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Message: 6455 Date: Wed, 12 Feb 2003 23:54:03 Subject: Re: vanishing diatonic semitone From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote: > i'd like to add one more row in this table, before the first row: > > Yahoo groups: /tuning/database? * > method=reportRows&tbl=10&sortBy=4&sortDir=down&start_at=0&query= > > this row would have 16:15 vanishing, and connect the family of ETs 5, > 8, 3. > > who can supply the necessary information? Aw, c'mon Paul. This isn't a 5-limit temperament, except perhaps in a musically-irrelevant purely-mathematical sense. This is the thing where the generator has to act as both the fourth and the major third (or the fifth and the minor sixth) and of course succeeds in doing neither. Next you'll be wanting the one where 9:10 vanishes. ;-) It's been a stretch for me to accept neutral thirds and pelogic as 5-limit temperaments. I think I have to draw the line at errors greater than 35 cents. So I have a similar objection to the one where 25:27 vanishes. But you should find what you want in http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Ok *
Message: 6456 Date: Wed, 12 Feb 2003 17:20:10 Subject: 5LimitTemp.xls From: Carl Lumma Dave, The degeneracy column seems broken. I've got Excel 2000. -Carl
Message: 6457 Date: Wed, 12 Feb 2003 17:28:39 Subject: Re: vanishing diatonic semitone From: Gene W Smith On Thu, 13 Feb 2003 01:27:39 -0000 "wallyesterpaulrus <wallyesterpaulrus@xxxxx.xxx>" <wallyesterpaulrus@xxxxx.xxx> writes: > the timbres that people like sethares talk about, even if they don't > always say so, start as harmonic and then each harmonic (up to 6, 8, > 12, whatever) is "tweaked" toward the nearest et (or whatever) > position. therefore, it's an approximation of an approximation of 5- > limit JI :) Csound lets you play with these, but I was disappointed to find that unless the inharmonic partials are close to harmonic, I find the timbres get on my nerves.
Message: 6458 Date: Wed, 12 Feb 2003 20:32:19 Subject: Re: 5LimitTemp.xls From: Carl Lumma >> The degeneracy column seems broken. I've got Excel 2000. >> >> -Carl > >I've only got Excel 97. Do you have the Analysis Toolpack (or >whatever) installed so the GCD function works? Look it up in Excel >Help. Ah, now it works. That is, if only rows 17-19, 27-29, 32-34, 44-46 are supposed to be degenerate, and the rest blank. -Carl
Message: 6459 Date: Wed, 12 Feb 2003 23:44:28 Subject: Re: poking monz (was: Re: naming temperaments( From: monz hi paul, > From: <wallyesterpaulrus@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, February 12, 2003 2:40 PM > Subject: [tuning-math] poking monz (was: Re: naming temperaments( > > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: > > > i'll try to get right on it. > > while we're at it, here's a more complete "small 5-limit intervals" > chart to replace the one on your equal temperament page: > > Yahoo groups: /tuning-math/files/Paul/small.gif * done. > due to the triangular/hexagonal geometry, it has the magical property > that each vector points in exactly the same direction -- that is, has > exactly the same slope -- as the green line for the corresponding > temperament in the graphs above. check it out! > > moreover, this chart could be used in conjunction with a set of > hexagonal bingo cards . . . as you know, i've made quite a few > already, and can make more in about 2 seconds apiece . . . so that > one can see exactly *how* a given small interval vanishes, or fails > to, in a given equal temperament. > > then this chart would have a dual function . . . > > let me know, > paul i'd like to include hexagonal graphs for *all* the EDOs on my "equal temperament" page *and* on the "bingo lattice" page. it's just a matter of me finding the opportunity to do it. keep sending me stuff ... i'll incorporate it as i have the chance. -monz
Message: 6460 Date: Wed, 12 Feb 2003 10:35:09 Subject: Re: notational specificity cont'd From: David C Keenan Aaron wrote: >Hi Dave. Thanks for your patience. Could one say that "comma inflected" >is a fairly accurate way to describe the notation? Yes! But of course it only describes one aspect of the notation. > I've seen the uploaded samples of the notation, and seen posts > referring to assignments of letter names and staff placements of > notes... I wonder: how many notes of various inflections - or, said > another way, how many discrete pitches - may conceivably occupy the same > position on a five line staff? We haven't counted them yet, and the more obscure ones using schisma accents are still in flux. But assuming the single or double symbol versions of the notation (considering any schisma accent marks to be part of a single symbol) we can go from double-flat to double-sharp in steps which are not more than 2 cents wide relative to just fifths, therefore we can do _at_least_ 233 discrete pitches, but I believe the actual figure is more like 400. However most of what anyone will ever want to do with the notation can be done with only 12 symbols (and their inversions) in conjunction with existing sharp and flat symbols. > Is it simply a matter of symbol combinatorics? In the single-symbol version (using the multi-shaft and X-shaft arrows) there are no combinations of symbols required. In the double-symbol version one uses only the single-shaft arrows in combination with conventional sharps and flats and their doubles. So the answer to your question is "No" for these versions of the notation. It's simply a matter of the number of discrete symbols (including any which appear as an arrow with a schisma accent mark). The symbols themselves have been derived as combinations of 8 flags or half arrowheads (at most two at a time), 2 accent marks (at most one used), 4 shafts (exactly one used) and 2 directions (up or down). But not all combinations are necessary or valid. There is also the possibility of using multiple symbols against a single note in a one-symbol-per-prime manner. This is what I've called the multi-symbol version of the notation. We would discourage this as being _much_ harder to read. This could involve up to 10 symbols to determine a single pitch! And would give 98,415 (= 5*3^9) discrete pitches on a single staff line! About 98,000 of which would be utterly indistinguishable from their neighbours by even the most expert listener. > How are positional boundaries determined? Again, I'm not sure what you mean, but I'll assume you mean how does one determine which staff line or space to place a note on. This is really not much different from the situation _without_ comma inflection, and is often referred to as the issue of "correct spelling". This is usually based on the structure of the scale or the structure of any chords the note might be part of. But in the absence of such context the default solution would be to use the position that requires the least pitch deviation from the natural note (in which case double-flats or sharps would not be used). -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 6462 Date: Wed, 12 Feb 2003 15:28:04 Subject: Re: A common notation for JI and ETs From: David C Keenan >--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" ><gdsecor@y...> wrote: >--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" ><d.keenan@u...> wrote: >I think that the term "comma" has been used in a broad sense to >denote smaller intervals (which we now call kleisma and schisma) more >often than larger ones, Possibly more often. But I expect it _has_ been used to cover larger ones often enough. > inasmuch as the term "diesis" has been used >for the latter since at least the 14th century. So I would be >inclined not to use the term "comma" for anything above ~37 cents, >even in a broader sense. What term do you suggest we use for all these intervals typically less than a scale step, from schisminas to small semitones? Here's what my Shorter Oxford (1959) has to say: Comma ... 3. Mus. A minute interval or difference of pitch 1597. ... Diesis ... 1. Mus. a. In ancient Gr. music, the pythagorean semitone (ratio 243:256). b. Now, the interval equal to the difference between three major thirds and an octave (ratio 125:128); usually called enharmonic diesis. ... > > > Unfortunately, the > > > particular dieses that we're using the o and m characters for are > > > both in the para category. > > > > I wouldn't place too much importance on this. But I note that in the > > three categories we have these symbols. > > > > small dq /|~ (|( ~|\ //| |~) > > middle unv /|) (|~ /|\ |)) (/| > > large owm |\) (|) (|\ > > > > But I find there is not much hope of making our prefixes match up >with > > any of these, except possibly in the large category. > > > > > Perhaps we could use meta for the largest > > > group (the meaning, "beyond," would still apply) and find a >couple of > > > other prefixes that wouldn't conflict with (and might even tie in > > > with) the letters q and n for the small and middle ranges. > > > > Good luck! > > > > There aren't very many prefixes starting with q. The only one that >is > > even slightly appropriate is "quasi-" but that means "almost but not > > quite" and would be better used for those things that have > > historically been called dieses but are smaller than 36.93 cents. > >I wasn't expecting to find anything appropriate for q, anyway. I'm >just trying to avoid names that might cause confusion. > > > "meso-" is _the_ Greek prefix meaning middle. > >Yes, I thought of that one, but would rather not use it, since it >begins with m. Given that meso- is such an obvious greek prefix for the job and it starts with the same letter as the English words middle, medium and mean, I don't feel we should avoid using it merely because the limtations of ASCII (which may not be relevant in a few years time) and the absence of a proper font, cause us to use the letter m to represent, in email, something which is not in the middle category. Someone might come up with a reason tomorrow that would cause us to change our single-character ASCII assignments. ASCII will never appear on the staff. And I should hope that the single-character ASCII approximations would never be used in teaching or explaining the notation. > > It is used with various > > other Greek pairs such as: > > > > hypo- under > > meso- middle > > hyper- over > > > > endo- inside > > meso- middle > > ecto- outside > > > > proto- (or pro-) earlier or to the front > > meso- middle > > meta- later or to the rear > > > > lepto- fine small thin delicate > > meso- middle > > hadro- thick stout > >I also found intra- (within or inside), neo- (new), and peri- (close >at hand, near, adjacent). In evaluating all of these, I tried to >identify what I would call the prototypical diesis in each group: Shouldn't you instead be looking at the primary interpretation of the most commonly ocurring sagittal symbol in each group? >37-45 cents -- 125:128, the meantone diesis, is not only in the group >with the *smallest size*, but is also the diesis by which three 4:5s >*fall short* of (i.e., on the near side of) an octave. So I thought >that peri- or intra- might be appropriate. Of these two I prefer >peri-. But proto- is also good, for a couple of reasons: it is >similar in meaning to peri-, and it is the opposite of meta- (should >we use that term for the large group). Besides, 125:128, which is >probably the best-known of any diesis in any group (and thus, on >account of its prominence, the one with the strongest claim to the >label proto-diesis), would validate an additional shade of meaning by >which the term could be applied to this group. But the minor diesis 125:128 is rarely used in the sagittal notation, having symbol .//|. By far the most common in this range will be the 25-<small>diesis //|. I can't find anywhere this has been previously named, presumably because it is simply a double syntonic comma. So, considered as a "comma" in its own right it is almost as "neo-" as the 11 and 13 commas below. And there are other commas in this group which are probably newer. >45-57 cents -- 32:33, the unidecimal diesis (or quartertone), >introduces some of the *strangest new* harmonies encountered in >alternative tunings. I thought neo- might be more descriptive of an >interval such as this, rather than some nondescript label (such as >meso-) that suggests that it might be average or middlin'. But it _is_ average as far as size goes, and that's what these prefixes are supposed to be about. > Even the >13 diesis (1024:1053, the second most prominent member of the group, >and the one that's actually symbolized by an "n") is new and strange. But their complements in the large-diesis group are just as new and strange. And anyway, how long does something remain "new"? Also, I should think that if 125:128 is prototypical of the small group then 243:250 would be that for the medium group. But again this is not a common comma to want to notate. It might be notated as /|) or (|~ . The 11-<medium>diesis /|\ will certainly be the most common in this group. >57-69 cents -- 625:648, besides being a *large* diesis (~27:28, or >1deg19) is also the amount by which four 5:6s *exceed* (i.e., go >beyond) an octave. I agree that the prototypical diesis in the large group is the major diesis 625:648, again not something we'd commonly use since it is '(|) . Clearly the 11-<large>diesis (|) will be the most common here. >I believe that we agree that meta- is a good >prefix for this group. Well, no. Only that it applies to this group better than it does to the medium group. The use of "meta-" to mean "beyond" is a recent departure from the Greek usage. As the Shorter Oxford puts it: "In supposed analogy to 'Metaphysics' (misaprehended as meaning 'the science of that which transcends the physical'), meta- has been prefixed to the name of a science, to form a designation of a higher science of the same nature but dealing with ulterior problems." But why not use prefixes that are a valid description of _all_ the commas in the group, rather than just ones that may be typical in any sense? i.e. ones that relate to size. > Need I say more? I'm afraid so. :-) My main objection is that neo- tells one nothing about the size. And if one adopts the modern sense of meta- one might take a meta-diesis to be a difference between dieses, in the same way that a diesis is a difference between other intervals. For example, we might well have used the term meta-comma to describe the differences between commas that we instead called schismas and now schisminas. If one takes the biological meanings of proto- front and meta- rear (of organisms) it is unclear that there is any correspondence with small and large. If one takes the temporal meaning of proto- before or early or primitive and meta- after or late or advanced, then it is only slightly more clear. In regard to having the right _meaning_, the best Greek set I can find are hypo- meso- hyper- If we were to depart from the Greek minor neutral major would be obvious enough, and so would small medium (or mean) large It is unfortunate that the word "diesis" already has two more syllables than we'd like it to have. This is presumably why we feel compelled to shorten any prefix we might add to it, down to a single syllable. We might instead shorten "diesis" to "di" for convenience when spoken (say in rehearsals) and then not need to shorten the prefixes. >With these labels, the boundaries (in cents) would then be: > >0 >schismina >0.98 >schisma >4.50 >kleisma >13.47 >comma >36.93 >protodiesis >45.11 >neodiesis >56.84 >metadiasis >68.57 Boundaries good. Labels still need work. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page *
Message: 6467 Date: Thu, 13 Feb 2003 11:29:55 Subject: scala show data From: Carl Lumma Manuel, With Scala 2.05f, I observe... equal 6 show data strictly proper roth stability 0 lumma stability 1 show data strictly proper roth stability 423799.833333 lumma stability 1 show data strictly proper roth stability 67041792.766666 lumma stability 1 The goofy stability value seems to max out at the 67 value despite further show data commands. -Carl
Message: 6469 Date: Thu, 13 Feb 2003 11:58:57 Subject: Re: scala show data From: Carl Lumma Manuel, I also notice that "Lumma stability" is the title of the value in the show data output, but "Lumma instability" is the title in the help for show data. Also in the help, the return type is given as n>1. But if it really is stability you're returning, it would be 0 <= n <= 1, right? Since it's the *portion* of the interval of equivalence not covered by the spans of the interval classes... -Carl
Message: 6470 Date: Thu, 13 Feb 2003 14:08:32 Subject: "Ultimate" 5-limit again From: Gene W Smith I returned to this, and added names, poptimal generators (this time using everyone's favorite defintion of the minimax generator) and "extensions". These are defined in a way which is very strict and perhaps a little arbitrary, but the results seem of some interest. I took the poptimal generator, found the corresponding val with the lowest badness, extended it in the way which gave lowest badness, and looked for the lowest badness 7-limit temperament compatible with this val and the 5-limit comma. I only go up to Monzimic with this list, which really seems far enough. To make up for that and make Paul happy, I tacked Miracle on the end despite the fact that as a 5-limit temperament it's nothing to get excited about. Bug 7/31 Extends 15/14 to Bug ([2, 3, 5, 0, 2, 3]) 27/25 [[1, 2, 3], [0, -2, -3]] [1200., 268.056438833948093748427143263] 3.739252 35.609240 1861.731473 Pelogic 10/23 Extends 36/35 to Pelogic (aka Hexadecimal) 135/128 [[1, 2, 1], [0, -1, 3]] [1200., 522.862345874111793591855751693] 4.132031 18.077734 1275.365360 Blackwood Universal 256/243 [[5, 8, 12], [0, 0, -1]] [240., 84.6637865678588914278600509674] 5.493061 12.759741 2114.877638 Dicot Universal 25/24 [[1, 1, 2], [0, 2, 1]] [1200., 350.977500432693708872243366367] 3.025593 28.851897 799.108711 Diminished Universal 648/625 [[4, 6, 9], [0, 1, 1]] [300., 94.1343573651111175944350240576] 6.437752 11.060060 2950.938432 Negri 2/19 Extends 49/48 to Negri (aka Tertiathirds) 16875/16384 [[1, 2, 2], [0, -4, 3]] [1200., 126.238272015257926746682149917] 8.172550 5.942563 3243.743713 Porcupine 11/81 Extends 64/63 to Porcupine 250/243 [[1, 2, 3], [0, -3, -5]] [1200., 162.996026370546548951179738408] 5.948286 7.975801 1678.609846 Augmented Universal 128/125 [[3, 5, 7], [0, -1, 0]] [400., 91.2018560670299909777049249654] 4.828314 9.677666 1089.323984 Magic 19/60 Extends 225/224 to Magic 3125/3072 [[1, 0, 2], [0, 5, 1]] [1200., 379.967949195094816842076920201] 7.741412 4.569472 2119.954990 Quadrafifths 26/177 Extends 245/243 to Octafifths 20000/19683 [[1, 1, 1], [0, 4, 9]] [1200., 176.282270436412295298990817071] 9.785568 2.504205 2346.540676 Pythagoric Universal 531441/524288 [[12, 19, 28], [0, 0, -1]] [100., 14.6637865678588914278600509674] 13.183347 1.382394 3167.444999 Meantone 34/81 Extends 126/125 to Meantone 81/80 [[1, 2, 4], [0, -1, -4]] [1200., 503.835154026035812053011163756] 4.132031 4.217731 297.556531 Diaschismic 10/114 Extends 245/243 to Shrutar 2048/2025 [[2, 3, 5], [0, 1, -2]] [600., 105.446531009812541696859310996] 6.271199 2.612822 644.408867 Tertiary 23/285 to Extends 3136/3125 to Tertiary ([3, -12, -30, -26, -56, -36]) 67108864/66430125 [[3, 5, 6], [0, -1, 4]] [400., 96.7879385616949726317268914802] 15.510107 .905187 3377.402314 Hemisixths 55/149 Various extensions, none much good 78732/78125 [[1, -1, -1], [0, 7, 9]] [1200., 442.979297439105373735900374126] 12.192182 1.157498 2097.802867 Wuerschmidt 53/164 Various extensions, none much good 393216/390625 [[1, -1, 2], [0, 8, 1]] [1200., 387.819673068349143521938606127] 12.543123 1.071950 2115.395301 Orwell 43/190 Extends 1029/1024 to Trifokker ([21, -9, -7, -63, -70, 9]) 2109375/2097152 [[1, 0, 3], [0, 7, -3]] [1200., 271.589599585245148575185388331] 12.772341 .800410 1667.723301 Septathirds 31/673 No good extensions 4294967296/4271484375 [[1, 2, 2], [0, -9, 7]] [1200., 55.2754932571412314963954609732] 18.573955 .483108 3095.692488 Kleismic 65/246 Extends 5120/5103 to Countercatakleismic ([6, 5, -31, -6, -66, -86]) 15625/15552 [[1, 0, 1], [0, 6, 5]] [1200., 317.079675185758890225628070818] 9.338935 1.029625 838.631548 Amity 58/205 Extends 5120/5103 to Amity 1600000/1594323 [[1, 3, 6], [0, -5, -13]] [1200., 339.508825625715624367834924710] 13.794200 .383104 1005.555381 Parakleismic 31/118 Extends 3136/3125 to Parakleismic ([13, 14, 35, -8, 19, 42]) 1224440064/1220703125 [[1, 5, 6], [0, -13, -14]] [1200., 315.250913337821936408197840098] 21.322672 .276603 2681.521263 Vulture 128/323 Extends 4375/4374 to Vulture ([4, 21, -56, 24, -100, -189]) 10485760000/10460353203 [[1, 0, -6], [0, 4, 21]] [1200., 475.542233398945960632986914825] 21.733049 .153767 1578.433204 Semisuper 30/506 Nothing much good 6115295232/6103515625 [[2, 4, 5], [0, -7, -3]] [600., 71.1460635722374759764193142621] 21.207625 .194018 1850.624306 Enneadecal Universal 19073486328125/19042491875328 [[19, 30, 44], [0, 1, 1]] [63.1578947368421052631578947368, 7.29225210195322285759291880280] 30.579320 .104784 2996.244873 Semitonic 95/1019 No good extensions 295578376007080078125/295147905179352825856 [[1, 0, 4], [0, 17, -18]] [1200., 111.875426120872633513689333181] 38.845486 .058853 3449.774562 Tricot 233/494 Extends 4375/4374 to Tricot ([3, 29, -95, 39, -159, -302]) 68719476736000/68630377364883 [[1, 3, 16], [0, -3, -29]] [1200., 565.988014913065527948022354197] 30.550812 .057500 1639.596150 Schismic 120/289 Extends 4375/4374 to Infraschismic ([1, -8, 39, -15, 59, 113]) 32805/32768 [[1, 2, -1], [0, -1, 8]] [1200., 498.272487171563819993901705714] 9.459948 .161693 136.885775 Counterschismic 237/571 No good extensions 2954312706550833698643/2951479051793528258560 [[1, 2, 21], [0, -1, -45]] [1200., 498.082318148218414995068857757] 48.911647 .026391 3088.065497 Hemithird 232/1441 Extends 4375/4374 to Infrahemithird ([15, -2, 113, -38, 137, 268]) 274877906944/274658203125 [[1, 4, 2], [0, -15, 2]] [1200., 193.199614933859969427837273253] 24.977022 .060822 947.732642 Minortone 196/1289 Extends 2460375/2458624 to Hemiminortone ([30, 70, 129, 32, 109, 103]. Minortone is [17, 35, -21, 16, -81, -147]) 50031545098999707/50000000000000000 [[1, -1, -3], [0, 17, 35]] [1200., 182.466089137089694182158775289] 38.845486 .025466 1492.763207 Ennnealimmal 68/1665 Extends 2401/2400 to Ennealimmal 7629394531250/7625597484987 [[9, 15, 22], [0, -2, -3]] [133.333333333333333333333333333, 49.0088197863290461293795242156] 33.653272 .025593 975.428947 Glum 103/935 Nothing much good 2475880078570760549798248448/2474715001881122589111328125 [[1, 5, 1], [0, -31, 12]] [1200., 132.194510561451335831533197063] 55.785793 .014993 2602.883149 Kwasy 182/1342 Extends 4375/4374 to Hemikwasy ([16, -10, 152, -53, 196, 381]) 9010162353515625/9007199254740992 [[2, 1, 6], [0, 8, -5]] [600., 162.741892126380267669129153916] 31.255737 .017725 541.228379 Mum 341/730 Extends 4375/4374 to Mum ([33, 25, 131, -37, 115, 234]) 116450459770592056836096/116415321826934814453125 [[1, 17, 14], [0, -33, -25]] [1200., 560.546969532517954992081849041] 50.788153 .012388 1622.898233 Bum 237/901 Extends 4375/4374 to Bum ([51, 52, 149, -36, 93, 200]) 444089209850062616169452667236328125/444002166576103304796646509039845376 [[1, 15, 16], [0, -51, -52]] [1200., 315.647874693157629813083932838] 82.462759 .004660 2613.109284 Monzimic 116/559 Extends 4375/4374 to Monzimic ([2, 37, -134, 54, -218, -415]) 450359962737049600/450283905890997363 [[1, 2, 10], [0, -2, -37]] [1200., 249.018447894645757478665305415] 39.665603 .005738 358.125500 ... Miracle 41/422 Extends 225/224 to Miracle 34171875/33554432 [[1, 1, 3], [0, 6, -7]] [1200., 116.578231256479] 14.2507126003310 1.98070788903097 5732.31669654049
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