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Message: 5575 - Contents - Hide Contents Date: Thu, 14 Nov 2002 19:54:35 Subject: Re: 43edo 7-limit periodicity-block From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:> i've just added some 7-limit lattices to my > Tuning Dictionary "meride" entry, showing the > "closest to 1/1" 7-limit periodicity-block > for 43edo. > > Definitions of tuning terms: meride, (c) 1998 ... * [with cont.] (Wayb.) > > (at the bottom of the page) > > just above the lattice, i refer to Gene's > "7-limit MT reduced bases for 43edo". but > i find that on these lattices, 225:224 is closer > than 126:125. is that because i'm using the > rectangular rather than triangular/hexagonal > taxicab metric?partially, yes. really, the "T" in MT refers to the Tenney Harmonic Distance function, in which the ratio with smaller numbers is always represented by a shorter distance than a ratio with larger numbers. geometrically, it *is* a rectangular lattice, but it (crucially) includes 2 as a factor, and the length of each rung along the axis for prime p is log(p). kees van prooijen's page, which i just referred you to in a private e-mail, presents an impressive attempt to incorporate the smaller-numbers-ratio->smaller-distance idea onto an octave-equivalent lattice (at least for the small, comma-like intervals), the octave-equivalence being necessary for representing periodicity blocks with a finite number of points. i tried very hard to get members of this list interested in kees' idea, and to help figure out what was going on with this metric, but i found it akin to beating my head against a wall.> so anyway, the bases i see are 81:80 and 225:224.these are not bases: a basis for an et in the 7-limit would have to consist of three unison vectors.> what's the third one?there are of course an infinite number of possibilities. note that 225:224 * 126:125 = 81:80. so these three are not linearly independent. any two imply the third. since gene's list was 81:80 126:125 12288:12005 we therefore know that 12288:12005 is one possible choice for forming a complete basis for 43 along with 81:80 and 225:224.

Message: 5576 - Contents - Hide Contents Date: Thu, 14 Nov 2002 19:55:32 Subject: Re: Osmium-Orwell-Secor From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> yup, gene mentioned that osmium is the densest metal, so i got it. >> now, if gene would help me with the graph density problem . . . >> Which graph are you talking about?it was my last post before this one: Yahoo groups: /tuning-math/message/5003 * [with cont.]

Message: 5577 - Contents - Hide Contents Date: Thu, 14 Nov 2002 13:19:28 Subject: bingo cards with period boundaries for monz From: wally paulrus these show the points closest to each "1/1" according to euclidean distance on the equilateral triangular (equilateral hexagonal) lattice: Yahoo groups: /tuning-math/files/Paul/10p.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/12p.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/19p.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/22p.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/31p.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/34p.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/41p.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/43p.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/53p.gif * [with cont.] Yahoo groups: /tuning-math/files/Paul/55p.gif * [with cont.] i can easily create more -- each only takes a few seconds . . . --------------------------------- Do you Yahoo!? Yahoo! Web Hosting - Let the expert host your site [This message contained attachments]

Message: 5578 - Contents - Hide Contents Date: Thu, 14 Nov 2002 04:09:51 Subject: Re: Osmium-Orwell-Secor From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> yup, gene mentioned that osmium is the densest metal, so i got it. > now, if gene would help me with the graph density problem . . .Which graph are you talking about?

Message: 5579 - Contents - Hide Contents Date: Fri, 15 Nov 2002 15:29:35 Subject: Re: A common notation for JI and ETs From: gdsecor I'd like to finalize the notation for the two remaining multiples of 12: --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4662]:> At 06:19 PM 17/09/2002 -0700, George Secor wrote:>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>>> At 10:24 AM 13/09/2002 -0700, George Secor wrote:>>>> 132a: ~|( /| |) |\ (|~ ~||( /|| ||) ||\(||~ /||\ (MS)>>>> 132b: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||~ /||\ (MS) >>>>>> I prefer 132b, but why not |( as 5:7-comma for 1deg132? >>>> I try to choose symbols that are as valid in as many roles as possible. >> |( is valid only as the 5:7 comma and not as the 11:13 or 17'-17 >> commas (1 out of 3), whereas ~|( needs to be valid only as the 17' >> comma (1 out of 1). This is another one that I don't have strong >> feelings about, and in the course of working on the spreadsheet I might >> change my mind. Even if we don't get any final agreement at this point >> about some of these less common divisions, at least our discussion of >> these will provide some examples from which I can arrive at general >> principles for choosing symbols. > > OKHere I've taken the single-shaft symbols of 132b and used their rational complements: 132c: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||( /||\ (RC) But I'm beginning to wonder if we should allow /|\ to exceed (|), which would give us a more meaningful 5deg symbol: 132d: ~|( /| |) |\ (|) /|\ /|| ||) ||\ (||( /||\ (RC) This might be justified on the same basis that we have allowed /| to exceed |) and even |\ in a few instances. After all, we are already used to seeing either sharps or flats higher in pitch in different octave divisions. Now for 144. I would really like to have its notation in the XH article, because it's been mentioned quite a bit on the tuning list. --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4654]:> At 10:24 AM 13/09/2002 -0700, George Secor wrote:>> 144: ~|( /| )|) |\ /|) /|\ (|\ /|| )||) ||\ /||) /||\ > Agreed.This is what we get if we use the above with rational complements: 144b: ~|( /| )|) |\ /|) /|\ (|\ /|| )/|| ||\ (||( /||\ (RC) I've now think that I wouldn't want to use )|) if I didn't have to -- it's a more unusual symbol (and therefore less memorable) than the 23- comma: 144c: ~|( /| |~ |\ /|) /|\ (|\ /|| ~||) ||\ (||( /||\ (RC) In the past you have used prime limit as a measure of simplicity, but I would justify using a 23-comma symbol on the basis of product complexity. This would also enable us to keep the notation for all the multiples of 12 up to 144 without going beyond the 18 single- shaft symbols that I am presenting in the article: )| |( ~| ~|( |~ )|~ /| |) |\ (| ~|) (|( //| /|) (|~ /|\ (|) (|\ These symbols are sufficient to notate all 17-limit consonances and all harmonics and subharmonics through 29, relative to the natural notes. Also, their rational complements collectively have the same combinations of flags as in the single-shaft set: (||~ /||) //|| (||( ~||) (|| ||\ ||) /|| )||~ ||~ ~||( ~|| (|\ )|| So I think that these 18 symbols could be a useful set for the moderately sophisticated user, just as the "starter set" of 7 symbols that I have in Table 3 of my article would be for the simpler ETs (including all multiples of 12 through 96): /| |) |\ /|) /|\ (|) (|\ and rational complements ||\ ||) /|| --George

Message: 5580 - Contents - Hide Contents Date: Fri, 15 Nov 2002 19:47:47 Subject: Re: 43edo 7-limit periodicity-block From: wallyesterpaulrus --- In tuning-math@y..., manuel.op.de.coul@e... wrote:> Paul wrote: >>> kees van prooijen's page, which i just >> referred you to in a private e-mail, presents an impressive attempt >> to incorporate the smaller-numbers-ratio->smaller-distance idea onto >> an octave-equivalent lattice (at least for the small, comma-like >> intervals), the octave-equivalence being necessary for representing >> periodicity blocks with a finite number of points. i tried very hard >> to get members of this list interested in kees' idea, and to help >> figure out what was going on with this metric, but i found it akin to >> beating my head against a wall. >> I haven't told yet that this metric is implemented in Scala: > SET ATTRIBUTE PROOIJEN > and there's a little text in tips.par. >>> we therefore know that 12288:12005 is one possible choice for forming >> a complete basis for 43 along with 81:80 and 225:224. >> This PB doesn't look like what Joe plotted on his page.i said a complete basis, not necessarily a set of edges for a fokker parallelepiped.> Is it really a PB? I'm not sure.is what really a PB? Joe's block? you have to resolve the ambiguous positions he indicated for different occurences of the same note at the same distance from 1/1, but once you've done that, it most certainly is a PB, since it contains one and only one instance of each of the 43 tones of 43-equal.

Message: 5581 - Contents - Hide Contents Date: Fri, 15 Nov 2002 10:40:19 Subject: Re: 43edo 7-limit periodicity-block From: manuel.op.de.coul@xxxxxxxxxxx.xxx Paul wrote:>kees van prooijen's page, which i just >referred you to in a private e-mail, presents an impressive attempt >to incorporate the smaller-numbers-ratio->smaller-distance idea onto >an octave-equivalent lattice (at least for the small, comma-like >intervals), the octave-equivalence being necessary for representing >periodicity blocks with a finite number of points. i tried very hard >to get members of this list interested in kees' idea, and to help >figure out what was going on with this metric, but i found it akin to >beating my head against a wall.I haven't told yet that this metric is implemented in Scala: SET ATTRIBUTE PROOIJEN and there's a little text in tips.par.>we therefore know that 12288:12005 is one possible choice for forming >a complete basis for 43 along with 81:80 and 225:224.This PB doesn't look like what Joe plotted on his page. Is it really a PB? I'm not sure. Manuel

Message: 5582 - Contents - Hide Contents Date: Sat, 16 Nov 2002 13:29:41 Subject: Re: 43edo 7-limit periodicity-block From: monz> From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Friday, November 15, 2002 11:47 AM > Subject: [tuning-math] Re: 43edo 7-limit periodicity-block > > > --- In tuning-math@y..., manuel.op.de.coul@e... wrote: >> Paul wrote: >>>>> we therefore know that 12288:12005 is one possible >>> choice for forming a complete basis for 43 along >>> with 81:80 and 225:224. >>>> This PB doesn't look like what Joe plotted on his page. >> i said a complete basis, not necessarily a set of edges > for a fokker parallelepiped.i don't know the difference between a "complete basis" and "a set of edges for a fokker parallelepiped", and would greatly welcome an explanation.>> Is it really a PB? I'm not sure. >> is what really a PB? Joe's block? you have to resolve > the ambiguous positions he indicated for different > occurences of the same note at the same distance from > 1/1, but once you've done that, it most certainly is > a PB, since it contains one and only one instance of > each of the 43 tones of 43-equal.paul's explanation here is exactly right. i left in the doubled and tripled instances of pitches which are an equal-number of taxicab steps away from 1/1 in any of the six directions (+/- 3/5/7), and connected them with lines showing their equivalence. but in any case, Manuel is absolutely correct that 12288:12005 is *not* one of the unison-vectors defining the periodicity-block in my graphic. the matrix i published on the webpage to represent Gene's "7-limit MT reduced bases" for 43-edo is: 2 3 5 7 [-4 4 -1 0] = 81/80 [ 1 2 -3 1] = 126/125 [12 1 -1 -4] = 12288/12005 the periodicity-block in my graphic only uses 7^(-1,0,+1). so my question still has not been answered: in addition to 81:80 and 225:224, which i can easily see in my diagram, what is the third necessary unison-vector which defines my periodicity-block? and how does one figure that out? -monz

Message: 5583 - Contents - Hide Contents Date: Sun, 17 Nov 2002 17:47:30 Subject: Re: 43edo 7-limit periodicity-block From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:>> From: "wallyesterpaulrus" <wallyesterpaulrus@y...> >> To: <tuning-math@y...> >> Sent: Sunday, November 17, 2002 12:19 AM >> Subject: [tuning-math] Re: 43edo 7-limit periodicity-block >> >> >> anyway, 12288:12005 is certainly going to work for this purpose. >> i don't see why you're denying it above. if you're worried that just >> because its 7s exponent is -4, while you're only using 3 different >> levels along the 7 direction in your plot, it isn't going to work, >> worry no further. it will work just fine. if you prefer to keep the 7s >> exponent within the plus-or-minus 3 range, you can always add >> or subtract (the exponents of ) any of the other unison vectors. for >> example, if you add the exponents of 126:125 ([2 -3 1]) to those >> of 12288:12005 ([1 -1 -4]), you get [3 -4 -3] . . . now how about >> adding the exponents of 224:225 ([-2 -2 1]), resulting in [1 -6 -2] . >> . . all of these are valid choices for the third unison vector as well. > >> ah, yes -- of course! duh, my bad. i just didn't see it at first. > got it.[1 -6 2] can actually be seen on your chart, separating two pairs of "0"s, each having one member on the 7^1 plane and one member on the 7^(-1) plane . . . see it? in any case, it's clear that the third unison vector, no matter what you decide to make it, is going to be a lot longer than the other two . . . so in a sense, it's the other two unison vectors that will determine what scales in 43 allow one to exploit its abundance of 7-limit harmony . . . and as we know from this list, the pair <81:80, 126:125>, or equivalently <81:80, 225:224>, defines the septimal meantone system, whose generator is the perfect fifth . . . thus, the most efficient 7-limit scales in 43 will tend to lie along the perfect fifth direction in the plot.

Message: 5584 - Contents - Hide Contents Date: Sun, 17 Nov 2002 08:19:27 Subject: Re: 43edo 7-limit periodicity-block From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:>> From: "wallyesterpaulrus" <wallyesterpaulrus@y...> >> To: <tuning-math@y...> >> Sent: Friday, November 15, 2002 11:47 AM >> Subject: [tuning-math] Re: 43edo 7-limit periodicity-block >> >> >> --- In tuning-math@y..., manuel.op.de.coul@e... wrote: >>> Paul wrote: >>>>>>> we therefore know that 12288:12005 is one possible >>>> choice for forming a complete basis for 43 along >>>> with 81:80 and 225:224. >>>>>> This PB doesn't look like what Joe plotted on his page. >>>> i said a complete basis, not necessarily a set of edges >> for a fokker parallelepiped. > >> i don't know the difference between a "complete basis" and > "a set of edges for a fokker parallelepiped", and would > greatly welcome an explanation.any set of unison vectors which get you a 43-tone periodicity block, which can be reasonably identified with 43-equal (in the ways you've been doing), are a complete basis for 43. the "edges of the fokker parallelepiped" refers to the method of creating periodicity blocks discussed in the gentle introduction (except the "excursion"). for example, you can see that (say, by comparing ramos' tuning vs. other 12-note tunings) there are different 12-tone periodicity blocks with different unison vectors serving as their edges -- but any of these pairs of unison vectors suffices as a basis for 12. if this still isn't making sense, look at the "excursion". note that *any* of the unison vectors: syntonic comma chromatic semitone greater limma describes how the first hexagonal periodicity block repeats itself in the lattice. any *two* of these unison vectors will form a basis for a 7-tone universe. similarly, *any* of the unison vectors syntonic comma diesis diaschisma describes how the second hexagonal periodicity block repeats itself in the lattice. any *two* of these unison vectors will form a basis for a 12-tone universe. in neither case, though, are two of the unison vectors representing the edges of a parallelogram -- because here, we have a hexagon instead of a parallelogram!>>> Is it really a PB? I'm not sure. >>>> is what really a PB? Joe's block? you have to resolve >> the ambiguous positions he indicated for different >> occurences of the same note at the same distance from >> 1/1, but once you've done that, it most certainly is >> a PB, since it contains one and only one instance of >> each of the 43 tones of 43-equal. > >> paul's explanation here is exactly right. i left in the > doubled and tripled instances of pitches which are an > equal-number of taxicab steps away from 1/1 in any of > the six directions (+/- 3/5/7), and connected them with > lines showing their equivalence. > > > but in any case, Manuel is absolutely correct that > 12288:12005 is *not* one of the unison-vectors defining > the periodicity-block in my graphic. > > the matrix i published on the webpage to represent > Gene's "7-limit MT reduced bases" for 43-edo is: > > 2 3 5 7 > > [-4 4 -1 0] = 81/80 > [ 1 2 -3 1] = 126/125 > [12 1 -1 -4] = 12288/12005 > > > the periodicity-block in my graphic only uses 7^(-1,0,+1). > > > so my question still has not been answered: in addition > to 81:80 and 225:224, which i can easily see in my > diagram, what is the third necessary unison-vector which > defines my periodicity-block?first of all, how are you "seeing" that 81:80 and 225:224 are two of the unison vectors defining your periodicity block?> and how does one figure > that out?just draw up a nice big block of the 7-limit lattice, divvy it up according to how your PB tiles it, and observe the vectors at which the tile repeats itself. simple! (there will be more than one answer, depending on which direction in the lattice you look.) i would have an easier time doing this by eye if you were to resolve the ambiguous positions one way or the other . . . anyway, 12288:12005 is certainly going to work for this purpose. i don't see why you're denying it above. if you're worried that just because its 7s exponent is -4, while you're only using 3 different levels along the 7 direction in your plot, it isn't going to work, worry no further. it will work just fine. if you prefer to keep the 7s exponent within the plus-or-minus 3 range, you can always add or subtract (the exponents of ) any of the other unison vectors. for example, if you add the exponents of 126:125 ([2 -3 1]) to those of 12288:12005 ([1 -1 -4]), you get [3 -4 -3] . . . now how about adding the exponents of 224:225 ([-2 -2 1]), resulting in [1 -6 -2] . . . all of these are valid choices for the third unison vector as well.

Message: 5585 - Contents - Hide Contents Date: Sun, 17 Nov 2002 01:06:28 Subject: Re: 43edo 7-limit periodicity-block From: monz> From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, November 17, 2002 12:19 AM > Subject: [tuning-math] Re: 43edo 7-limit periodicity-block > > > anyway, 12288:12005 is certainly going to work for this purpose. > i don't see why you're denying it above. if you're worried that just > because its 7s exponent is -4, while you're only using 3 different > levels along the 7 direction in your plot, it isn't going to work, > worry no further. it will work just fine. if you prefer to keep the 7s > exponent within the plus-or-minus 3 range, you can always add > or subtract (the exponents of ) any of the other unison vectors. for > example, if you add the exponents of 126:125 ([2 -3 1]) to those > of 12288:12005 ([1 -1 -4]), you get [3 -4 -3] . . . now how about > adding the exponents of 224:225 ([-2 -2 1]), resulting in [1 -6 -2] . > . . all of these are valid choices for the third unison vector as well.ah, yes -- of course! duh, my bad. i just didn't see it at first. got it. -monz

Message: 5586 - Contents - Hide Contents Date: Wed, 20 Nov 2002 11:25 +0 Subject: Re: A common notation for JI and ETs From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <5.1.1.6.1.20021120144249.01b20b38@xx.xxx.xx> David C Keenan wrote:> We might propose a standard format for that. I imagine something like > the tempo specification at the start of some scores that says "crotchet > = 120" or some such. e.g. "C:G = 700 c" or "~2:3 = 700 c" or "P5 = 700 > c". Additional words might say things like "7-limit JI" or "Miracle > temperament", or "22-ET", or "Blackjack tuning", but if the reader has > never heard of Blackjack at least they have the size of the fifth, and > can proceed to play it correctly."~2:3" wouldn't be appropriate. You're saying something that's written a certain way is heard as a certain interval. What you're writing is C:G, not ~2:3. For JI, you could say C:G = 2:3. For generality, you could specify the octave as well. Or any other interval that would be helpful. Graham

Message: 5587 - Contents - Hide Contents Date: Wed, 20 Nov 2002 14:43:27 Subject: Re: A common notation for JI and ETs From: David C Keenan At 07:32 AM 15/11/2002 -0800, you wrote:>From: George Secor, 11/15/2002 (#5015) >Subject: A common notation for JI and ETs > >I'd like to finalize the notation for the two remaining multiples of >12: > >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4662]:>> At 06:19 PM 17/09/2002 -0700, George Secor wrote:>>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>>>> At 10:24 AM 13/09/2002 -0700, George Secor wrote:>>>>> 132a: ~|( /| |) |\ (|~ ~||( /|| ||) ||\ (||~ /||\ >(MS)>>>>> 132b: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||~ /||\ >(MS) >>>>>>>> I prefer 132b, but why not |( as 5:7-comma for 1deg132? >>>>>> I try to choose symbols that are as valid in as many roles as >possible.>>> |( is valid only as the 5:7 comma and not as the 11:13 or 17'-17 >>> commas (1 out of 3), whereas ~|( needs to be valid only as the 17' >>> comma (1 out of 1). This is another one that I don't have strong >>> feelings about, and in the course of working on the spreadsheet I >might>>> change my mind. Even if we don't get any final agreement at this >point>>> about some of these less common divisions, at least our discussion >of>>> these will provide some examples from which I can arrive at general >>> principles for choosing symbols. >> >> OK >>Here I've taken the single-shaft symbols of 132b and used their >rational complements: > >132c: ~|( /| |) |\ (|~ /|\ /|| ||) ||\ (||( /||\ (RC) > >But I'm beginning to wonder if we should allow /|\ to exceed (|), which >would give us a more meaningful 5deg symbol: > >132d: ~|( /| |) |\ (|) /|\ /|| ||) ||\ (||( /||\ (RC) > >This might be justified on the same basis that we have allowed /| to >exceed |) and even |\ in a few instances. After all, we are already >used to seeing either sharps or flats higher in pitch in different >octave divisions.Yes. I approve of allowing (|) to be smaller than /|\ in the larger multiples of 12-ET. It's what I had earlier but only starting with 204-ET. However, I think I was using /|) in its place as the 5+7 comma for the smaller multiples, which I've now agreed we should only do if it's also the 13-comma. I can accept either 132c or 132d. You (or your spreadsheet) should decide. I'm mentally too distant from such details at present.>Now for 144. I would really like to have its notation in the XH >article, because it's been mentioned quite a bit on the tuning list. > >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4654]:>> At 10:24 AM 13/09/2002 -0700, George Secor wrote: >>>> 144: ~|( /| )|) |\ /|) /|\ (|\ /|| )||) ||\ /||) /||\ > >> Agreed. >>This is what we get if we use the above with rational complements: > >144b: ~|( /| )|) |\ /|) /|\ (|\ /|| )/|| ||\ (||( /||\ >(RC) > >I've now think that I wouldn't want to use )|) if I didn't have to -- >it's a more unusual symbol (and therefore less memorable) than the >23-comma: > >144c: ~|( /| |~ |\ /|) /|\ (|\ /|| ~||) ||\ (||( /||\ >(RC) > >In the past you have used prime limit as a measure of simplicity, but I >would justify using a 23-comma symbol on the basis of product >complexity.I used prime limit for much of our discussion, but with the introduction of 5:7 commas etc, I started using product complexity, perhaps inconsistently and without making a point about it. So I agree that product complexity is more meaningful.> This would also enable us to keep the notation for all the >multiples of 12 up to 144 without going beyond the 18 single-shaft >symbols that I am presenting in the article: > >)| |( ~| ~|( |~ )|~ /| |) |\ (| ~|) (|( //| /|) (|~ /|\ >(|) (|\ > >These symbols are sufficient to notate all 17-limit consonances and all >harmonics and subharmonics through 29, relative to the natural notes. >Also, their rational complements collectively have the same >combinations of flags as in the single-shaft set: > >(||~ /||) //|| (||( ~||) (|| ||\ ||) /|| )||~ ||~ ~||( ~|| >(|\ )|| > >So I think that these 18 symbols could be a useful set for the >moderately sophisticated user, just as the "starter set" of 7 symbols >that I have in Table 3 of my article would be for the simpler ETs >(including all multiples of 12 through 96): > >/| |) |\ /|) /|\ (|) (|\ and rational complements ||\ ||) /||Yes. That sounds very sensible to me. I agree with 144c. Just a few more thoughts before going "public". We must point out that the saggital notation on a score, by itself is not enough. The score must also have something to tell the reader what tuning it is in. In fact the minimum piece of information required is what size the notational fifths are (e.g. in cents). We might propose a standard format for that. I imagine something like the tempo specification at the start of some scores that says "crotchet = 120" or some such. e.g. "C:G = 700 c" or "~2:3 = 700 c" or "P5 = 700 c". Additional words might say things like "7-limit JI" or "Miracle temperament", or "22-ET", or "Blackjack tuning", but if the reader has never heard of Blackjack at least they have the size of the fifth, and can proceed to play it correctly. Something else that may need standardising is how one pronounces the saggital symbols when reading a score out loud. It seems to me that one should say "5-comma up", "11-diesis down" etc., but these are a bit of a mouthful (compared to e.g. "sharp" and "flat") and I can see them being a problem with composers who just want to use an ET without having to know anything about JI. Note that Sims-notation users say "twelfth up, sixth down, quarter up, etc., referring to that fraction of a 12-ET whole tone. We shouldn't make extreme claims about the universality of this notation. We can do a lot of JI and ETs (and linears by mapping to ETs), but how do we deal, for example, with non-octave tunings such as Bolen-Pierce or 88-cET or planar temperaments, or randomly chosen pitches. Can we give an algorithm, that Manuel might implement in Scala, to give a notation for say 90% of the tunings in the Scala archive, that will be accurate to within +-0.5 c? Assume it is allowed to consult a table of all our agreed ET notations. Just some thorts. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)

Message: 5588 - Contents - Hide Contents Date: Wed, 20 Nov 2002 23:47:16 Subject: Re: A common notation for JI and ETs From: monz hi Dave (and Graham)> From: "Dave Keenan" <d.keenan@xx.xxx.xx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, November 20, 2002 6:51 PM > Subject: [tuning-math] Re: A common notation for JI and ETs > > > --- In tuning-math@y..., graham@m... wrote: >>>> For generality, you could specify the octave as well. Or any other >> interval that would be helpful. >

Message: 5589 - Contents - Hide Contents Date: Thu, 21 Nov 2002 02:51:28 Subject: Re: A common notation for JI and ETs From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> "~2:3" wouldn't be appropriate. You're saying something that's written a > certain way is heard as a certain interval. What you're writing is C:G, > not ~2:3. For JI, you could say C:G = 2:3.Good point. And P5 is probably not appropriate in those extreme tunings where it would seem odd to refer to the notational fifth as "perfect". But should it be C:G? Some tunings will not contain the notes C or G natural. Why not D:A since D is the natural centre of a chain of fifths and A is the pitch standard.> For generality, you could specify the octave as well. Or any other > interval that would be helpful.Another excellent idea. A full spec might look like this. A = 440 Hz, A:A = 1:2, D:A = 2:3 or A = 440 Hz, A:A = 1200 c, D:A = 700 c or perhaps more usefully A = 440 Hz, A:A = 1:2, D:A = 2:3 - 2 c

Message: 5590 - Contents - Hide Contents Date: Fri, 22 Nov 2002 20:18:52 Subject: 5-limit comma names From: Gene Ward Smith Since the monzisma (450359962737049600/450283905890997363) is named and appears on Manuel's list, there is a prima facie case for naming at least the significant 5-limit commas of lesser height which don't yet have names. Here's a proposal: 16875/16384 "Negrisma" The comma of Negri's temperament. 78732/78125 "hemisixths comma" 1600000/1594323, "amitisma" Or "amt comma", but because of the complaints about amt I'm wondering if we could call it "amity" instead. 2109375/2097152 "Georgema" After George Orwell (sorry, George. :)) 4294967296/4271484375 "septathirds comma" 1224440064/1220703125 "parakleisma" 6115295232/6103515625 "semisuper comma" 19073486328125/19042491875328 "enneadecima" Manuel called this the "19-tone comma", but that sounds like it should be3^19/2^30. If the temperament is enneadecimal, this seems like a good name for the comma ((5/3)^19 2^(-14)) 274877906944/274658203125 "hemithirds comma", 50031545098999707/50000000000000000 "heptidecatonma" This comma is 6 (9/10)^17; I propose "minortone" for the corresponding temperament, with generator almost exactly 10/9. 9010162353515625/9007199254740992 "quasiseptima" I propose "quasiseptimal" for the corresponding temperament, since it's generator is an excellent approximation to 9/7.

Message: 5591 - Contents - Hide Contents Date: Fri, 22 Nov 2002 20:31:11 Subject: Re: 5-limit comma names From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> 19073486328125/19042491875328 "enneadecima" > > Manuel called this the "19-tone comma", but that sounds like it >should be3^19/2^30.from one point of view, the most notable feature of 19-tone are its pure minor thirds, so constructing a chain of 19 6:5s comes to mind rather readily.

Message: 5592 - Contents - Hide Contents Date: Fri, 22 Nov 2002 22:12:46 Subject: Re: 5-limit comma names From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: > >> 19073486328125/19042491875328 "enneadecima" >>>> Manuel called this the "19-tone comma", but that sounds like it >> should be3^19/2^30. >> from one point of view, the most notable feature of 19-tone are its > pure minor thirds, so constructing a chain of 19 6:5s comes to mind > rather readily.I was thinking of the context of his list, where we also have the 41-tone comma and Mercator's comma.

Message: 5593 - Contents - Hide Contents Date: Fri, 22 Nov 2002 22:20:11 Subject: Re: 5-limit comma names From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:>> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: >> >>> 19073486328125/19042491875328 "enneadecima" >>>>>> Manuel called this the "19-tone comma", but that sounds like it >>> should be3^19/2^30. >>>> from one point of view, the most notable feature of 19-tone are its >> pure minor thirds, so constructing a chain of 19 6:5s comes to mind >> rather readily. >> I was thinking of the context of his list, where we also have the > 41-tone commaisn't the most notable feature of 41-tone its pure perfect fifth?> and Mercator's comma.same for 53-tone, of course.

Message: 5594 - Contents - Hide Contents Date: Fri, 22 Nov 2002 22:48:11 Subject: Re: Adaptive JI notated on staff From: wallyesterpaulrus --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:>> right, but i'd like to see this actually notated, on a staff. >this notation . . . personally, it doesn't do much for me -- for example, looking at this 217-equal example, Yahoo groups: /tuning-math/files/Dave/Adaptive... * [with cont.] only a few of the pure thirds are immediately recognizable from the notation, unless you've memorized all the symbols and the order in which they occur in 217-equal. the symbol for a syntonic comma alteration will quickly be learned by any user of the system, but all the sets of symbols whose difference is a syntonic comma in a given tuning?

Message: 5595 - Contents - Hide Contents Date: Fri, 22 Nov 2002 23:20:15 Subject: Re: 5-limit comma names From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:> isn't the most notable feature of 41-tone its pure perfect fifth? >>> and Mercator's comma. >> same for 53-tone, of course.We could say the term "n-tone comma" is reserved for the case where m/n is a convergent to the log base two of a consonant interval. The comma you were asking about, (7/6)^9 / 4, would then be the "9-tone comma". This does not disambiguate the term completely, but it's probably good enough. Other commas would be: 11-tone comma (7/9)^11 * 16 26-tone comma (8/7)^26 / 32 28-tone comma (5/4)^28 * 2^(-9) 33-tone comma (7/5)^33 * 2^(-16) 35-tone comma (5/7)^35 * 2^17 59-tone comma (4/5)^59 * 2^19 68-tone comma (7/5)^68 * 2^(-33) 80-tone comma (9/7)^80 * 2^(-29) 171-tone comma (7/9)^171 * 2^62

Message: 5596 - Contents - Hide Contents Date: Fri, 22 Nov 2002 23:34:17 Subject: Re: 5-limit comma names From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >>> isn't the most notable feature of 41-tone its pure perfect fifth? >>>>> and Mercator's comma. >>>> same for 53-tone, of course. >> We could say the term "n-tone comma" is reserved for the case where > m/n is a convergent to the log base two of a consonant interval.is there no n for which this would be ambiguous?

Message: 5597 - Contents - Hide Contents Date: Sat, 23 Nov 2002 02:15:35 Subject: Re: Adaptive JI notated on staff From: monz hmmm ... the adaptive-JI example reminds me very much of another musical notation which is nearly-unique in the literature: that "Daseian" notation used in the _musica enchiriadis_ and _scolia enchiriadis_ treatises of c. 800 AD. (i was writing a paper about my speculations on the possible intonational meanings of that notation back around 1997, but never finished it.) -monz ----- Original Message ----- From: "Dave Keenan" <d.keenan@xx.xxx.xx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Friday, November 22, 2002 8:01 PM Subject: [tuning-math] Re: Adaptive JI notated on staff> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> > wrote:>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>>> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:>>>> right, but i'd like to see this actually notated, on a staff. >>>>> this notation . . . personally, it doesn't do much for me -- for >> example, looking at this 217-equal example, >> >> Yahoo groups: /tuning-math/files/Dave/Adaptive... * [with cont.] >> >> only a few of the pure thirds are immediately recognizable from the >> notation, unless you've memorized all the symbols and the order in >> which they occur in 217-equal. the symbol for a syntonic comma >> alteration will quickly be learned by any user of the system, but all >> the sets of symbols whose difference is a syntonic comma in a given >> tuning? >> Hi Paul. Thanks for your belated response. I totally agree with you re > the adaptive JI example. But surely you're not rejecting all possible > uses of the notation on the basis of that? > > I gave that example, not because I thought it was a particularly good > use of the notation, but in response to your request in message 3993: >>>> i think it would be cool if someone notated the adaptive-ji > version>>> of the chord progression >>> >>> Cmajor -> A minor -> D minor -> G major -> C major >>> >>> in 217-equal. then we could all look at it and see if we have any >>> major problems with it. >> Can you tell us what you expect of a notation for 217-ET? How might it > be done better so the pure thirds could all be immediately > recognisable? Surely any notation for something as large as 217-ET > will require a significant learning curve? > > Why not tell us instead how you feel about the way the notation would > work in your old favourite, 22-ET. It only needs one pair of new > symbols /| (for the 5-comma), and its semantics are the same as the > standard Scala one I've been promoting for ages, and I think it has > the same semantics as the one Alison Monteith uses. Or in 31-ET, where > there is also only one new pair of symbols /|\ which are > simultaneously the 7-comma and the 11-comma (a semi-sharp in this > case). Or in 72-ET where its semantics are identical to the Sims > notation. Only the symbols change. /| |) /|\ > > > To unsubscribe from this group, send an email to: > tuning-math-unsubscribe@xxxxxxxxxxx.xxx > > > > Your use of Yahoo! Groups is subject to Yahoo! Terms of Service * [with cont.] (Wayb.) > >

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