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Message: 4475 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 atall 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 - Contents - Hide Contents 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: 4485 - Contents - Hide Contents Date: Mon, 01 Apr 2002 20:50:03 Subject: Re: A common notation for JI and ETs From: Herman Miller On Mon, 01 Apr 2002 07:42:04 -0000, "dkeenanuqnetau" <d.keenan@xx.xxx.xx> 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+vROn 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).
Message: 4486 - Contents - Hide Contents 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 enforcecompatibility 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, 67797/65536I'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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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+xRAi, 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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 - Contents - Hide Contents 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+vRI 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 - Contents - Hide Contents 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.
Message: 4498 - Contents - Hide Contents Date: Wed, 03 Apr 2002 19:16:47 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> Considering that the semantics of the notation have already put us >> past the 19 limit, that 217 is not 23-limit consistent, and that 311 >> is such an excellent division, I'd say let's go for it! > >> I guess I just have a knack for finding useful commas (even before I >> start looking for them). Are you ready for the next one? It's a >> honey: 20735:20736 (5*11*13*29:2^8*3^4, ~0.083 cents). And it turns >> out that we don't need any new flags to get the 29 factor: its >> defining interval is 256:261 (2^8:3^2*29, ~33.487 cents), and the >> convex left flag that we already have (715:729) is ~33.571 cents. >> 256:261 is also the 29-comma I had settled on. > > That's awesome! But don't forget that the convex left flag also has > the meaning 45056:45927 (2^12*11:3^8*7) from its use in combination > with the convex right flag to give the large 11-comma 704:729 (what > I've taken to abbreviating as the 11'-comma. Fortunately this still > differs by less than 0.5 c (5103:5104) from the 29-comma.With the 29 factor we have passed the point (in both 217 and 311) where all of the ratios within the harmonic limit are a unique number of degrees, so some bridging is inevitable. However, what's most important is that the number of degrees for the 29 comma is consistent with this flag in both 217 (6 deg) and 311 (9 deg).> I personally think primes beyond 11 are of very limited use musically, > but I know there are people who claim to have sucessfully used up to > 31.Perhaps you would change your mind if you tried a few isoharmonic chords (i.e., chords having a common difference tone between consecutive tones) containing higher primes, such as 16:19:22:25 or 17:21:25:29. They are consonant in the sense that, if you mistune one of the tones in the chord, you will easily hear it go out of tune due to the perception of difference tones beating against one another. (The JI heptatonic scales that I refer to below were constructed with this principle in mind.)> I think we can take the 11-limit (at least the semantics) as set in > stone now. > > sL 80:81 21.51 c > sR 54:55 31.77 c > xL 45056:45927 33.15 c > xR 63:64 27.26 c > > And the 13-limit is set in stone in so far as it uses no new flags but > gives the existing ones additional meanings. > > sL 65536:66339 21.08 c 6/311-ET > sR 22113:22528 32.19 c 8/311-ET > xL 715:729 33.57 c 9/311-ET > xR 64:65 26.84 c 7/311-ETI prefer to consider the xL flag as 715:729 with the additional meaning of 45056:45927. Not only is this a much simpler ratio, but it also gives an exact 26:27 for the 13 diesis. This makes the defining commas yield exact dieses for both 11 and 13 (rather than two different 11 dieses). The 4095:4096 schisma then defines alternate dieses for both 11 and 13, which better maintains the one- comma-per-prime (or in this case, one-flag-per-prime) concept. I realize that 26:27 is more than half an apotome, but it is not that much more, still falling within the neighborhood of half an apotome; in any case, its symbol is a combination of two comma-flags (rather than a single flag), and we have already specified that it will never be combined with a sharp or flat that alters in the opposite direction. Or am I just splitting hairs, inasmuch as conflating 4095:4096 would allow us to look at it either way (or both ways at once)?> But I think we're still free to fiddle around with 17, 19, 23, 29, 31 > with the proviso that we introduce no more than one new flag as we > introduce each new prime in order.Maybe. I have some comments about this below.> One thing that annoys me is that the 23-comma that works so well re > no-new-flags (16384:16767) is not necessarily the most useful one. I > prefer 729:736 since it spans the same number of fifths (-6 instead of > +6) and is smaller by a pythagorean comma.Upon reading this, I decided to go through some of my papers to find all of the JI heptatonic scales in which I used 23, and I found that I had three that (relative to C as 1/1) require an F-sharp and only one that requires a G-flat, which does indeed make 729:736 more useful. These are all *diatonic* usages, in which one has no liberty to "misspell" a note by using its equivalent sharp or flat (as might be allowable in a *chromatic* usage). But if there is even a single case that uses the flat (which there is), then we need 16384:16767 as well as 729:736. Whatever the case, what is significant here is that neither comma introduces a new flag unless we disallow 4096:4131 as a defining comma for 17 (on the basis of one comma per prime) and replace the definition of 3deg217 and/or 4deg311 with the 23-comma, 729:736. We then have the alternate 23 comma (16384:16767) as a freebie. In the process of counting the number of sharp-vs.-flat occurrences for 23, I also kept count for some other high odd-number ratios and came to some conclusions which I will give below.> The 31-comma I favour is 243:248 (3^5:2^3*31) 35.26 c.I agree. I also found what should have been a useful schisma for this (59024:59049, or 2^4*7*17*31:3^10, ~0.733 cents), but its usage is inconsistent in both 217 and 311, since 243:248 is not the same number of degrees as 238:243 (vL+xR) in either division. (So with 31 my luck with schismas has run out.) However, with 31 we have reached the point where the defining comma does not introduce an interval having a new number of degrees in either 217 or 311; in other words, the 31 comma is totally unnecessary for the notation. This should come as no surprise, since with 31 we have passed the point where the ratios can be represented uniquely (as both 29:30 and 30:31 are 15deg311). In fact, it happened sooner than this, starting with 24:25 and 25:26 (which are both 18deg311). So, while we can consistently notate 31 in *either* 217 *or* 311, it must be done with the realization that ratios of 31 are not unique and that some intervals may occasionally bridge to other primes. (This makes me begin to wonder why we would want to use 311 instead of 217; more about this below.)> We have a bunch of commas between 20 and 35 cents which can correspond > to a single flag. It really seems to me that the 17-comma (8.73 c) > should be represented by something noticeably smaller. Using a concave > flag goes some of the way, but maybe not small enough. And certainly > the 19-comma (3.38 c) should be represented by something fairly > insignificant in size, being 1/7th to 1/10th the size of the others > and 1/3rd the size of the 17-comma. > > What if we make the 19-comma just a blob on the end of the shaft. > Neither right nor left but able to be combined with any flags. Then > maybe we can get from 19 to 31 with only the two concave flags.I tried something of this sort for both the 17 and 19 commas prior to presenting the 17 and 23-limit approaches in message #3793, but I didn't come up with any way of symbolizing them that was simple enough to satisfy me, so I dropped it. Among the things I tried was a small filled-in triangle on the end of the shaft, but it just didn't seem to stand out well enough to work. I also tried both open and filled-in triangles (which could also point in either direction) on the other end of the shaft, but this required a longer shaft and started to get a bit cumbersome (so that it might be unclear which note of a chord is being altered if the symbol were too large vertically). Perhaps you have a better idea?> What if we leave concave-left as 2176:2187 (the 17-comma) but make > concave-right 19683:19840, so that we have: > > vL 2176:2187 8.73 c 3/311-ET > vR 19683:19840 13.75 c 3/311-ET > > sL+vR 243:248 35.26 c 9/311-ETThat's interesting, but desiring a complete range of intervals in 311, I would prefer to see an interval of 2deg311, which in combination with 2176:2187, would also supply one of 5deg311.> By the way, I think that two straight left flags, one above the other > on the same shaft, is the best thing for two 5-commas.I was thinking about that also. The interval of 12deg311 that it supplies is one that we need.> And do we > really need the 17'-comma, 4096:4131 (14.73 c)?I guess we could consider the 17-as-flat interval as approximated by the sum of the 17 and 19 commas, as long as the number of system degrees matches (as it does in both 217 and 311) and as long as we could still notate that number of degrees as their combination. This would also lead to the question, do we really need the 23-comma (729:736) of that same number of degrees?> The above doesn't give us all the steps of 311-ET from 1 to 17, but I > don't think that matters. We don't need to actually be able to notate > 311-ET. The gaps are 2, 5 and 15 steps (and 12 if you don't accept my > suggestion for two 5-commas).I think that the gaps are unacceptable for a couple of reasons. As I mentioned above, in the process of counting sharp-vs.-flat occurrences for 23 in my JI heptatonic scales, I also counted the number of sharp-vs.-flat occurrences for 17 (3#, 3b), 19 (4#, 5b), 25 (5#, 3b), and 29 (no#, 4b). I conclude that there has to be a provision for spelling any interval in at least two different ways, which is a compelling reason for providing a complete set of symbols for whatever division we settle on for the JI notation. In addition, being able to notate all of the degrees would ensure that no matter how much modulation is done in JI, at least one would never run out of symbols. If this were not done, then we would lose one of the principal advantages of mapping the JI notation to a specific division. So I think that we are either going to have to fill in the gaps in 311 or go with 217. To further complicate things in 311, I also noticed in my notes that, while I favored using 19 as an E-flat (where 1/1 is C), there were almost as many instances where a heptatonic scale called for a D- sharp (using the 19-comma 19456:19683, 3^9:2^14*19, ~20.082 cents). This usage turns out to be *inconsistent* in 311 (but consistent in 217), a problem that I didn't expect to find. While I compared the inconsistency of a few ratios of 23 in 217 with driving a car slightly onto the shoulder, the 311 problem (affecting all ratios of 19) is more like attempting to drive in the less-traveled direction on route 19 and finding yourself in the wrong lane moving against traffic. So I am beginning to have serious doubts about going to 311. Taking another look at 217, I see that it does uniquely represent all of the 19-limit consonances and, except for the slight inconsistency previously noted with the 23 factor, is otherwise consistent through the 31 limit. We have been able to define usable commas all the way through the 29 limit (past the point where we even need them), and we can even notate ratios of 31 consistently (if not uniquely) with intervals of the appropriate numbers of degrees. And the next two odd numbers are not primes, so we are actually getting a 35-odd-limit capability with 217. So if 311 proves to be a bit unwieldly, at least I believe we have a workable solution in 217. (By the way, I noticed that Manuel in message #3930 indicated that 217-ET is probably the highest system that he is going to support.) Something else in favor of 217: I noted earlier that it is 7 times 31. If you make instruments for 31-ET (or train string players to play 31-ET), then your JI can be reckoned in alterations of +/- 1 to 3 increments of 217-ET (i.e., multiples of ~5.5 cents), to a maximum of 16.6 cents, which is not an unreasonable amount of intonation adjustment for instruments of flexible pitch. This would be a decent practical alternative to alterations reckoned relative to 12-ET, whether for (too-coarse) 72-ET or Johnny Reinhard's (too-small) one- cent increments. Plus you get 31-ET in the bargain. And I should mention that a series of 41 fifths in 217 (kept within an octave) brings you only one degree away from your starting point, giving you a very close approximation of 41-ET. So the question now becomes: Are we left with any good reason for basing the JI notation on 311 instead of 217?
Message: 4499 - Contents - Hide Contents Date: Wed, 03 Apr 2002 19:22:36 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- 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?One of the main features of the sagittal notation is the incorporation of Bosanquet's slanted comma lines with the appropriate slope indicating an up or down (Didymus) comma. While the lines by themselves do not provide the easiest way to distinguish up from down, when one of these is used in combination with at least one vertical line (as it is in the sagittal notation), an arrow is formed, which clearly points in the appropriate direction, thus allowing us to be doubly sure of its meaning. This symbol is basic to the notation, being used for this single purpose in the overwhelming majority of ET's, as well as in the JI notation. A straight flag is preferred over a curved 5-flag for two reasons: 1) A curved line does not have a constant slope and therefore tends to obscure the direction of pitch alteration that it is supposed to be signifying. 2) Straight flags are simpler to make than curved flags, so they should be used more frequently. This is consistent with my guidelines (in message #3817) to keep things simple by: << [minimizing] the use of curved flags in the ET notations, introducing them only as it is necessary or helpful: 1) to avoid lateral confusibility (in 72-ET); 2) to distinguish 32:33 from 1024:1053 (in 46 and 53-ET, *without* lateral confusibility!); and 3) to notate increments smaller than 80:81 (in 94-ET). >> Finally, the 5-comma flag must be on the *left side* in order to have the desired slope. Note that the Sims notation uses these symbols (which are 1 degree in the 72-ET sagittal notation) for 2 degrees of 72. Ezra Sims' failure to take into account any prior use of certain notational features when he devised his 72-tone notation is most unfortunate. However, I feel no obligation to corrupt the sagittal notation in order to make it Sims-compatible, because I believe that he is the odd-man-out in this game. It would be far more reasonable to ask the users of the Sims notation to change the symbols for 2deg72 by reversing them left- to-right, which should cause them no disorientation, but I doubt that anything like that is going to happen.
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