _small_ angle between them when projected onto > the 5-limit >> plane. >> Hmm . . . can you explain the thinking behind that? Of course, the > fact that you're projecting them makes it very different from the > idea I was thinking about.Yes. A different idea. This was how I found Canasta's planar-temperament precursor. I started at a note in the 5-limit lattice and I knew if I grew the scale in a particular approximate direction (pair of opposing directions) I would get both aproximate 7's and approximate 11's.>> But yes, I'm sure you could do something like this, but why > bother,>> when we have a "near-JI filter" on the end of the pipeline. >> To make the process more transparent and intuitive for those > who prefer to look at, and work with, JI lattices.That's a worthy aim, but it can be acheived by finding the linear-temperaments by existing methods and working backwards to the unison vectors.>> If the question is not quite rhetorical: >> 1. No one thinks that all linear temperaments are equally > interesting. > > Erv Wilson?Doesn't it seem to you, from his diagrams, that he at least considers noble generators to be more interesting or useful or special in some way? He also obviously believes, as we do, that SWNRs (and nearby) are special.>> 2. Masses of people over centuries have effectively given us a > short>> list of those they found useful. (Popularity of Partch's scales > would>> in effect tell us that MIRACLE is useful) >> wha . . . wha . . . what??I assume you're not objecting to the first sentence? I'll adress the second. Graham Breed (and George Secor) have shown that MIRACLE_41 is almost identical to several of Partch's scales. I can't help seeing Partch's various scales as gropings towards either Canasta or MIRACLE-41. I expect Partch would not have been able to distinguish his scales from the corresponding MIRACLE-temperament of them, since I understand someone said he couldn't distinguish one of them from 41-EDO. I think the fact that Partch, doing it mostly by ear, and we, doing it mostly by math, (and George Secor doing it by ???), essentially converged on the same thing, is no accident.>> 3. There is wide acceptance (even by Dan Stearns :-) that >> approximation of small whole-number ratios contributes > _something_>> towards making a linear temperament useful. It's certainly one > of>> _your_ key assumptions Paul. It's presumably the reason why > you're>> interested in unison vectors in the first place. >> Yup! I just thought this paper would be better if it were capable of > unifying different fields of tuning theory, and presenting a few > new interesting scales with descriptions according to this new > unified theory, than being some sort of attempt to crown a few > scales with the title of "best".Gimme a break Paul. Dan's already slapped me on the wrist for that. The "political correctness police" are getting a little tedious. I thought I made it clear that by "best" I wasn't trying to claim something which is _obviously_ a matter of personal taste. I first set up some criteria (which incidentally an awful lot of people find, align well to their personal taste, at least some of the time) and then I talk about what is "best" according to those criteria. Surely I don't have to re-state these criteria in every post I make, especially when it's to the tuning-math list?> Of course, mentioning these > searches would be very relevant in the context of the paper, but I > see it as more of a footnote than as the main subject of the > paper.Me too. Just enought to say that we ran these searches and we found the previouly mentioned scales (which you are welcome to introduce in the manner of "Forms of Tonality" using unison vectors) to be the "best".> Every scale has its unique properties, so ruling out any > just because others are "better" means blocking off many > potentially interesting musical effects.Who is ruling out such scales. You mean you don't think I should have ruled out a MOS with a 2 cent generator. Oh. Well sorry.>> But they don't "flow naturally", do they? What is the definition of >> "well-formedness"? Is it simply MOS/Myhill's? >> Yes. So why don't they "flow naturally"?Because there are zillions of MOS scales that have no relationship with small unison vectors. Sure you could probably always find a corresponding periodicity block, but these will have "unison vectors" so large as not to merit the name.>> But presumably all you want to do is show that the current > definition>> of "well-formedness" isn't enough, and that additional criteria > of>> near-JI-ness should be added. >> More than that -- I want to show that well-formedness should not > be an "axiom" at all but could instead be derived from more > "fundamental considerations". A JI-friendly underpinning to much > modern scale theory. One might even include a case where > _two_ of the unison vectors are not tempered out, and related > this to a second-order ME scale, such as the Indian 7-out-of-22.See response to previous paragraph. You can't derive MOS from JI or vice versa. One is a horizontal melodic property, the other vertical harmonic. Periodicity blocks may give you MOS approx-JI scales but they won't give you the MOS non-approx-JI scales. Regards, -- Dave Keenan
Message: 323 - Contents - Hide Contents Date: Mon, 25 Jun 2001 03:15:37 Subject: Re: Hypothesis revisited From: Dave Keenan I wrote:> Because there are zillions of MOS scales that have no relationship > with small unison vectors. Sure you could probably always find a > corresponding periodicity block, but these will have "unison vectors" > so large as not to merit the name.Try this one: A chain of 10, 369c generators, octave period.
Message: 324 - Contents - Hide Contents Date: Mon, 25 Jun 2001 03:24:43 Subject: 41 "miracle" and 43 tone scales From: jpehrson@r...> almost identical to several of Partch's scales. I can't help seeing > Partch's various scales as gropings towards either Canasta or > MIRACLE-41. I expect Partch would not have been able to distinguish > his scales from the corresponding MIRACLE-temperament of them, since I > understand someone said he couldn't distinguish one of them from > 41-EDO. I think the fact that Partch, doing it mostly by ear, and we, > doing it mostly by math, (and George Secor doing it by ???), > essentially converged on the same thing, is no accident. >I'm getting a little confused here... Did Harry Partch use a 41-tone scale in addition to his 43-tone scale?? He never actually used "Miracle 41" did he?? _________ _______ _______ Joseph Pehrson
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
300 - 325 -