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Message: 6500 - Contents - Hide Contents Date: Sun, 16 Feb 2003 00:44:17 Subject: Re: A common notation for JI and ETs From: monz hi paul,> From: <wallyesterpaulrus@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Saturday, February 15, 2003 1:02 PM > Subject: [tuning-math] Re: A common notation for JI and ETs > > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> > wrote: >>> and i think you missed this -- which you'd also probably >> be interested in at least in passing: >> Onelist Tuning Digest # 483 message 26, (c)200... * [with cont.] (Wayb.) >> >> >> >> -monz > > hi monz, >> i brought this page up to dave and george very recently (last > week) here on this list, and they indeed found it very useful for > their discussion.oops, my bad. OK, i haven't really been following this particular list closely, but have only glanced more-or-less randomly at posts which looked interesting. i'm glad my page was found useful.> unfortunately, several erroneous statements > persist on this page, most notably: > > "It can be seen easily from the lattice that all the intervals are > made up of various combinations of the ones described by > Paul." > > of course, we all know you're very busy right now, and i at least > appreciate your brief and all too infrequent visits to this list. thanks, paul.OK, if you tell me *exactly* what i should do with that sentence (remove it, edit it, change it? -- and if the latter two, then replace it with exactly what?), i'll just copy and paste what you write into the page to replace my sentence. (i really do try to stay on top of the accuracy of my webpages. sorry about falling behind sometimes. ... i know you still have a slew of stuff that you want me to fix. i'll do them one page at a time.) -monz

Message: 6501 - Contents - Hide Contents Date: Mon, 17 Feb 2003 02:25:53 Subject: Re: A common notation for JI and ETs From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi paul, > > > >> From: <wallyesterpaulrus@y...> >> To: <tuning-math@xxxxxxxxxxx.xxx>>> Sent: Saturday, February 15, 2003 1:02 PM >> Subject: [tuning-math] Re: A common notation for JI and ETs >> >> >> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> >> wrote: >>>>> and i think you missed this -- which you'd also probably >>> be interested in at least in passing: >>> Onelist Tuning Digest # 483 message 26, (c)200... * [with cont.] (Wayb.) >>> >>> >>> >>> -monz >> >> hi monz, >>>> i brought this page up to dave and george very recently (last >> week) here on this list, and they indeed found it very useful for >> their discussion. > > >> oops, my bad. OK, i haven't really been following > this particular list closely, but have only glanced > more-or-less randomly at posts which looked interesting. > i'm glad my page was found useful. > > >>> unfortunately, several erroneous statements >> persist on this page, most notably: >> >> "It can be seen easily from the lattice that all the intervals are >> made up of various combinations of the ones described by >> Paul." >> >> of course, we all know you're very busy right now, and i at least >> appreciate your brief and all too infrequent visits to this list. > > > > thanks, paul. >> OK, if you tell me *exactly* what i should do with that > sentence (remove it, edit it, change it? -- and if the > latter two, then replace it with exactly what?), i'll just > copy and paste what you write into the page to replace > my sentence.please remove the sentence, and replace it with this: ' It can be seen easily from the lattice that these intervals, as well as some lesser-known 'commas' like 243:250 and 3072:3125, cannot made up of various combinations of the ones described by Paul. Western triadic music prior to Beethoven requires "bridging" solely through the syntonic comma, and hence is often performed in meantone temperament. Since Beethoven, "bridging" through syntonic comma and *any* (and therefore, all) of the other 'commas' paul mentions above (in connection with mathieu) has been a feature of western triadic music, hence the use of 12-tone equal (or well) temperament. The other 'commas' can be used for bridging in other, "invented" musical systems, motivating certain corresponding tuning systems as shown at: Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) for example, you can see from the first chart and table on that page that "bridging" through 243:250 is characteristic of porcupine temperament, through 3072:3125 of magic temperament, and through both of them (and thus also any combination of the two) of 22-tone equal temperament. ' to understand why, consider the following: the first graph on your page shows the commas that vanish in 12- equal. this can also be displayed by taking the 12-equal bingo card, Yahoo groups: /tuning-math/files/Paul/12p.gif * [with cont.] superimposing it with the "small 5-limit intervals" graph, Yahoo groups: /tuning-math/files/Paul/small.gif * [with cont.] with this result: Yahoo groups: /tuning-math/files/Paul/superimp... * [with cont.] and noting which commas fall on "0"s. hopefully, this will help you understand my original claim, as quoted by you: "Basically, there are only two independent commas that come into play when trying to analyze . . . music of the Western conservatory tradition". now you yourself write: "From the 'practical' point of view of "trying to analyze or render music of the Western conservatory tradition in just intonation", Paul is probably right that these 'commas' (the smallest and largest of which are actually forms of skhisma and diesis, respectively) are the only ones that need be considered." the reason for this is that only meantone temperament and 12-equal (or well) reflect the forms of "bridging" that hav been used in the western tradition. "But strictly speaking, depending on how far one takes the powers of 3 and 5 in any given 2-dimensional lattice or periodicity-block, there are all sorts of 'commas' that may come into play." in fact, these commas will come into play if you're working from a *different* temperament than meantone or 12-equal.> (i really do try to stay on top of the accuracy of my webpages. > sorry about falling behind sometimes. ... i know you still > have a slew of stuff that you want me to fix. i'll do them > one page at a time.)ok, let me know when you're ready with this page, and then we'll move on . . . your faithful scrutinizer, paul

Message: 6502 - Contents - Hide Contents Date: Mon, 17 Feb 2003 15:39:04 Subject: Huron Voice Leading From: Graham Breed I've finished reading "A Derivation of the Rules of Voice-leading from Perceptual Principles". It seems to be good sense in so far as it goes. Not really a derivation, but a good place for scientists to start when learning counterpoint. Janata et al isn't so interesting. All it says is that notes closer to the key center are closer to the key center, or something. And there are some brain images I don't understand. That "Pitch Schemata" link I gave might have the background details. The first time I looked at it I recognized a lot of Rothenberg's ideas, but they were credited to Balzano. I'll have another look sometime. Graham

Message: 6503 - Contents - Hide Contents Date: Mon, 17 Feb 2003 05:29:24 Subject: poking monz (was: Re: naming temperaments( From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> i'd like to include hexagonal graphs for *all* the EDOs on my > "equal temperament" page *and* on the "bingo lattice" page. > > keep sending me stuff ... i'll incorporate it as i have the chance.i've done them for all ets from 7 through 80 which are both 5-limit consistent and "non-torsional". they're the first 50 or so files on this page: Yahoo groups: /tuning-math/files/Paul/ * [with cont.] enjoy!

Message: 6504 - Contents - Hide Contents Date: Mon, 17 Feb 2003 09:19:20 Subject: Re: Huron Voice Leading From: Carl Lumma>That "Pitch Schemata" link I gave might have the background details. >The first time I looked at it I recognized a lot of Rothenberg's ideas, >but they were credited to Balzano. I'll have another look sometime.Balzano independently came up with a lot of Rothenberg's ideas, years later, and along with some other mush. -Carl

Message: 6505 - Contents - Hide Contents Date: Mon, 17 Feb 2003 05:31:31 Subject: Re: A common notation for JI and ETs From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> please remove the sentence, and replace it with this: > > ' > It can be seen easily from the lattice that these intervals, as well > as some lesser-known 'commas' like 243:250 and 3072:3125, cannot made > up of various combinations of the ones described by Paul.oops: that should say "cannot be made up of" . . .

Message: 6506 - Contents - Hide Contents Date: Mon, 17 Feb 2003 17:34:48 Subject: Pitch Schemata (Was briefly: Huron Voice Leading From: Graham Breed Carl Lumma wrote:> Balzano independently came up with a lot of Rothenberg's ideas, years > later, and along with some other mush.So why doesn't Rothenberg get any credit for them? And it's not only Balzano -- Browne's given some of them as well. Another weird thing is that a Forte 1973 paper is mentioned as a "formulation" of Balzano's 1982. So what was Forte formulating? It must be peculiar if major and minor triads are considered identical. Pitch Schemata * [with cont.] (Wayb.) It says that Balzano's coherence is different to Rothenberg's propriety because it only considers adjacent pairs of intervals. So a Pythagorean diatonic is still coherent because the conflicting sizes of tritones aren't considered. Is that right? Graham

Message: 6507 - Contents - Hide Contents Date: Mon, 17 Feb 2003 10:05:09 Subject: Re: Huron Voice Leading From: Carl Lumma [I wrote...]>> That "Pitch Schemata" link I gave might have the background details. >> The first time I looked at it I recognized a lot of Rothenberg's ideas, >> but they were credited to Balzano. I'll have another look sometime. >>Balzano independently came up with a lot of Rothenberg's ideas, years >later, and along with some other mush.While many of their criteria were the same, many were not. So it must be considered a strange coincidence that R. and B. wind up recommending the same scale, R in 31-tET, and B in 20-tET. B missed the 31-tET version because it didn't have all the other nonsense properties that he was so fascinated with (such as the product of the sizes of the 3rds giving the number of notes in the embedding et). Dan Stearns further independently suggested this scale in 20-tET for his own reasons. But AFAIK I'm the first to notice its excellent approximations to 5:3 and 7:4 nicely interleaved on its 8ths. The 31-tET version gets closer to JI, but the 20-tET version has higher Lumma stability and already gets you closer than 12-tET to these intervals. The lower stability of Rothenberg's scale kicks it down to position 6 on my gd spreadsheet -- Balzano's version is at position 3, just below the pentatonic and diatonic scales... * [with cont.] (Wayb.) All the scales are available as scala files... * [with cont.] (Wayb.) -Carl

Message: 6508 - Contents - Hide Contents Date: Mon, 17 Feb 2003 10:19:29 Subject: Re: Pitch Schemata From: Carl Lumma>Pitch Schemata * [with cont.] (Wayb.) > >It says that Balzano's coherence is different to Rothenberg's propriety >because it only considers adjacent pairs of intervals. So a Pythagorean >diatonic is still coherent because the conflicting sizes of tritones >aren't considered. Is that right?I think that's wrong. ;) No, I remember there being small differences like that. Here's a bit of a message I sent to tuning some years ago... """ Balzano wants // the scale to be covered with three-note chords that fall on every-other degree of the scale. Which means they'll be made of thirds and fifths. It is actually a huge mistake to consider them chords, tho, since Balzano hasn't given any property that defines "chords" (he's deliberately thrown out the usual one: harmony). Which means that his whole idea amounts to a lot of nothing. Well, not quite. He does require that the same interval appears as a fifth in exactly n-1 modes of the scale, when the scale has n notes per 2:1. Which is no more and no less than MOS when the generator turns out to be a fifth and the interval of equivalence a 2:1. So the final list of stuff is now... (1) // "coherence" of scale degrees across modes. Search list archives for "Rothenberg". (2) tuning coverage; rank order matrix and interval matrix are the same. I don't think this has much to do with anything. // (3) linear connectivity; transposing by generator changes one note. But not less than one --- closed chains are not allowed. Makes for symmetry at the generator, and prevents the (Rothenberg) efficiency from becoming very low (see TD 262.14). (4) fifths; the generator is a fifth in exactly n-1 modes of the scale. // Choice of fifths is completely arbitrary. """ -Carl

Message: 6509 - Contents - Hide Contents Date: Mon, 17 Feb 2003 19:54:33 Subject: Re: Pitch Schemata From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Pitch Schemata * [with cont.] (Wayb.) >> >> It says that Balzano's coherence is different to Rothenberg's propriety >> because it only considers adjacent pairs of intervals. So a Pythagorean >> diatonic is still coherent because the conflicting sizes of tritones >> aren't considered. Is that right? >> I think that's wrong. ;) >it is wrong, because balzano, like clough and too many academic theorists, considers all scales to be subsets of some discrete cyclic "universe": 12-tone, 20-tone, etc. balzano defines coherence in terms of units of the "universe set". which means that it, like maximal evenness, is undefined for a random scale given in cents (or ratios, or whatever), without any assumed "universe" in which it is embedded. this is a serious weakness of academic scale theory, in my opinion.

Message: 6510 - Contents - Hide Contents Date: Mon, 17 Feb 2003 12:18:38 Subject: Re: Pitch Schemata From: Carl Lumma>>> >t says that Balzano's coherence is different to Rothenberg's >>> propriety because it only considers adjacent pairs of intervals. //>> I think that's wrong. ;) >>it is wrong, because balzano, like clough and too many academic >theorists, considers all scales to be subsets of some discrete >cyclic "universe": 12-tone, 20-tone, etc. balzano defines coherence >in terms of units of the "universe set". which means that it, like >maximal evenness, is undefined for a random scale given in cents (or >ratios, or whatever), without any assumed "universe" in which it is >embedded. this is a serious weakness of academic scale theory, in my >opinion. For sure.But further, it's wrong unless Balzano has a model that justifies coherence (which I don't remember him having). Rothenberg assumes that listeners can rank intervals by size, and gives the condition required for them to be able to repeatably map what they hear to a fixed scale. It's obvious, it's simple, and it's probably true. I can't imagine how one could doctor this so that only adjacent intervals matter... -Carl

Message: 6511 - Contents - Hide Contents Date: Mon, 17 Feb 2003 12:02:58 Subject: Re: Pitch Schemata From: Carl Lumma>So why doesn't Rothenberg get any credit for them?Presumably because "Mathematical Systems Theory" isn't one of the journals that the exclusive circle of music theorists watch.>And it's not only Balzano -- Browne's given some of them as well.From the details available in the schemata paper, Browne rediscovers not only something like propriety ("pattern matchng"), but also something like efficiency ("position finding"). -Carl

Message: 6512 - Contents - Hide Contents Date: Mon, 17 Feb 2003 21:17:57 Subject: Re: Pitch Schemata From: Graham Breed Carl Lumma wrote:> But further, it's wrong unless Balzano has a model that justifies > coherence (which I don't remember him having). Rothenberg assumes > that listeners can rank intervals by size, and gives the condition > required for them to be able to repeatably map what they hear to a > fixed scale. It's obvious, it's simple, and it's probably true. > I can't imagine how one could doctor this so that only adjacent > intervals matter...Then did Balzano so doctor it? So that the Pythagorean diatonic embedded in 53-equal would still be coherent? I was only asking if the Pitch Schemata paper had Balzano correct. Graham

Message: 6513 - Contents - Hide Contents Date: Mon, 17 Feb 2003 13:23:41 Subject: Re: Pitch Schemata From: Carl Lumma>> >ut further, it's wrong unless Balzano has a model that justifies >> coherence (which I don't remember him having). Rothenberg assumes >> that listeners can rank intervals by size, and gives the condition >> required for them to be able to repeatably map what they hear to a >> fixed scale. It's obvious, it's simple, and it's probably true. >> I can't imagine how one could doctor this so that only adjacent >> intervals matter... >>Then did Balzano so doctor it? So that the Pythagorean diatonic >embedded in 53-equal would still be coherent? I was only asking if >the Pitch Schemata paper had Balzano correct.I was assuming so. I don't remember that from the Balzano paper, and my copy is in Montana, so... -Carl

Message: 6514 - Contents - Hide Contents Date: Mon, 17 Feb 2003 14:13:50 Subject: Re: A common notation for JI and ETs From: monz hi paul,> From: <wallyesterpaulrus@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, February 16, 2003 6:25 PM > Subject: [tuning-math] Re: A common notation for JI and ETs > > > Onelist Tuning Digest # 483 message 26, (c)200... * [with cont.] (Wayb.) > > > please remove the sentence, and replace it with this: > > ' > It can be seen easily from the lattice that these intervals, as well > as some lesser-known 'commas' like 243:250 and 3072:3125, cannot made > up of various combinations of the ones described by Paul. > > Western triadic music prior to Beethoven requires "bridging" solely > through the syntonic comma, and hence is often performed in meantone > temperament. Since Beethoven, "bridging" through syntonic comma and > *any* (and therefore, all) of the other 'commas' paul mentions above > (in connection with mathieu) has been a feature of western triadic > music, hence the use of 12-tone equal (or well) temperament. The > other 'commas' can be used for bridging in other, "invented" musical > systems, motivating certain corresponding tuning systems as shown at: > > Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) > > for example, you can see from the first chart and table on that page > that "bridging" through 243:250 is characteristic of porcupine > temperament, through 3072:3125 of magic temperament, and through both > of them (and thus also any combination of the two) of 22-tone equal > temperament. > 'OK, i added that. when i have more time i'd also like to include what you wrote after that. -monz

Message: 6515 - Contents - Hide Contents Date: Tue, 18 Feb 2003 19:00:43 Subject: Re: scala show data From: manuel.op.de.coul@xxxxxxxxxxx.xxx That bug is fixed now, along with some other ones. There's a new feature which may be interesting, in the Chromatic Clavier you can now arpeggiate or hold a chord, and play with the mouse at the same time. The chord is entered using the right mouse button, like it could be done before. Also new now is that when you open the chord list, the selected chord is used (actually the nearest approximation of it in the current scale), so you can quickly change chords without having to click-enter them. Please click on the Help button in the clavier window first if there's some trouble. http://www.huygens-fokker.org/software/Scala_Setup.exe - Type Ok * [with cont.] (Wayb.) Manuel

Message: 6516 - Contents - Hide Contents Date: Tue, 18 Feb 2003 11:04:50 Subject: Re: scala show data From: Carl Lumma>That bug is fixed now, along with some other ones.With equal 6, I get .2 for Rothenberg stability. How are you getting that? If I delete the 3rd degree, it goes up to .4! -Carl

Message: 6517 - Contents - Hide Contents Date: Tue, 18 Feb 2003 13:23:38 Subject: lattice diagram "levels" of complexity From: monz is there an accepted method of nomenclature for describing the "level" of complexity of lattice diagrams? i know that "stellation" has something to do with this, but here i'm talking pretty much about the algorithm used by Partch to fill out the Tonality Diamond. in other words, say we have a 7-limit (3-D) lattice. (use "Expand Messages" if viewing on Yahoo website for proper formatting of diagram) here are the two basic tetrads (otonal and utonal) which have the 1/1 ratio as their 1-identity: 5:4 /|\ / | \ / | \ / 7:4 \ /. ' ' .\ 4:3---------1:1---------3:2 \ '. .' / \ 8:7 / \ | / \ | / \|/ 8:5 this would be "level" 1. now if we build complete otonal tetrads on all of the notes in the "basic" utonality tetrad (i.e., 4/3, 8/5, and 8/7 all become 1-odentities of their respective tetrads), and complete utonal tetrads on all of the notes in the "basic" otonality tetrad (i.e, 3/2, 5/4, and 7/4 all become 1-udentities of their respective tetrads), we get this: 5:3---------5:4 /|\ '. .' /|\ / | \ 10:7 / | \ / | \ /|\ / | \ / 7:6--/-|-\--7:4 \ /. ' '/.\|/.\' ' .\ 4:3------/--1:1--\------3:2 \ '. /.' /|\ '.\ .'/ \ 8:7--/-|-\--12:7 / \ | / | \ | / \ | / 7:5 \ | / \|/ .' '. \|/ 8:5---------6:5 which would be "level" 2. is there already an accepted term for what i'm calling "level"? and can someone give a very clear and lucid explanation of how stellation differs from this, if it does? thanks. -monz

Message: 6518 - Contents - Hide Contents Date: Tue, 18 Feb 2003 14:49:48 Subject: Re: lattice diagram "levels" of complexity From: Carl Lumma heya monz,>is there already an accepted term for what i'm >calling "level"?Not to my knowledge, though we often talk about regions of the lattice within some taxicab radius. The diamond is r=1, and it sounds like your levels correspond to successively higher r values. You might want to check that and tell me if it's the case. I said "lattice region". Gene has used the term "ball", and I think that's a convex hull, plus everything inside. zthat right, Gene?>and can someone give a very clear and lucid explanation >of how stellation differs from this, if it does?Stellation is Wilson's term. He borrowed it from geometry, where the term often refers to the process of adding points above the faces of a polyhedron, turning *them* into polyhedra. This is indeed what happens when stellating the hexany -- it's an octahedron being extended so that each face becomes a tetrahedron. I'm sure there's a more precise definition on a geometry website somewhere. Post it here if you are interested and find it... The one I'm remembering is 'the compound of a polyhedron and its dual'. Back when, there was some debate over what stellation should include when extending other CPSs (the eikosany was the main inquiry). I said that each existing face should be completed into a saturated n-limit chord, and that's it. Others, apparently including Wilson, wanted to include other points, basically out to the power set (the compound of all CPSs of a given limit). I was never clear on why they wanted to do this. Maybe Paul remembers. I don't think it's related to your levels, really. -Carl

Message: 6519 - Contents - Hide Contents Date: Tue, 18 Feb 2003 23:14:00 Subject: Re: lattice diagram "levels" of complexity From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Others, apparently including Wilson, wanted > to include other points, basically out to the power > set (the compound of all CPSs of a given limit).that isn't the case. look over the old discussions on this list about stellation. the compound of all CPSs of a given limit can be constructed in many ways, but most typically (as d'allessandro) as an euler genus, and this (or any of the other ways) clearly does not have the symmetry of the original, unstellated CPS, which the stellation must have by definition.

Message: 6520 - Contents - Hide Contents Date: Tue, 18 Feb 2003 23:25:14 Subject: Re: A common notation for JI and ETs From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi paul, > > >> From: <wallyesterpaulrus@y...> >> To: <tuning-math@xxxxxxxxxxx.xxx>>> Sent: Sunday, February 16, 2003 6:25 PM >> Subject: [tuning-math] Re: A common notation for JI and ETs >> >> >> Onelist Tuning Digest # 483 message 26, (c)200... * [with cont.] (Wayb.) >> >> >> please remove the sentence, and replace it with this: >> >> ' >> It can be seen easily from the lattice that these intervals, as well >> as some lesser-known 'commas' like 243:250 and 3072:3125, cannot made >> up of various combinations of the ones described by Paul. >> >> Western triadic music prior to Beethoven requires "bridging" solely >> through the syntonic comma, and hence is often performed in meantone >> temperament. Since Beethoven, "bridging" through syntonic comma and >> *any* (and therefore, all) of the other 'commas' paul mentions above >> (in connection with mathieu) has been a feature of western triadic >> music, hence the use of 12-tone equal (or well) temperament. The >> other 'commas' can be used for bridging in other, "invented" musical >> systems, motivating certain corresponding tuning systems as shown at: >> >> Definitions of tuning terms: equal temperament... * [with cont.] (Wayb.) >> >> for example, you can see from the first chart and table on that page >> that "bridging" through 243:250 is characteristic of porcupine >> temperament, through 3072:3125 of magic temperament, and through both >> of them (and thus also any combination of the two) of 22-tone equal >> temperament. >> ' > >> OK, i added that. when i have more time i'd also like to > include what you wrote after that. > > > > > -monzthanks monz. shall we move on to the next correction now?

Message: 6521 - Contents - Hide Contents Date: Tue, 18 Feb 2003 23:34:42 Subject: Re: lattice diagram "levels" of complexity From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> is there an accepted method of nomenclature for > describing the "level" of complexity of lattice > diagrams? > > i know that "stellation" has something to do > with this, but here i'm talking pretty much about > the algorithm used by Partch to fill out the > Tonality Diamond. > > in other words, say we have a 7-limit (3-D) lattice. > > (use "Expand Messages" if viewing on Yahoo website > for proper formatting of diagram) > > > here are the two basic tetrads (otonal and utonal) > which have the 1/1 ratio as their 1-identity: > > > 5:4 > /|\ > / | \ > / | \ > / 7:4 \ > /. ' ' .\ > 4:3---------1:1---------3:2 > \ '. .' / > \ 8:7 / > \ | / > \ | / > \|/ > 8:5 > > this would be "level" 1. > > > now if we build complete otonal tetrads > on all of the notes in the "basic" utonality > tetrad (i.e., 4/3, 8/5, and 8/7 all become > 1-odentities of their respective tetrads), and > complete utonal tetrads on all of the notes in > the "basic" otonality tetrad (i.e, 3/2, 5/4, and > 7/4 all become 1-udentities of their respective > tetrads), we get this: > > 5:3---------5:4 > /|\ '. .' /|\ > / | \ 10:7 / | \ > / | \ /|\ / | \ > / 7:6--/-|-\--7:4 \ > /. ' '/.\|/.\' ' .\ > 4:3------/--1:1--\------3:2 > \ '. /.' /|\ '.\ .'/ > \ 8:7--/-|-\--12:7 / > \ | / | \ | / > \ | / 7:5 \ | / > \|/ .' '. \|/ > 8:5---------6:5 > > which would be "level" 2. > > > is there already an accepted term for what i'm > calling "level"? > > and can someone give a very clear and lucid explanation > of how stellation differs from this, if it does? > > thanks. > > > > -monzlet me suggest a different, cleaner method. level 1 will be a *single* tetrad (*either* utonal or otonal, it doesn't matter, but not both). level 2 results from building a tetrad that is the inverse of the original one, on each tone of the original tetrad. this is again a tonality diamond, but is closer to how many people (e.g., john chalmers) actually think of the diamond -- the cartesian product of a tetrad with its inverse.

Message: 6522 - Contents - Hide Contents Date: Tue, 18 Feb 2003 23:11:50 Subject: Re: lattice diagram "levels" of complexity From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> is there an accepted method of nomenclature for > describing the "level" of complexity of lattice > diagrams? > > i know that "stellation" has something to do > with this, but here i'm talking pretty much about > the algorithm used by Partch to fill out the > Tonality Diamond. > > in other words, say we have a 7-limit (3-D) lattice. > > (use "Expand Messages" if viewing on Yahoo website > for proper formatting of diagram) > > > here are the two basic tetrads (otonal and utonal) > which have the 1/1 ratio as their 1-identity: > > > 5:4 > /|\ > / | \ > / | \ > / 7:4 \ > /. ' ' .\ > 4:3---------1:1---------3:2 > \ '. .' / > \ 8:7 / > \ | / > \ | / > \|/ > 8:5 > > this would be "level" 1. > > > now if we build complete otonal tetrads > on all of the notes in the "basic" utonality > tetrad (i.e., 4/3, 8/5, and 8/7 all become > 1-odentities of their respective tetrads), and > complete utonal tetrads on all of the notes in > the "basic" otonality tetrad (i.e, 3/2, 5/4, and > 7/4 all become 1-udentities of their respective > tetrads), we get this: > > 5:3---------5:4 > /|\ '. .' /|\ > / | \ 10:7 / | \ > / | \ /|\ / | \ > / 7:6--/-|-\--7:4 \ > /. ' '/.\|/.\' ' .\ > 4:3------/--1:1--\------3:2 > \ '. /.' /|\ '.\ .'/ > \ 8:7--/-|-\--12:7 / > \ | / | \ | / > \ | / 7:5 \ | / > \|/ .' '. \|/ > 8:5---------6:5 > > which would be "level" 2. > > > is there already an accepted term for what i'm > calling "level"? > > and can someone give a very clear and lucid explanation > of how stellation differs from this, if it does? > > thanks. > > > > -monzi don't really get the appeal of this (what would level 3 be?), but it's somewhat similar to paul hahn's diameter measure. a *single* tetrad has a diameter of 1, while *either* a hexany *or* a diamond would have a diameter of 2, since you'd need no more than 2 consonant intervals to connect any two pitches . . .

Message: 6523 - Contents - Hide Contents Date: Tue, 18 Feb 2003 16:26:00 Subject: Re: lattice diagram "levels" of complexity From: Carl Lumma>> >thers, apparently including Wilson, wanted >> to include other points, basically out to the power >> set (the compound of all CPSs of a given limit). >>that isn't the case. look over the old discussions on this list about >stellation. the compound of all CPSs of a given limit can be >constructed in many ways, but most typically (as d'allessandro) as an >euler genus, and this (or any of the other ways) clearly does not >have the symmetry of the original, unstellated CPS, which the >stellation must have by definition.Right, right, you're extending the tones out to a EF Genus in all directions. A stellated EF Genus is what I called it in that thread. -Carl

Message: 6524 - Contents - Hide Contents Date: Tue, 18 Feb 2003 00:44:31 Subject: Re: A common notation for JI and ETs From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> OK, i added that. when i have more time i'd also like to > include what you wrote after that.This has turned into quite a page from the 5-limit point of view. Have you considered using the dual zoomers as well?

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