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Message: 4475 Date: Sun, 31 Mar 2002 10:53:57 Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?) From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: > > > i don't like this kind of constraint because it makes 11/7 seem > as > > > complex as 77/64. > > > > Remember that the purpose of this temperament is to make a notation > > with a minimum number of symbols (or sagittal flags) that can > notate > > rational scales so even Johnny Reinhard can't tell the difference, > and > > notate all ETs below 100-ET and many above it. > > even so. So what would you suggest?
Message: 4476 Date: Sun, 31 Mar 2002 11:06:13 Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > So what would you suggest? What was wrong with the plan of using 2^a 3^b p commas? I'm not at all clear why you want to abandon it.
Message: 4477 Date: Mon, 01 Apr 2002 05:25:02 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau One of the possibilities of George's new sagittal notation is that one can ignore the meaning of the individual flags and simply take it as a set of symbols for 2^a*3^b*p commas, one per prime p from 5 to 31, to be used in conjunction with # and b and to be simply placed side-by side. As such it presently looks something like this (with the symbols for the primes above 13 still being negotiated). Joseph, I'm sorry to have to point out that these symbols bear a much greater resemblance to the "European" symbols, than the Sims symbols. I've also proposed single-ASCII-character substitutes for some of them. [If you're reading this on the yahoogroups website you will need to choose Message Index, Expand Messages, to see the following symbols rendered correctly.] 5-comma 80;81 /| / | | / \ | | 7-comma 63;64 _ | \ | | | P L | | 11-comma 32;33 /|\ / | \ | ^ v | | 13-comma 1024;1053 _ /| \ / | | | | | 17-comma 2176;2187 | _/| | | | 19-comma 512;513 O | | * o | | 23'-comma 16384;16767 (unfortunately not 729;736) |\ _/| \ | | | 29-comma 256;261 _ / | | | | q d | | 31-comma 243;248 /| / |\_ | | | For the down versions of these, flip them vertically (don't rotate them 180 degrees). The smaller 23-comma _can_ be rendered as, the unfortunately complicated: 23-comma 729;736 O _/O | | | Notice that lateral confusability only occurs beyond the 23-limit, and this might be eliminated by adding blobs to the end of the curved strokes of the 29 and 31 commas like this. 29-comma 256;261 _ / | b | | | | 31-comma 243;248 /| / |\o | | | We also have optional symbols for larger 11, 13 and 17 commas. 11'-comma 704;729 _ _ / | \ | | | | ] [ | | 13'-comma 26;27 _ / |\ b | \ | | | Only if you start combining multiple symbols into a single symbol, do you begin to assume the vanishing of the following pair of sub-half-cent schismas: 4095:4096 (13-limit), 3519:3520 (23-limit using large 23-comma). If you use the symbols for the large 11-comma and large 13-comma as well as the small ones, you are also assuming the vanishing of the sub-half-cent schisma 5103:5104 (29-limit). This is all George Secor's work (apart from the liberties I've taken with the post 13-limit symbols), I'm just trying to explain one delightful aspect of it.
Message: 4478 Date: Mon, 01 Apr 2002 05:59:04 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- I wrote: > We also have optional symbols for larger 11, 13 and 17 commas. The astute reader will have noticed that I did not in fact give a symbol for a larger 17-comma (which would have been 4096:4131). That's because it can't be done without either adding another type of flag, disallowing the smaller 23-comma, or assuming the vanishing of a schisma of about 1.6 cents (1543503872:1544898987) which is too big for a JI notation. I don't really see a need for the larger 17 comma. It would merely allow one to notate, for example, a 16:17 above C as a slightly raised (pythagorean) Db instead of a slightly lowered (pythagorean) C#. If we disallowed the smaller 23-comma (729:736) the larger 17-comma could have a symbol like this. [Message Index, Expand messages] 17'-comma 4096:4131 | _/O | | |
Message: 4479 Date: Mon, 01 Apr 2002 07:42:04 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > Joseph, I'm sorry to have to point out that these symbols bear a much > greater resemblance to the "European" symbols, than the Sims symbols. > The one thing I've always found unjustifiable and now find irredeemable about the Sims notation is the use of arrows with full heads to indicate something smaller than the arrows with half heads. I could almost make a version of this notation that is compatible with the Sims notation, if it wasn't for the twelfth-tone arrows. Joseph, remind me what you don't like about slashes again, assuming the up slash has a short vertical stroke thru the middle of it and the down slash doesn't? George, remind me why the 5-flag is on the left and the 7-flag on the right, and why the 5-flag is straight and the 7-flag curved? Why couldn't either of these properties be switched between 5 and 7? Here's a more complete "single-ASCII-character substitutes" proposal. Symbol dn up Comma Abbrev. descr. of actual symbol ------------------------------------------------------------- \ / 5-comma 80;81 sL L P 7-comma 63;64 xR v ^ 11-comma 32;33 sL+sR [ ] 11'-comma 704;729 xL+xR { } 13-comma 1024;1053 sL+xR ; | 13'-comma 26;27 xL+sR j f 17-comma 2176;2187 vL * o 19-comma 512;513 cO w m 23-comma 729;736 vL+cO+cI W M 23'-comma 16384;16767 vL+sR q d 29-comma 256;261 xL y h 31-comma 243;248 sL+vR If it turns out we allow the large 17-comma instead of the small 23-comma then this could be: J F 17'-comma 4096;4131 vL+cI The abbreviated descriptions above refer to single-shaft arrows with various "flags" making up the arrow head: sL straight left sR straight right xL convex left xR convex right vL concave left vR concave right cO small filled circle, outer (on tip of arrow) cI small filled circle, inner (away from tip of arrow)
Message: 4480 Date: Mon, 01 Apr 2002 07:54:33 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau By the way, we can actually notate 311-ET with combinations of these flags, so that no note has more than one arrow next to it in addition to a sharp or flat. Not that this is of any particular importance. The values of the flags in steps of 311-ET are: sL 6 sR 8 xL 9 xR 7 vL 3 vR 3 cO 1 cI 1
Message: 4481 Date: Mon, 01 Apr 2002 00:14:25 Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?) From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > > So what would you suggest? > > What was wrong with the plan of using 2^a 3^b p commas? I'm not at all clear why you want to abandon it. We're not abandoning that plan at all. Sorry I didn't explain. These sub-half-cent thingies are better thought of, not as commas, but as schismas, where a schisma is defined (for our purposes here at least) as a small difference between commas (just as _the_ schisma is the difference between the syntonic and pythagorean commas). We want to reduce the number of symbols to less than the number of primes, if possible, so we're interested in sub-half-cent schismas which correspond to a very simple relationship between commas like x + y ~= z, where x,y and z are some of our 2^a*3^b*p commas. The most striking schisma proposed so far, found by George Secor, is 4095:4096 which says that the 13-comma is the 5-comma plus the 7-comma.
Message: 4484 Date: Mon, 01 Apr 2002 21:50:41 Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?) From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> > wrote: > > > The most striking schisma proposed so far, found by George > Secor, is > > 4095:4096 which says that the 13-comma is the 5-comma > plus the > > 7-comma. > > how do you get 13 from 5 and 7? 5-comma 80:81 21.51 c 7-comma 63:64 27.26 c 5-comma + 7-comma = 48.77 c 13-comma 1024:1053 48.35 c It doesn't matter for notating ETs less than 100 that the 4095:4096 doesn't always vanish, because one never needs to use all three notational commas together (5, 7 & 13).
Message: 4486 Date: Tue, 02 Apr 2002 13:18:38 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > Here's a more complete "single-ASCII-character substitutes" proposal. > > > > Symbol > > dn up Comma Abbrev. descr. of actual symbol > > ------------------------------------------------------------- > > \ / 5-comma 80;81 sL > > L P 7-comma 63;64 xR > > v ^ 11-comma 32;33 sL+sR > > [ ] 11'-comma 704;729 xL+xR > > { } 13-comma 1024;1053 sL+xR > > ; | 13'-comma 26;27 xL+sR > > j f 17-comma 2176;2187 vL > > * o 19-comma 512;513 cO > > w m 23-comma 729;736 vL+cO+cI > > W M 23'-comma 16384;16767 vL+sR > > q d 29-comma 256;261 xL > > y h 31-comma 243;248 sL+vR > > I don't know if anyone cares about 12-et-compatibility up to the > 31-limit, but in case they do here is what you get if you enforce compatibility with the "standard" h12 and h7 mappings: > > 2187/2048, 256/243, 81/80, 64/63, 729/704, 1053/1024, 4131/4096, > 513/512, 16767/16384, 261/256, 67797/65536 I'm not sure I know what you mean by "enforce compatibility with the "standard" h12 and h7 mappings", because I can't see why you would get 67797/65536 as the 31-comma when this contains 3^7 and 248/243 only contains 3^-5. This is the difference between a 1:31 from C being a Cb or a B respectively (which are of course the same thing in 12-tET). I'm also unclear how this method chooses 729/704 over 8192/8019 for the 11-comma, and 16767/16384 over 736/729 for the 23-comma. Since both these choices involve 3^6 versus 3^-6. I'm guessing these have something to do with h7 or 7-ET. Could you please explain your method in more detail? Perhaps instead of 7-ET it would make more sense to use the C major scale in 12-ET. It seems there is no argument over which commas to use for 3,5,7,19,29 - the same choices for these can be arrived at in any number of ways - but 11,13,17,23,31 are not yet settled. Even if our final system has symbols for more than one comma per prime, we will still want to specify a "preferred" comma for each prime. I believe we decided long ago that we were basing this notation on pythagorean (i.e a chain of Just fifths) not 12-ET, although we might favour 12-ET to the extent that all the odd numbers, up to an odd-limit determined by our highest prime, should be expressible without requiring enharmonics such as G# and Ab to be used simultaneously, and more strictly, without requiring anyone to go outside of a chain of fifths having 12 notes, e.g. Eb to G# or Ab to C#. And further, not to use any commas (at least as "preferred" commas) that fail to vanish in 12-ET, such as 26:27, 16:17, 18:19 or 23:24. 32:33 is only considered to pass this test because it vanishes in the case of 3:11 and 9:11 even though it doesn't vanish in the case of 1:11. This last requirement is more accurately expressed as requiring all preferred commas to be smaller than half an apotome (i.e. smaller than 56.84 cents). It might seem obvious that we should limit the 12 base notes to which the commas must be applied, and hence the range of exponents of 3 in these commas, to a range of either -6 to +5 fifths from the 1/1, or a range of -5 to +6. However 25 is within our odd-limit and presumably we want the "25-comma" to simply be two 5-commas, which means two syntonic commas (6400:6561), which means 3^8, which means our range of allowed-exponents-of-3 must be shifted at least as far as -3 to +8. The only alternatives to this are either to go outside a 12 note chain of fifths within our odd-limit, or to invoke a different comma for 25, requiring its own symbol, such as the diaschisma (2025:2048) which has 3^-4. A separate 25-comma symbol seems like a bad idea to me, unless it is obviously made up of two 5-comma symbols, in which case it should _be_ two 5-commas. Here's a table showing all the 2,3,prime commas (and 2,3,25 commas) that could possibly be of any interest in this notational endeavour. [On the website, use Message Index, Expand Messages to see the columns formatted correctly.] 3 Note exp Odd Comma Cents Odd Comma Cents -------------------------------------------------- -7 Cb 29 65536:63423 -56.74 31 65536:67797 58.72 -6 Gb 11 8192:8019 -36.95 23 16384:16767 40.00 -5 Db 17 4096:4131 14.73 -4 Ab 13 1024:1053 48.35 25 2048:2025 -19.55 -3 Eb 19 512:513 3.38 -2 Bb 7 64:63 -27.26 29 256:261 33.49 -1 F 11 32:33 53.27 21 0 C 1 1 G 3 2 D 9 3 A 27 13 27:26 -65.34 4 E 5 81:80 -21.51 5 B 31 243:248 35.26 15 6 F# 23 729:736 16.54 11 729:704 -60.41 7 C# 17 2187:2176 -8.73 8 G# 13 6561:6656 24.89 25 6561:6400 -43.01 Within a range of -3 to +8 fifths we have the following commas for the disputed primes, smaller than half an apotome. 11 32:33 13 6561:6656 (note: not 1024:1053) 17 2187:2176 23 729:736 31 243:248 I would be very sorry not to have 1024:1053 as the 13-comma because: (a) it has that neat sub-half-cent schisma with the 5 and 7 commas (4095:4096). (b) 8:13 makes more sense as a neutral sixth than a superaugmented fifth. There is a completely unrelated reason why we should go to at least a 3-exponent of 7, and that is so that we have 2187:2176 as the preferred 17-comma. We need a comma of about this size (8.7 cents) to help us in notating the larger ETs, because it fills in a huge gap between the 19-comma of 3.4 cents and the 5-comma of 21.5 cents. I expect the other 17-comma (4096:4131) at 14.7 cents would be nowhere near as useful. Within a range of -4 to +7 fifths we would have the same commas as -3 to +8 fifths, except for the 13-comma, which would become the beloved 1024:1053. But rather than introduce the disachisma as a 25-comma I'm inclined to allow the 3-exponents to range from -4 to +8 so that, from C, an 8:11 is a variety of Ab (or A, using the non-preferred 13-comma) and a 16:25 is a variety of G#. Too bad about the possibility of simultaneous enharmonics. How do others feel about this?
Message: 4487 Date: Tue, 02 Apr 2002 13:27:35 Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?) From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> > wrote: > > --- In tuning-math@y..., "emotionaljourney22" <paul@s...> > wrote: > > > how do you get 13 from 5 and 7? > > > > 5-comma 80:81 21.51 c > > 7-comma 63:64 27.26 c > > 5-comma + 7-comma = 48.77 c > > 13-comma 1024:1053 48.35 c > > > > It doesn't matter for notating ETs less than 100 that the > 4095:4096 > > doesn't always vanish, because one never needs to use all > three > > notational commas together (5, 7 & 13). > > i'm not following. I'm not sure what it is you're not following. Do you follow how to get 13 from 5 and 7 (within 0.5 cents)? Please ask a specific question, or put up a specific objection.
Message: 4488 Date: Tue, 02 Apr 2002 13:32:56 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote: > On Mon, 01 Apr 2002 07:42:04 -0000, "dkeenanuqnetau" <d.keenan@u...> > wrote: > > >Here's a more complete "single-ASCII-character substitutes" proposal. > > > >Symbol > >dn up Comma Abbrev. descr. of actual symbol > >------------------------------------------------------------- > >\ / 5-comma 80;81 sL > >L P 7-comma 63;64 xR > >v ^ 11-comma 32;33 sL+sR > >[ ] 11'-comma 704;729 xL+xR > >{ } 13-comma 1024;1053 sL+xR > >; | 13'-comma 26;27 xL+sR > >j f 17-comma 2176;2187 vL > >* o 19-comma 512;513 cO > >w m 23-comma 729;736 vL+cO+cI > >W M 23'-comma 16384;16767 vL+sR > >q d 29-comma 256;261 xL > >y h 31-comma 243;248 sL+vR > > On Windows systems, the | character appears as an unbroken vertical line. > It might make more sense to use : for the 13'-comma up (Windows does have a > broken vertical line character, but it's not ASCII). Thanks Herman, I had forgotten that the vertical bar is in two pieces in some fonts (where?). I proposed that pair of characters ; | for compatibility with Scala's JI notation.
Message: 4489 Date: Tue, 02 Apr 2002 14:09:23 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > But rather than introduce the disachisma as a 25-comma I'm > inclined to allow the 3-exponents to range from -4 to +8 so that, from > C, an 8:11 is a variety of Ab (or A, using the non-preferred 13-comma) > and a 16:25 is a variety of G#. That should have been "... an 8:13 is a variety of Ab ...".
Message: 4490 Date: Tue, 2 Apr 2002 18:03:05 Subject: Re: A common notation for JI and ETs From: manuel.op.de.coul@xxxxxxxxxxx.xxx Dave wrote: >v ^ 11-comma 32;33 sL+sR >[ ] 11'-comma 704;729 xL+xR Ai, I had recently removed v ^ from the JI notation systems to make them exclusively denote the diaschisma, and replaced them with [ ] for the undecimal comma 33/32. They are also used in the new E217 system which is probably the highest ET system I'm going to support. 33/32 is smaller anyway so I feel it's more logical to have the symbols exchanged. Manuel
Message: 4492 Date: Tue, 02 Apr 2002 21:40:34 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., manuel.op.de.coul@e... wrote: > Dave wrote: > >v ^ 11-comma 32;33 sL+sR > >[ ] 11'-comma 704;729 xL+xR > > Ai, I had recently removed v ^ from the JI > notation systems to make them exclusively > denote the diaschisma, and replaced them with > [ ] for the undecimal comma 33/32. I'm sorry. I wasn't aware of this. Maybe it's a good idea, but I would have used u and n for the diaschisma since Rappoport's symbols are convex arrow-heads (with no shaft). > They are also > used in the new E217 system which is probably > the highest ET system I'm going to support. How many steps of E217 are they? I assume by "they" you mean [ ]. > 33/32 is smaller anyway so I feel it's more > logical to have the symbols exchanged. I don't follow this. v ^ seem like smaller symbols to me than [ ]. When considering the ASCII symbols alone, and other notation systems, I wouldn't have a problem with using [ ] for 32:33 and v ^ for 704:729. It would be nice to have the preferred (smaller) 11 and 13 commas (dieses) be [] and {}. The trouble is that George's symbol for 32:33 is an arrow with straight head-flags (an ordinary arrow) and the symbol for 704:729 has convex head-flags (like Rappoport's diaschisma symbols with a vertical shaft added). Anyway, things are still in flux, so lets not worry about it too much just yet.
Message: 4493 Date: Tue, 02 Apr 2002 21:54:17 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > That should have been "... an 8:13 is a variety of Ab ...". > > i'm much more comfortable with 8:13 as a variety of C:A. the dominant > 13th chord in 12-equal definitely gains a lot of tonalness because > 13:11 and even 13:7 are represented by their best approximations in > 12-equal. and simpler chords in 12-equal simply fail to evoke the 13- > limit, under any circumstances. Paul, Don't panic. I took that on board some time ago, thanks to you. The complete corrected sentence reads. "But rather than introduce the disachisma as a 25-comma I'm inclined to allow the 3-exponents to range from -4 to +8 so that, from C, an 8:13 is a variety of Ab (or A, using the non-preferred 13-comma) and a 16:25 is a variety of G#." Which is one reason we have both 26:27 ; |, and 1024:1053 { } as 13-commas. Are you saying we shouldn't have 1024:1053 at all? The notation is pythagorean-based not 12-ET based and so to avoid multiple accidentals one might wish to notate an 8:13 from A as A:F} rather than A:F#;
Message: 4495 Date: Tue, 02 Apr 2002 06:27:51 Subject: Re: A common notation for JI and ETs From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > Here's a more complete "single-ASCII-character substitutes" proposal. > > Symbol > dn up Comma Abbrev. descr. of actual symbol > ------------------------------------------------------------- > \ / 5-comma 80;81 sL > L P 7-comma 63;64 xR > v ^ 11-comma 32;33 sL+sR > [ ] 11'-comma 704;729 xL+xR > { } 13-comma 1024;1053 sL+xR > ; | 13'-comma 26;27 xL+sR > j f 17-comma 2176;2187 vL > * o 19-comma 512;513 cO > w m 23-comma 729;736 vL+cO+cI > W M 23'-comma 16384;16767 vL+sR > q d 29-comma 256;261 xL > y h 31-comma 243;248 sL+vR I don't know if anyone cares about 12-et-compatibility up to the 31-limit, but in case they do here is what you get if you enforce compatibility with the "standard" h12 and h7 mappings: 2187/2048, 256/243, 81/80, 64/63, 729/704, 1053/1024, 4131/4096, 513/512, 16767/16384, 261/256, 67797/65536
Message: 4496 Date: Wed, 03 Apr 2002 08:22:51 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote: > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: > > > Which is one reason we have both 26:27 ; |, and 1024:1053 { } as > > 13-commas. Are you saying we shouldn't have 1024:1053 at all? > > so the rule is that every comma and its 2187:2048 complement has a > unique symbol? No, not every comma. So far George and I have only agreed on the desirability of apotome complements of those commas which are close to the half-apotome, say those between 1/3 and 2/3 apotome. So far that's only the 11 and 13 commas (dieses). > if so, then the symbols should reflect that in a > natural way . . . Read George Secor's two most recent posts to this forum (and mine with the ASCII graphics of his symbols) and let us know whether you find the system sufficiently natural. The two 17-commas that have been mentioned are pythagorean comma complements, and of the two 23-commas mentioned, one is a pythagorean comma larger than the other; similarly the two 31-commas mentioned ; although I'm still waiting to hear from Gene (or anyone) why anyone would want 65536:67797 (2^16:3^7*31) 59.7 c as a 31-comma when we have 243:248 (3^5:2^3*31) 35.3 c.
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