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Message: 1025 - Contents - Hide Contents Date: Thu, 28 Jun 2001 01:37:51 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich> --- In tuning-math@y..., "M. Edward (Ed) Borasky" <znmeb@a...> wrote: >>>> Outside of Partch, yes -- Otonal/Major is *musically* more> important than Utonal/Minor *in common practice Western music*.On what basis do you make that claim? They seem to be equal enough in importance in this music to "fool" Riemann, Partch, and many other theorists to give them equal footing a priori.
Message: 1026 - Contents - Hide Contents Date: Thu, 28 Jun 2001 21:47:35 Subject: Re: questions about Graham's matrices From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> monz wrote:>> This is along the lines of what I was trying to get Paul to >> understand a couple of different times in the past. It's not >> necessary to always use prime-factors as the basis for lattice >> metrics... any numbers that give even, consistent divisions >> of the pitch-space *in SOME way* will do. The different ways >> of dividing (and multiplying) produce different kinds of lattices.Funny -- I would have said that this is something I was trying to get Monz to understand, rather than something Monz was trying to get me to understand . . . Maybe Monz could restate the context in which he was trying to get me to understand this?
Message: 1027 - Contents - Hide Contents Date: Thu, 28 Jun 2001 21:55:38 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> Paul Erlich wrote: >>> I'm still confused about how there can be two different MIRACLE- 41s. >> Are there two different Canastas too, or does the divergence only >> happen at 41? >> There are two Canstas, 10+31n and 21+31n.Hmmm . . . what's the _real_ difference between these two?> > Are the FPBs different in this sense?Yes -- look back a few days -- I showed that there was a schisma difference between a few corresponding pitches in the two FPBs, even though you're claiming the schisma as a chromatic unison vector (hence one that isn't tempered out).>> How does it work? Certainly the scale doesn't repeat itself every >> 41st of an octave. >> Yes, it would do. If you try tuning a 12-note meantone in cents relative > to 12-equal, you'll see the pattern.I see the pattern, but that doesn't make 1/12 octave the period of a 12-note meantone . . . ?>> Well you _should_ be able to find the generator without specifying >> the chroma, but you need the chroma to select a particular MOS. >> Indeed so! But the octave invariant matrix doesn't give you that > particular MOS.Sure it does! Just take the determinant (usually)! (Assuming you already know the generator.)> Although it gives you enough of a clue to work it out > from the determinant,A big clue!> the main result is the mapping in terms of > generators.Well, that does seem to be something very interesting you've found. How can we get that without plugging in a chroma at all?
Message: 1028 - Contents - Hide Contents Date: Thu, 28 Jun 2001 02:36:46 Subject: Re: 41 "miracle" and 43 tone scales From: Paul Erlich --- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:> I was paraphrasing Partch ... I can probably find the line in _Genesis_, but > one of his goals was to restore Untonality to equal footing with Otonality, > thus implying an existing *in*equality.In musical _theories_ -- not in any of the musical _practice_ that he liked, as he understood it.
Message: 1029 - Contents - Hide Contents Date: Thu, 28 Jun 2001 22:29:28 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., Graham Breed <graham@m...> wrote:>>>> One is 10+41n, the other 31+41n. >>>>>> What do you mean by this notation? >>>> Temperements including the ETs with 10+41n or 31+41n notes, where n > is a >> non-negative integer. Graham,I still don't understand this. So 10+41n includes ETs 10 51 61 71 ... and 31+41n includes 31 72 113 ... Only the second looks anything like Miracle to me. Paul Erlich wrote:> I'm still confused about how there can be two different MIRACLE-41s. > Are there two different Canastas too, or does the divergence only > happen at 41? Graham replied: > There are two Canstas, 10+31n and 21+31n.What could this mean when 31 and 72 aren't members of either of these series? I'm very confused. -- Dave Keenan
Message: 1030 - Contents - Hide Contents Date: Thu, 28 Jun 2001 03:52:15 Subject: Re: 41 "miracle" and 43 tone scales From: jpehrson@xxx.xxx --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: Yahoo groups: /tuning-math/message/428 * [with cont.]> > But remember these are only conventions or conveniences. Themusical specialness is in "big number versus little number", not "numerator versus denominator".> > Regards, > -- Dave KeenanGot it! Thanks, Dave! ________ _______ ______ Joseph Pehrson
Message: 1031 - Contents - Hide Contents Date: Thu, 28 Jun 2001 22:35:32 Subject: Re: questions about Graham's matrices From: graham@xxxxxxxxxx.xx.xx To: <tuning-math@xxxxxxxxxxx.xxx> Cc: <graham@xxxxxxxxxx.xx.xx> Sent: Thursday, June 28, 2001 2:18 PM Subject: [tuning-math] Re: questions about Graham's matrices>> So all thru the rest of the explanation when you refer to >> "steps of 19=, 22=, 41="... they're all approximations to >> the generated scale. Right? >> There isn't "a generated scale". The generator can be used for a whole > family of scales. 19, 22 and 41-equal are merely special cases of the > scales it can generate.OK... how about "the *potential* generated scales" instead? But my point is... they're *approximately* 19, 22, and 41-equal, right?>> I really think it's much more intuitive to have it the other way around >> (your mistake here shows the persistence of that way of thinking). >> >> Put the generator first and the octave second, consistently. >> I agree with you that the number of generators is the more important >> figure, and to me it makes sense to *see* that number first. >> (I think your unconscious switch in the original post shows that.) >> No, because that would contradict the usual way of writing vectors from > low to high primes.Huh? In a general sense, generators can be any interval, so what does prime ordering have to do with it? Clarification on this would be appreciated.>>> And the "octave less two major thirds" translates into >> approximate ratios as ~2:1 / ~(5/4)^2 = ~32/25 . >> When I did the matrix calculation I got >> >> (0, 1) - 2*(1, 2) = (-2, -3) . >> >> Hmmm... the important number, the generator, works out to be >> the same -2, which is correct. But why is the period calculation >> not working out when the octave is included? Is is because >> there is no zero period? >> You're using 5:1 instead of 5:4.Thanks, Graham... I had figured that out by the time I sent this, but neglected to delete that bit before sending. (I *did* delete a lot of it as I learned-by-figuring-out...)>>> I think you're complicating it by bringing ratios and exact pitches >>> back >>> into it. >>>> Agreed... but using the ratios allowed *me* to do the math in >> an Excel spreadsheet so that I could follow your reasoning. >> I went thru it step by step, looking at the cents values all >> along the way. >> >> If I understood better how to manipulate the matrices, I certainly >> would have done it that way too. I can see that it's *much* more >> elegant that way, even tho I've been having trouble understanding it. >> If you've got Excel, you can do that! I explain it on my website. You > use MINVERSE, MDETERM and MMULT, pressing CTRL-SHIFT-RETURN to enter the > formulae.Thanks for that. Since buying my new system, and no longer having the CD of Excel, I can only run it with the minimal set of formulae. I recall having to add some extra files to Excel on my old machine to make it do some more esoteric math, and have avoided even trying on the new PC. I'll try it and see if these will work.>>> This is along the lines of what I was trying to get Paul to >> understand a couple of different times in the past. It's not >> necessary to always use prime-factors as the basis for lattice >> metrics... any numbers that give even, consistent divisions >> of the pitch-space *in SOME way* will do. The different ways >> of dividing (and multiplying) produce different kinds of lattices. >> Yes, I think this is what Pierre Lamothe was trying to get across before > he left the list as well.Wow... now *that's* a revelation, because I've been struggling to understand Pierre's work. It looks to me to be very closely related to some of my own, and I'm anxious to cut thru the comprehension barrier with respect to his work, as I've finally done with yours.>>> Of course, in the case of an octave-invariant scale >> like this it's much simpler to just omit the period. So >> ignoring the first column in "H" because it's powers of 2, >> >> ratio prime vector 380.391-cent generators >> >> 2:1 = ( 1 0 0 0)H = ~ 0 >> 3:2 = (-1 1 0 0)H = ~ 5 >> 5:4 = (-2 0 1 0)H = ~ 1 >> 7:4 = (-2 0 0 1)H = ~12 >> 9:7 = ( 0 2 0 -1)H = ~(2*5)-12 = ~-2. >> If you're thinking octave invariantly, you can simplify it further. > > ratio prime vector 380.391-cent generators > > 3:2 = (1 0 0)H = ~ 5 > 5:4 = (0 1 0)H = ~ 1 > 7:4 = (0 0 1)H = ~12 > 9:7 = (2 0 -1)H = ~(2*5)-12 = ~-2.Right, I knew that and just didn't bother to omit the "2" column.>>> My other program starts with the unison vectors, but that's more >>> complex. >>> A third program could go from unison vectors to a mapping in terms of >>> generators within a period that's a fraction of an octave. In that >>> case, >>> it'd have to get the equal temperaments by optimizing for the best >>> generator/period ratio, and walking the scale tree. >>>> Hmmm.... that last algorithm sounds like a good one! Exactly the >> kind of thing I always wanted to include in my JustMusic software, >> applicable to rational systems as well as irrational. >> Try it. Type the octave-invariant unison vectors into the spreadsheet as > an array, with the chromatic one at the top. Then select an equal sized > square, type "=minverse(?:?)*mdeterm(?:?)" where ?:? is that original > array, and the left hand column will be the mapping by generators.Cool... I can't wait to check this out. Hope it works.>>>>> So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal >>>>> and p is 1 step in 41-or 22-equal, and no steps in 19-equal. >>>> >>>> Getting foggier... >>>>>> | 19= | 22= | 41= >>> --------------------------------- >>> x | 0 | 1 | 1 >>> y | 1 | 0 | 1 >>> p | 0 | 1 | 1 >>> q | 1 | 1 | 2 >>> >> >>>> Uh-oh... still not getting this part. Please elaborate. >> For some reason I'm not seeing the connection between >> p and q and the steps of the EDOs. >> >> Too many layers of abstraction for me to follow ...I was hoping for another remedial-algebra lesson on how this table could be calculated from the other data you gave. :)> If you tune to a given ET, the table shows you how many steps will be in > each interval. >>> So your parenthetical lists are elegant, but IMO could use >> a little bit more of a legend explaining what those lists >> represent. Otherwise one has to learn the sequences beforehand >> and keep them in mind. I suggest adding a label giving the >> parameter list before each line. >> The "parenthetical lists" are the stringifications of the objects the > program uses. I could get it to write out an HTML file for each > temperament, but for the moment it's cutting-edge data.Keep up the good work, Graham! Please come on over to JustMusic when you're interested in helping us graph this stuff. (And that goes for you two too, Dave and Paul.) -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: 1033 - Contents - Hide Contents Date: Thu, 28 Jun 2001 22:39:42 Subject: Re: questions about Graham's matrices From: Paul Erlich To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Thursday, June 28, 2001 2:47 PM Subject: [tuning-math] Re: questions about Graham's matrices> --- In tuning-math@y..., graham@m... wrote: >> monz wrote: >>>> This is along the lines of what I was trying to get Paul to >>> understand a couple of different times in the past. It's not >>> necessary to always use prime-factors as the basis for lattice >>> metrics... any numbers that give even, consistent divisions >>> of the pitch-space *in SOME way* will do. The different ways >>> of dividing (and multiplying) produce different kinds of lattices. >> Funny -- I would have said that this is something I was trying to get > Monz to understand, rather than something Monz was trying to get me > to understand . . . Maybe Monz could restate the context in which he > was trying to get me to understand this?Oh... one time was back around when I asked you to show me how to prime-factorize my meantone formulae. As I said, I don't have the mathematical understanding to even imagine accurately, let alone describe, some of the vague latticing ideas I have. I'm trying. But I'm glad to see that somehow we managed to agree on this, without understanding each other! Cool. -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: 1034 - Contents - Hide Contents Date: Fri, 29 Jun 2001 12:27 +0 Subject: Re: Hypothesis revisited From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9hgb48+f78j@xxxxxxx.xxx> Dave Keenan wrote:> I still don't understand this. So 10+41n includes ETs 10 51 61 71 ... > and 31+41n includes 31 72 113 ... Only the second looks anything like > Miracle to me.They both cover this part of the scale tree 31 10 41 72 51 93 113 91 61 but branch differently at 41. So one is more closely associated with the Miracle family, but I don't think there's anything special about one unison vector as compared to the other. Probably I should ignore the generalisation, and take 31+10 and 10+31 as different ways of writing the same MOS.> Graham replied:>> There are two Canstas, 10+31n and 21+31n. >> What could this mean when 31 and 72 aren't members of either of these > series? I'm very confused.31 is the member where n=infinity. It enshrines the relationship 10+21=31, and hence this part of the scale tree 21 10 31 52 41 73 83 72 51 There are two different ways you can move on from Canasta, and different chromas suggest different branchings, but I've yet to see a deep reason for it. Graham
Message: 1035 - Contents - Hide Contents Date: Fri, 29 Jun 2001 12:27 +0 Subject: Re: questions about Graham's matrices From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <003101c1005d$50edba00$4448620c@xxx.xxx> monz wrote:> OK... how about "the *potential* generated scales" instead? > > But my point is... they're *approximately* 19, 22, and 41-equal, right?They all contain MOS subsets with these numbers of notes.> Huh? In a general sense, generators can be any interval, so > what does prime ordering have to do with it? Clarification on > this would be appreciated.The period is a stand-in for the octave, and the generator is a stand-in for the twelfth (or fifth or whatever but not always). So it makes sense to put them in the same positions as the octave and fifth. In some common cases they really are an octave and fifth. Although you can define them the other way round, it's simpler not to.>>>> | 19= | 22= | 41= >>>> --------------------------------- >>>> x | 0 | 1 | 1 >>>> y | 1 | 0 | 1 >>>> p | 0 | 1 | 1 >>>> q | 1 | 1 | 2> I was hoping for another remedial-algebra lesson on how this > table could be calculated from the other data you gave. :)An octave is 22x+19y steps. So when x=0 you have 19-equal and when y=0 you have 22-equal. When x=y you have 41-equal. As p is the same as x, it must have the same size in those temperaments. As q is the sum of x and y it'll either be 1 (from 1+0 or 0+1) or 2 (from 1+1). In algebraic terms (x) = (1)oct/22 or (0)oct/19 or (1)oct/41 (y) (0) (1) (1) (q) = (1 1)(x) = (1 1)(1)oct/22 = (1)oct/22 (p) (1 0)(y) (1 0)(0) (1) or (q) = (1 1)(0)oct/19 = (1)oct/19 (p) (1 0)(1) (0) or (q) = (1 1)(1)oct/41 = (2)oct/41 (p) (1 0)(1) (1) Graham
Message: 1036 - Contents - Hide Contents Date: Fri, 29 Jun 2001 12:27 +0 Subject: Re: Hypothesis revisited From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9hg94q+1es0@xxxxxxx.xxx> Paul wrote:>> There are two Canstas, 10+31n and 21+31n. >> Hmmm . . . what's the _real_ difference between these two?How are you defining reality?>> Are the FPBs different in this sense? >> Yes -- look back a few days -- I showed that there was a schisma > difference between a few corresponding pitches in the two FPBs, even > though you're claiming the schisma as a chromatic unison vector > (hence one that isn't tempered out).So does that mean the schisma isn't a valid chroma?>> Yes, it would do. If you try tuning a 12-note meantone in cents > relative>> to 12-equal, you'll see the pattern. >> I see the pattern, but that doesn't make 1/12 octave the period of a > 12-note meantone . . . ?It makes it the period of a linear temperament that includes 12-note meantone as a subset.>>> Well you _should_ be able to find the generator without > specifying>>> the chroma, but you need the chroma to select a particular MOS. >>>> Indeed so! But the octave invariant matrix doesn't give you that >> particular MOS. >> Sure it does! Just take the determinant (usually)! (Assuming you > already know the generator.)Usually isn't good enough, we're looking for proof here. Besides, taking the determinant's cheating. It doesn't mean anything for octave-invariant matrices, but happens to be part of the result for octave-specific matrices.>> the main result is the mapping in terms of >> generators. >> Well, that does seem to be something very interesting you've found. > How can we get that without plugging in a chroma at all?I'm hoping that always using a fifth for the top row will work. If not, framing the problem might help. We want to find a generator consistent with the simplest mapping, I suppose. Which means minimizing the determinant. We don't want it to go to zero, but that follows from the matrix being invertible. Is there any established theory of integer matrices, or discrete vector spaces, we can latch on to? Graham
Message: 1037 - Contents - Hide Contents
Date: Fri, 29 Jun 2001 15:06 +0
Subject: Vector update
From: graham@xxxxxxxxxx.xx.xx
Dividing through by the common factor of the whole normalised, inverted
matrix does do the trick for my multiple-29 vectors.
I think it's the *commatic* unison vectors that mean you have to do this.
So my unison vector finder needs to be improved (surprise!)
The second column of the normalised octave-specific inverse is always the
same as the first column of the octave-invariant one, with an extra zero.
I was forgetting the extra zero before. It may be this doesn't always
work for really silly unison vectors, but it does for all the examples
I've tried.
The octave-specific column of the octave-specific matrix is important for
getting the right scale-step mapping. This may be what was going wrong
with the multiple-29 before. Whatever, it works now.
I've got a rough and ready Excel spreadsheet showing this at
<404 Not Found * [with cont.] Search for http://www.microtonal.co.uk/vectors.xls in Wayback Machine>.
You need to install the Analysis ToolPack for the GCD function to work.
Matrix operations work with the standard install.
The Exchange Server at work is currently flaky, and although I have
offline folders I don't seem to be able to get at them offline. So
although I did read an e-mail from Monzo this morning, I can't reply to
it.
I think any commatic unison vector will do to get the generator mapping,
so long as it's orthogonal to the other vectors. One good way of finding
such is to try a 1 in every column until the determinant is non-zero.
I'll try to include these changes in my Python code tonight. Python with
the Numerical extensions is a good way of hacking this stuff, but the
latter had disappeared from the FTP server last I checked, so I don't
know how you'll get hold of them.
The source code to MIDI Relay should include a matrix library for C++.
Graham
Message: 1038 - Contents - Hide Contents Date: Fri, 29 Jun 2001 19:25:19 Subject: Re: agreeing without understanding (was: questions about Graham's matrices) From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > Sure, these are all good. But I was thinking along the > lines of, say, an axis representing a meantone generator, > such as (3/2)/((81/80)^(1/4)) , for example.Well that, of course, would be the _right_ way to plot meantone tunings. Which is what I was trying to tell you while grudgingly figuring out for you how to plot meantone on the usual (3,5) axes using fractional exponents.
Message: 1039 - Contents - Hide Contents Date: Fri, 29 Jun 2001 19:32:52 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9hg94q+1es0@e...> > Paul wrote: >>>> There are two Canstas, 10+31n and 21+31n. >>>> Hmmm . . . what's the _real_ difference between these two? >> How are you defining reality? Tuning. >>>> Are the FPBs different in this sense? >>>> Yes -- look back a few days -- I showed that there was a schisma >> difference between a few corresponding pitches in the two FPBs, even >> though you're claiming the schisma as a chromatic unison vector >> (hence one that isn't tempered out). >> So does that mean the schisma isn't a valid chroma?I'm not saying that . . . but first, can you determine which of the two scales (if either) is the "real" MIRACLE-41 (to within commatic unison vectors)?>>>> Yes, it would do. If you try tuning a 12-note meantone in cents >> relative>>> to 12-equal, you'll see the pattern. >>>> I see the pattern, but that doesn't make 1/12 octave the period of a >> 12-note meantone . . . ? >> It makes it the period of a linear temperament that includes 12- note > meantone as a subset.Oh -- but a very strange subset. Any "normal" subset should repeat exactly at the period . . . or that's how I've been thinking about this stuff.>>>>> Well you _should_ be able to find the generator without >> specifying>>>> the chroma, but you need the chroma to select a particular MOS. >>>>>> Indeed so! But the octave invariant matrix doesn't give you that >>> particular MOS. >>>> Sure it does! Just take the determinant (usually)! (Assuming you >> already know the generator.) >> Usually isn't good enough, we're looking for proof here. Besides, > taking the determinant's cheating. It doesn't mean anything for > octave-invariant matrices,It doesn't mean anything?? It means a lot -- see the "Gentle Introduction" again . . .> but happens to be part of the result for > octave-specific matrices. >Part of the __________ result?>>> the main result is the mapping in terms of >>> generators. >>>> Well, that does seem to be something very interesting you've found. >> How can we get that without plugging in a chroma at all? >> I'm hoping that always using a fifth for the top row will work. If not, > framing the problem might help. We want to find a generator consistent > with the simplest mapping, I suppose.The simplest mapping? Not following you. The generator of an MOS is unique.
Message: 1040 - Contents - Hide Contents Date: Fri, 29 Jun 2001 19:34:59 Subject: Re: Vector update From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> I think any commaticYou mean chromatic?> unison vector will do to get the generator mapping, > so long as it's orthogonal toYou mean linearly independent from?> the other vectors.
Message: 1041 - Contents - Hide Contents Date: Fri, 29 Jun 2001 22:45 +0 Subject: Re: Vector update From: graham@xxxxxxxxxx.xx.xx Paul wrote:>> I think any commatic >> You mean chromatic? Yes.>> unison vector will do to get the generator mapping, >> so long as it's orthogonal to >> You mean linearly independent from?I think so, but I didn't take good notes in that lecture.>> the other vectors. > Graham
Message: 1042 - Contents - Hide Contents Date: Fri, 29 Jun 2001 22:45 +0 Subject: Re: Hypothesis revisited From: graham@xxxxxxxxxx.xx.xx Paul wrote:>>>> There are two Canstas, 10+31n and 21+31n. >>>>>> Hmmm . . . what's the _real_ difference between these two? >>>> How are you defining reality? > > Tuning.There's no difference.>> So does that mean the schisma isn't a valid chroma? >> I'm not saying that . . . but first, can you determine which of the > two scales (if either) is the "real" MIRACLE-41 (to within commatic > unison vectors)?Don't know, it's all on my Linux partition.>> It makes it the period of a linear temperament that includes 12- > note>> meantone as a subset. >> Oh -- but a very strange subset. Any "normal" subset should repeat > exactly at the period . . . or that's how I've been thinking about > this stuff.Yes, it takes 144 notes to get the 12 note scale. But it does prove the hypothesis that every set of vectors gives some linear temperament.>> Usually isn't good enough, we're looking for proof here. Besides, >> taking the determinant's cheating. It doesn't mean anything for >> octave-invariant matrices, >> It doesn't mean anything?? It means a lot -- see the "Gentle > Introduction" again . . .I'll look it up.>> but happens to be part of the result for >> octave-specific matrices. >>> Part of the __________ result?Octave-specific. It's the top left-hand corner.>> I'm hoping that always using a fifth for the top row will work. If > not,>> framing the problem might help. We want to find a generator > consistent>> with the simplest mapping, I suppose. >> The simplest mapping? Not following you. The generator of an MOS is > unique.As long as it's unique, there's no problem. (Technically, it'll be +/-, but that's all negotiable) Graham
Message: 1044 - Contents - Hide Contents Date: Fri, 29 Jun 2001 23:48:35 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9hgb48+f78j@e...> > Dave Keenan wrote: >>> I still don't understand this. So 10+41n includes ETs 10 51 61 71 ... >> and 31+41n includes 31 72 113 ... Only the second looks anything like >> Miracle to me. >> They both cover this part of the scale tree > > 31 10 > > > 41 > > 72 51 > > 93 113 91 61Ok. I can see that you are the one who is confused here. Miracle does not go outside of 31 41 72 93 113 Well Ok, it does go a tiny bit past 41, but nowhere near all the way to 10. Just as 5-EDO is nothing like a meantone. This is because, outside of 31 to 41(and-a-bit) there are better 7 or 11-limit approximations than the ones used by MIRACLE.>> Graham replied:>>> There are two Canstas, 10+31n and 21+31n. >>>> What could this mean when 31 and 72 aren't members of either of these >> series? I'm very confused. >> 31 is the member where n=infinity.Huh? When n=oo 10+31n and 21+31n also go to oo. I think you must be talking your own language here. It enshrines the relationship> 10+21=31, and hence this part of the scale tree > > > 21 10 > > > 31 > > 52 41 > 73 83 72 51 > > > There are two different ways you can move on from CanastaMaybe so. But only one of them is MIRACLE.
Message: 1046 - Contents - Hide Contents Date: Fri, 29 Jun 2001 05:45:37 Subject: Re: questions about Graham's matrices From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> But I'm glad to see that somehow we managed to agree on this, > without understanding each other! Cool. >Yes, and I think it's a very important point. Rather than the basis being 3, 5, 7, it could just as easily be, say, 5/4, 6/5, 7/6, or any other basis that spans the 3D lattice(though consonant intervals are somewhat preferable here).
Message: 1047 - Contents - Hide Contents Date: Fri, 29 Jun 2001 01:45:10 Subject: agreeing without understanding (was: questions about Graham's matrices) From: monz> From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, June 28, 2001 10:45 PM > Subject: [tuning-math] Re: questions about Graham's matrices > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> But I'm glad to see that somehow we managed to agree on this, >> without understanding each other! Cool. >>> Yes, and I think it's a very important point.I agree. We were both frustrated that we couldn't understand each other, in both cases, but it turns out that we were thinking basically the same thing all along.> Rather than the basis being 3, 5, 7, it could just as > easily be, say, 5/4, 6/5, 7/6, or any other basis that > spans the 3D lattice (though consonant intervals are > somewhat preferable here).Sure, these are all good. But I was thinking along the lines of, say, an axis representing a meantone generator, such as (3/2)/((81/80)^(1/4)) , for example. This is why, a few months back, I was so interested in Regener's work. It seems to be along these lines... transformations of ratio-space. -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: 1048 - Contents - Hide Contents Date: Sat, 30 Jun 2001 15:03:32 Subject: Re: Hypothesis revisited From: Graham Breed Dave Keenan wrote:>> They both cover this part of the scale tree >> >> 31 10 >> >> >> 41 >> >> 72 51 >> >> 93 113 91 61 >> Ok. I can see that you are the one who is confused here. Miracle does > not go outside of > > 31 > > > 41 > > 72 > > 93 113I see a 41 there.> Well Ok, it does go a tiny bit past 41, but nowhere near all the way > to 10. Just as 5-EDO is nothing like a meantone.So decimal notation is now invalid? And blackjack isn't part of the family? 5-EDO may not be a meantone, but pentatonic scales certainly are.> This is because, outside of 31 to 41(and-a-bit) there are better 7 > or 11-limit approximations than the ones used by MIRACLE. >>> Graham replied:>>>> There are two Canstas, 10+31n and 21+31n. >>>>>> What could this mean when 31 and 72 aren't members of either of > these>>> series? I'm very confused. >>>> 31 is the member where n=infinity. >> Huh? When n=oo 10+31n and 21+31n also go to oo. I think you must be > talking your own language here.Canasta is made up of 31 steps. For 10+31n, there are 10 of those at (n+1)/(10+31n) octaves and the other 21 are n/(10+31n) octaves. As n tends to infinity, both steps tend to 1/31 octaves.> It enshrines the relationship>> 10+21=31, and hence this part of the scale tree >> >> >> 21 10 >> >> >> 31 >> >> 52 41 >> 73 83 72 51 >> >> >> There are two different ways you can move on from Canasta >> Maybe so. But only one of them is MIRACLE.I've changed the way the temperaments are written to sweep all this under the carpet. Graham "I toss therefore I am" -- Sartre
Message: 1049 - Contents - Hide Contents Date: Sat, 30 Jun 2001 23:29:39 Subject: Re: Hypothesis revisited From: Dave Keenan --- In tuning-math@y..., Graham Breed <graham@m...> wrote:> So decimal notation is now invalid?Of course not.> And blackjack isn't part of the family?Of course it is part of the family.> 5-EDO may not be a meantone, but pentatonic scales certainly are.Indeed, this is the crux of the confusion. The Stern-Brocot tree (considered as fractions of an octave) knows nothing about odd-limits (or any other kind), while the definition of Miracle, or meantone or any other temperament, must refer to them. The tree can tell us two different things about a temperament. (a) The number of notes in its MOS (b) The EDOs that are included in that temperament But we look up these things on the scale tree in two different ways. We need to know the range of generator sizes that are within the temperament. First we determine what limit we are using (say 7-odd for Miracle, 5-odd for meantone). Then we consider the maximum number of generators we are willing to chain to approximate these just intervals. (say 20 for Miracle and 11 for meantone). From this we can determine the range of generator sizes for which the temperament's mapping from primes to generators (Miracle [6, -7, -2] and meantone [1, 4]) gives us the best approximation. It is really the mapping from primes to generators that is the definition of the temperament. Once we have the two extreme generator sizes, we express these as fractions of an octave and mark them at the "bottom" of the tree (where the reals live). Draw straight lines up from these and the denominator of any fraction between those bounds gives us an ET within that temperament. The denominator of any fraction reachable by going up the tree from these, gives us the cardinality of a MOS in the temperament. So 10 and 11 and 21 are MOS cardinalities in Miracle temperament but certainly not EDO cardinalities. If that were the case, why stop at 10 and 11, why not go all the way back to 0 and 1? And remember that the SB tree has numerators and denominators. For convenience when talking about a particular temperament we drop the numerators. This might lead to confusion if we join together what are really disjoint parts of the tree, based on the denominators only.> I've changed the way the temperaments are written to sweep all this under > the carpet. I'm glad.-- Dave Keenan
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