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Message: 7376 Date: Mon, 01 Sep 2003 01:02:38 Subject: Re: x^4 - x^2 - 1 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> The subject line is a polynomial whose big root ...

The subject line is wrong, alas; I meant x^4-x^3-1.

Message: 7377 Date: Mon, 01 Sep 2003 07:45:16 Subject: Graham-style recurrence polynomials, degree three From: Gene Ward Smith 1.324718 x^3-x-1 1.465571 x^3-x^2-1 1.521380 x^3-x-2 1.618034 x^3-2*x-1 1.618034 x^3-x^2-x 1.695621 x^3-x^2-2 1.769292 x^3-2*x-2 1.796322 x^3-x-4 2.114908 x^3-4*x-1 2.166313 x^3-x-8 2.205569 x^3-2*x^2-1 2.214320 x^3-4*x-2 2.330746 x^3-2*x-8 2.359304 x^3-2*x^2-2 2.382976 x^3-4*x-4 2.394859 x^3-x^2-8 2.414214 x^3-2*x^2-x 2.561553 x^3-x^2-4*x 2.594313 x^3-2*x^2-4 2.649436 x^3-4*x-8 2.732051 x^3-2*x^2-2*x 2.888969 x^3-8*x-1 2.931142 x^3-2*x^2-8 2.945995 x^3-8*x-2 3.051374 x^3-8*x-4 3.236068 x^3-2*x^2-4*x 3.236068 x^3-8*x-8 3.372281 x^3-x^2-8*x 3.414214 x^3-4*x^2+2*x 3.709275 x^3-4*x^2+4 3.732051 x^3-4*x^2+x 3.866198 x^3-4*x^2+2 3.935432 x^3-4*x^2+1 4.060647 x^3-4*x^2-1 4.117942 x^3-4*x^2-2 4.224170 x^3-4*x^2-4 4.236068 x^3-4*x^2-x 4.411139 x^3-4*x^2-8 4.449490 x^3-4*x^2-2*x 4.828427 x^3-4*x^2-4*x 5.464102 x^3-4*x^2-8*x 6.828427 x^3-8*x^2+8*x 7.464102 x^3-8*x^2+4*x 7.741657 x^3-8*x^2+2*x 7.870865 x^3-8*x^2+8 7.872983 x^3-8*x^2+x 7.936496 x^3-8*x^2+4 7.968502 x^3-8*x^2+2 7.984314 x^3-8*x^2+1 8.015564 x^3-8*x^2-1 8.031009 x^3-8*x^2-2 8.061549 x^3-8*x^2-4 8.121294 x^3-8*x^2-8 8.123106 x^3-8*x^2-x 8.242641 x^3-8*x^2-2*x 8.472136 x^3-8*x^2-4*x 8.898979 x^3-8*x^2-8*x

Message: 7378 Date: Tue, 02 Sep 2003 15:29:34 Subject: The Forms of Tonality now Online! From: Carl Lumma All; Paul Erlich's fantastic paper, *The Forms of Tonality*, is now available on the web... Music Theory from Paul Erlich * [with cont.] (Wayb.) Getting it down to 400K is an achievement, I'm sure you'll agree, Paul. Strongly recommend Acrobat 6 to view it. -Carl

Message: 7379 Date: Tue, 02 Sep 2003 04:42:26 Subject: Brats for temperaments From: Gene Ward Smith Below I give 40 commas and the associated mapping, optimal generators for exponents 2,3,4 and infinity (ie., minimax), the associated brats, a reasonable choice for a brat on the basis of the above, and the brat as a rational function of the generator. By solving for the generator, we can find what corresponds to a particular choice of brat, such as my "reasonable choice". 27/25 [[1, 2, 3], [0, -2, -3]] [268.056438833948, 267.435479555972, 268.056438834223, 271.228762045052] [7.41821754512150, 6.28707119757482, 7.41821754573784, 2026193647657.34] b = infinity 2*(-6+5*g)/(-8+5*g^3) 135/128 [[1, 2, 1], [0, -1, 3]] [522.862345874114, 523.070610486922, 522.862345843404, 521.089678249860] [-.595712670824538, -.689980196352613, -.595712657415004, .145521127181207e-11] b = 0 (3*g^4-10)/g/(2*g^3-5) 256/243 [[5, 8, 12], [0, 0, -1]] [84.6637865678555, 84.6637865678584, 84.6637865678614, 84.6637865678580] [-1.50783780618484, -1.50783780618484, -1.50783780619035, -1.50783780618484] b = -3/2 2^(3/5)*(-3*2^(4/5)+5*g)/(-4*2^(2/5)+5*g) 25/24 [[1, 1, 2], [0, 2, 1]] [350.977500432694, 350.977500432688, 350.977500431322, 350.977500432690] [1.50000000000012, 1.49999999999890, 1.49999999988252, 1.50000000000012] b = 3/2 -g*(-6+5*g)/(4*g-5) 648/625 [[4, 6, 9], [0, 1, 1]] [94.1343573651068, 94.1343573651113, 94.1343573651289, 94.1343573651120] [2.99323921078598, 2.99323921078598, 2.99323921077971, 2.99323921078598] b = 3 -2*(-6+5*2^(1/4))*g/(8*g-5*2^(3/4)) 16875/16384 [[1, 2, 2], [0, -4, 3]] [126.238272015258, 126.205405491448, 126.238272048931, 126.336958999920] [.136091735477899, .179098457309447, .136091690828154, .780659798090136e-11] b = 0 2*(3*g^7-5)/g^4/(4*g^3-5) 250/243 [[1, 2, 3], [0, -3, -5]] [162.996026370546, 163.069650366763, 162.996026374613, 162.737257227031] [12.0314476213430, 9.49916182634974, 12.0314474417337, 683526730904.000] b = infinity 2*(-6+5*g^2)/(-8+5*g^5) 128/125 [[3, 5, 7], [0, -1, 0]] [91.2018560670246, 91.2018560670318, 91.2018560671082, 91.2018560670318] [.751482280984136, .751482280984136, .751482280993138, .751482280984136] b = 3/4 1/3*(25*2^(1/3)+20*2^(2/3)+32)*(6*g-5*2^(1/3))/g 3125/3072 [[1, 0, 2], [0, 5, 1]] [379.967949195095, 379.932993406921, 379.967950000885, 380.391000173076] [.999388966181635, .961015450204820, .999389855671830, 1.50000000000604] b = 1 -1/2*g*(-12+5*g^4)/(4*g-5) 20000/19683 [[1, 1, 1], [0, 4, 9]] [176.282270436412, 176.291430912569, 176.282270741616, 176.257079318316] [-19.5178863924243, -14.0908094624819, -19.5176398261720, 333148837511.168] b = infinity g^4*(3*g^5-5)/(2*g^9-5) 531441/524288 [[12, 19, 28], [0, 0, -1]] [14.6637865678600, 14.6637865678561, 14.6637865678567, 14.6637865678560] [-1.49915329944943, -1.49915329944943, -1.49915329944943, -1.49915329944943] b = -3/2 2^(7/12)*(-3*2^(3/4)+5*g)/(-4*2^(1/3)+5*g) 81/80 [[1, 2, 4], [0, -1, -4]] [503.835154026037, 503.883744407432, 503.835153661954, 503.421571533790] [-3.74373500460168, -3.23201443400256, -3.74373928931788, 372097502304.000] b = infinity 2*(-12+5*g^3)/(-16+5*g^4) 2048/2025 [[2, 3, 5], [0, 1, -2]] [105.446531009815, 105.492093791319, 105.446530999138, 105.213762333520] [-.375378300113750, -.463999206632087, -.375378280006185, 0.] b = 0 2^(1/2)*(-6+5*g^3)/(-4*2^(1/2)+5*g^2) 67108864/66430125 [[3, 5, 6], [0, -1, 4]] [96.7879385616958, 96.7810140152084, 96.7879385539879, 96.8717425998888] [-.75027233388, -.84003132898, -.75027243052, -.3e-10] b = 0 (6*g^5-5*2^(2/3))/g/(4*g^4-5) 78732/78125 [[1, -1, -1], [0, 7, 9]] [442.979297439105, 442.984696404155, 442.979314219411, 442.923745984982] [4.79847558521374, 4.40299365566162, 4.79712734696446, -47754600097.7616] b = infinity 1/2*g^7*(3*g^2-5)/(g^9-10) 393216/390625 [[1, -1, 2], [0, 8, 1]] [387.819673068348, 387.822889697592, 387.819654171468, 387.744375108173] [.900156573210775, .875861107340688, .900299608507285, 1.50000000004430] b = 1 -1/4*g*(-24+5*g^7)/(4*g-5) 2109375/2097152 [[1, 0, 3], [0, 7, -3]] [271.589599585246, 271.593697588711, 271.589601548604, 271.564128700056] [.352856800255913, .405031315378923, .352882080754974, -.752603975517185e-10] b = 0 1/2*(-24+5*g^10)/(-8+5*g^3) 4294967296/4271484375 [[1, 2, 2], [0, -9, 7]] [55.2754932571411, 55.2765687202447, 55.2754940620164, 55.2724195624650] [.120019601606355, .160053073802616, .120049929465304, .113835249438497e-9] b = 0 2*(3*g^16-5)/g^9/(4*g^7-5) 15625/15552 [[1, 0, 1], [0, 6, 5]] [317.079675185760, 317.085190264962, 317.079673835428, 316.992500144230] [2.35749900106746, 2.44007011449707, 2.35747939320971, 1.49999999997026] b = 2 -1/2*g^5*(-6+5*g)/(2*g^5-5) 1600000/1594323 [[1, 3, 6], [0, -5, -13]] [339.508825625716, 339.507387256433, 339.508875042446, 339.514329702704] [-9.00130202082944, -6.94510840078312, -9.09121293421452, 10500688224.0000] b = infinity 4*(-24+5*g^8)/(-64+5*g^13) 1224440064/1220703125 [[1, 5, 6], [0, -13, -14]] [315.250913337821, 315.250648792898, 315.250774706137, 315.263306152511] [3.37478861493051, 3.30649638615107, 3.33864128093987, 5239355206.80000] b = infinity 16*(-6+5*g)/(-64+5*g^14) 10485760000/10460353203 [[1, 0, -6], [0, 4, 21]] [475.542233398946, 475.542500964170, 475.537866896123, 475.538748279278] [-2.88479361344442, -2.59252603021660, 17.4229369490071, -3465801815.30269] b = infinity 1/2*g^4*(3*g^17-320)/(g^21-320) 6115295232/6103515625 [[2, 4, 5], [0, -7, -3]] [71.1460635722381, 71.1449905180548, 71.1460673212410, 71.1492855906600] [1.36364524574015, 1.32056202037312, 1.36379772876677, 1.49999999998592] b = 3/2 -2^(1/2)*(6*g^4-5*2^(1/2))/g^4/(-4*2^(1/2)+5*g^3) 19073486328125/19042491875328 [[19, 30, 44], [0, 1, 1]] [7.29225210195604, 7.29225210195535, 7.29225210203665, 7.29225210195600] [3.00006419857425, 3.00006419857425, 3.00006420206449, 3.00006419857425] b = 3 -2*(-6+5*2^(5/19))*g/(8*g-5*2^(13/19)) 295578376007080078125/295147905179352825856 [[1, 0, 4], [0, 17, -18]] [111.875426120873, 111.875408100894, 111.873024328626, 111.875465342873] [-.288455519924944e-1, -.422903799249676e-1, -4.55207607290570, .600359663647405e-9] b = 0 1/2*(-48+5*g^35)/(-16+5*g^18) 68719476736000/68630377364883 [[1, 3, 16], [0, -3, -29]] [565.988014913065, 565.987975909345, 565.997677421647, 565.989182280523] [-2.08700529757088, -1.97604825636316, 1.81643654799886, 1099510569.80000] b = infinity 4*(-24576+5*g^26)/(-65536+5*g^29) 32805/32768 [[1, 2, -1], [0, -1, 8]] [498.272487171564, 498.272868834532, 498.272049258382, 498.262079222160] [-1.04993101558579, -1.11384696853000, -.980096572950770, .299047265951068e-10] b = -1 1/2*(3*g^9-40)/g/(g^8-10) 2954312706550833698643/2951479051793528258560 [[1, 2, 21], [0, -1, -45]] [498.082318148220, 498.082321198085, 498.072699083572, 498.081917469671] [-1.60463387655409, -1.58143258093245, 1.60014916042977, -479153673.714286] b = infinity 2*(-1572864+5*g^44)/(-2097152+5*g^45) 274877906944/274658203125 [[1, 4, 2], [0, -15, 2]] [193.199614933860, 193.199695616494, 193.197870548564, 193.197571352909] [.609383934851494, .632249777329436, .930137288618768e-1, .159458989868289e-9] b = 0 2*(3*g^17-20)/g^15/(4*g^2-5) 50031545098999707/50000000000000000 [[1, -1, -3], [0, 17, 35]] [182.466089137089, 182.466081068180, 182.468384070474, 182.466106110424] [-77.9993086006430, -52.6184224616755, 1.35836621555422, 211999114.193699] b = infinity 1/2*g^17*(3*g^18-20)/(g^35-40) 7629394531250/7625597484987 [[9, 15, 22], [0, -2, -3]] [49.0088197863282, 49.0092660738190, 49.0088122512589, 49.0065398228312] [7.50005927378393, 6.35414803641991, 7.52327055663510, -7526906337.33860] b = infinity 2^(2/3)*(-3*2^(7/9)+5*g)/(-4*2^(4/9)+5*g^3) 2475880078570760549798248448/2474715001881122589111328125 [[1, 5, 1], [0, -31, 12]] [132.194510561451, 132.194527882796, 132.191628330792, 132.194388674406] [.385136835303054, .435434635544873, 12.5616518433724, -.142174950967159e-7] b = infinity (3*g^43-80)/g^31/(2*g^12-5) 9010162353515625/9007199254740992 [[2, 1, 6], [0, 8, -5]] [162.741892126380, 162.741958792709, 162.744739215164, 162.741637461582] [.214284486343732, .266546184643909, 1.61679609019013, -.698683814530209e-9] b = 1 1/2*2^(1/2)*(-12*2^(1/2)+5*g^13)/(-8+5*g^5) 116450459770592056836096/116415321826934814453125 [[1, 17, 14], [0, -33, -25]] [560.546969532518, 560.546989674304, 560.550415366598, 560.546818155593] [2.12194817527459, 2.23544091312893, -.903028139088175, 1.50000000092280] b = 3/2 -8192*(3*g^8-40)/g^8/(-16384+5*g^25) 444089209850062616169452667236328125/444002166576103304796646509039845376 [[1, 15, 16], [0, -51, -52]] [315.647874693155, 315.647874725983, 315.649938234086, 315.647813194906] [3.09000325776413, 3.08836970087491, .117431939393260, -46321886.6176000] b = infinity 16384*(-6+5*g)/(-65536+5*g^52) 450359962737049600/450283905890997363 [[1, 2, 10], [0, -2, -37]] [249.018447894646, 249.018446056434, 249.013842268974, 249.018548273923] [-1.77273144947357, -1.71533485435417, 1.35084773049401, -50081536.7142858] b = infinity 2*(-768+5*g^35)/(-1024+5*g^37) 162285243890121480027996826171875/162259276829213363391578010288128 [[1, -1, 11], [0, 14, -47]] [221.567865486044, 221.567867194692, 221.562607414796, 221.567889950828] [-.659999130523142, -.599704461945995, 1.98303836002560, -.102372548206321e-6] b = 0 1/4*(-12288+5*g^61)/(-2048+5*g^47) 22300745198530623141535718272648361505980416/22297583945629639856633730232715606689453125 [[1, -2, 4], [0, 47, -22]] [91.5310212152784, 91.5310188118782, 91.5331152603505, 91.5310331449356] [.323276236582079, .383101227190776, 5.10076534940776, -.152099898191020e-6] b = infinity 1/8*(-192+5*g^69)/(-16+5*g^22) 381520424476945831628649898809/381469726562500000000000000000 [[1, 11, 19], [0, -35, -62]] [322.801386591773, 322.801392385865, 322.810340953662, 322.801391711857] [18.1874711880147, -132.546574791409, .643135139215846, 37459128.8888889] b = infinity 1024*(-768+5*g^27)/(-524288+5*g^62) 1292469707114105741986576081359316958696581423282623291015625/1292388115393055295535123767426518253869322213731913416835072 [[2, 14, 19], [0, -43, -57]] [151.117308517283, 151.117310901324, 151.120216709418, 151.117303265530] [5.17241093569130, 3.67251559849278, .377067159030008, -14320613.1809524] b = infinity 64*(-12*2^(1/2)+5*g^14)/(-512*2^(1/2)+5*g^57)

Message: 7380 Date: Tue, 02 Sep 2003 19:53:49 Subject: uniqueness From: Carl Lumma Regular temperaments enforce consistency. I'm struggling once again for Gene's way of saying this... if the consonant intervals are a group closed under multiplication, then there's a group homomorphism to elements of the temperament that preserves the closure. How's that? Anyway, what about uniqueness... can the cardinality (do groups have cardinality, or is that only sets?) of the group decrease after the mapping to the temperament? If, after figuring out what I'm asking and answering, I can go back and phrase it correctly, I'd be tremendously pleased. -Carl

Message: 7381 Date: Tue, 02 Sep 2003 19:27:25 Subject: the chord-finding problem again From: Carl Lumma Heya Gene,

>I would work in 3-dimensions for the 9-limit, and just make 3 >half the size of 5 or 7. In other words, > >||3^a 5^b 7^c|| = sqrt(a^2 + 4b^2 + 4c^2 + 2ab + 2ac + 4bc) > >would be the length of 3^a 5^b 7^c. Everything in a radius of 2 >of anything will be consonant.

Any suggestions for how to find all integer solutions for a,b,c... in something like this?

>If we like, we may adjust matters by multiplying through by the >lcm of the exponents of the largest prime powers, so as to be >able to work with integers.

Not sure if this is a start at answering the above question.

>This sort of thing is what I meant when I said I came upon >hexanies and the like geometrically. This metric is useful partly >because two octave equivalence classes separated by a distance of >one or less are o-consonant, and by a distance of greater than >one are o-dissonant.

1 or less? That's radius 1/2. Above you say radius 2!

>How's this as a method: using the standard o-limit metric, take >everything in a radius of 1 of the unison, //

Now it's radius 1!

> // which should give you the o-limit diamond. Now take all > subsets of size k, find the centroid by averaging the coordinates >(which should be in the prime-power basis, so that in the 9-limit >5/3 would be 9^(-1/2) * 5^1 * 7^0 = [-1/2, 1, 0], for instance)

9 isn't prime!

>and test if everything is within a radius of 1/2 of the centroid, >in which case put it on your list. For larger values of o, this >would be faster than simply testing for pairwise consonance.

Radius 1/2 again. I assume this metric works whether one starts on a lattic note or not (I assume the centroid could be on a point at which no pitch class is found, or even a point not lying on a consonant dyad line)... -Carl

Message: 7382 Date: Wed, 03 Sep 2003 22:00:58 Subject: Re: uniqueness From: Graham Breed hstraub64 wrote:

>Huh? Groups _are_ sets, and therefore have cardinality, of course. >And in a mapping that preserves the group operations (homomorphism), >the cardinality of the image can be lower if the mapping is not >injective. You could call that "descreasing cardinality". >Looks like elemetary math stuff - or did I again miss something? > >

I've been reading up on group theory lately. I don't think it makes sense for a temperament to be an injective mapping. Meaning that at least one interval in the tempered scale should represent more than one interval in the ideal scale. Otherwise, all you have is a subset. Like the 5-limit is a subset of the 7-limit, but not a temperament of it (unless you map the 7:4 in some way). Furthermore, I suggest a temperament should strictly be a surjective homomorphism, or an epimorphism. That means each interval in the tempered scale represents at least one ideal interval. This definition excludes things like 24-equal in the 5-limit where every other tempered interval isn't a temperament of anything. As for the cardinality issue, that's the number of elements in the set, isn't it? Only octave-specific equal temperaments have a finite number of elements. Every other regular scale is a countable infinity, which I think means the cardinality is the same (aleph 1). This is assuming they're fully fledged groups -- models that take into account the limits of human perception will have a finite number of elements, but they won't be groups. I think this is what Lindley and Turner Smith were getting at in the paper Gene doesn't like. Graham

Message: 7383 Date: Wed, 03 Sep 2003 00:42:46 Subject: Re: The Forms of Tonality now Online! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> All; > > Paul Erlich's fantastic paper, *The Forms of Tonality*, is > now available on the web... > > Music Theory from Paul Erlich * [with cont.] (Wayb.) > > Getting it down to 400K is an achievement, I'm sure you'll > agree, Paul. Strongly recommend Acrobat 6 to view it. > > -Carl

thanks so much carl. hoping to correct the typos soon.

Message: 7384 Date: Wed, 03 Sep 2003 21:19:04 Subject: Re: uniqueness From: Carl Lumma

> Huh? Groups _are_ sets, and therefore have cardinality, > of course.

Yes, I should have looked up "groups" first.

> And in a mapping that preserves the group operations > (homomorphism), the cardinality of the image can be lower > if the mapping is not injective. You could call that > "descreasing cardinality". Looks like elemetary math > stuff - or did I again miss something?

My question is, can a "regular temperament" be injective. -Carl

Message: 7385 Date: Wed, 03 Sep 2003 14:28:49 Subject: Re: uniqueness From: Carl Lumma

>I don't think it makes sense for a temperament to be an injective >mapping. Meaning that at least one interval in the tempered scale >should represent more than one interval in the ideal scale.

What about things like dicot?

>Otherwise, all you have is a subset. Like the 5-limit is a subset >of the 7-limit, but not a temperament of it (unless you map the 7:4 >in some way).

I don't think there's necessarily a homomorphism from a group to all its subsets. You have to keep closure. For example, the integers are closed under addition but the subset (1 2 3) is not. Am I getting this right?

>Furthermore, I suggest a temperament should strictly be a >surjective homomorphism, or an epimorphism. That means each >interval in the tempered scale represents at least one ideal >interval.

I don't think that's possible. Unless you allow compound intervals to be ideal. But then you're always going to be injective, since any comma will confound two compound intervals (wheras only commas like 25:24 confound two primary intervals).

>As for the cardinality issue, that's the number of elements in >the set, isn't it?

I wasn't sure if card. was defined on groups. Turns out that groups are special types of sets.

>Only octave-specific equal temperaments have a finite number >of elements. Every other regular scale is a countable infinity, >which I think means the cardinality is the same (aleph 1).

Wait, are temperaments groups of intervals or pitches? I assumed intervals. -Carl

Message: 7387 Date: Thu, 04 Sep 2003 13:06:23 Subject: Re: uniqueness From: Carl Lumma

>I don't know, what's dicot? It's not in the catalog.

It's in Paul's database. IIRC it's the new name for neutral thirds.

>Well, I don't see what stops it being injective. If it's >only 5-limit then it isn't surjective, and so not strictly >a temperament by the definition below.

//

>Mathworld has a diagram that explains it: > >Injection -- from MathWorld * [with cont.]

Gene's linking to surjection on his regular temperaments page. But I don't see any definition of "rank" on the abelian group mathworld page, so I can't follow the bit about icons -- it looks like Gene's spiced up his def. of "regular temperament". Actually, I don't see a def. on this page. And it doesn't seem to be in monz's site like I thought it was. -Carl

Message: 7388 Date: Thu, 04 Sep 2003 21:54:21 Subject: Re: uniqueness From: Graham Breed Carl Lumma wrote:

>It's in Paul's database. IIRC it's the new name for >neutral thirds. > >

It's a 5-limit temperament that maps both 6:5 and 5:4 to the neutral third generator. It looks like an epimorphism as good as any other.

>Gene's linking to surjection on his regular temperaments >page. But I don't see any definition of "rank" on the >abelian group mathworld page, so I can't follow the bit >about icons -- it looks like Gene's spiced up his def. >of "regular temperament". Actually, I don't see a def. >on this page. And it doesn't seem to be in monz's site >like I thought it was. > >

Oh, yes, there are quite a few new things there. He's been quiet round here lately. He also says the icon is an epimorphism, which is the same as a surjection, at least for groups. (Sets can have surjections as well, which might be what I was thinking of before. Well temperaments can be thought of as sets but not groups, and they have an epimorphism of intervals onto an equal temperament. I terms of pitches, there's an epimorphism from JI into a well temperament.) I can't follow the chain of definitions to see if a regular temperament is an epimorphism from JI to something simpler, but it's probably something like that. Rank is a property of a free abelian group http://mathworld.wolfram.com/FreeAbelianGroup.... * [with cont.] It's the number of (linearly independent) generators -- 2 for 5-limit JI, 3 for 7-limit, 4 for 11-limit, 1 for an equal temperament, 2 for a linear temperament, 3 for a planar temperament. Graham

Message: 7389 Date: Thu, 04 Sep 2003 21:36:28 Subject: Re: uniqueness From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Gene's linking to surjection on his regular temperaments > page. But I don't see any definition of "rank" on the > abelian group mathworld page, so I can't follow the bit > about icons -- it looks like Gene's spiced up his def. > of "regular temperament". Actually, I don't see a def. > on this page. And it doesn't seem to be in monz's site > like I thought it was.

I think I defined rank somewhere or other. The MathWorld definition seems to be written for mathematicians, but here it is: Group Rank -- from MathWorld * [with cont.] The tensor product Q tensor G makes the torsion part go away and turns the free part into a vector space, giving us a vector space of dimension equal to the number of copies of Z in G. This number is the rank of an abelian group G; when G is free it doesn't need to be tensored with Q in order to exterminate the torsion part. Concretely, if your group is free and group elements are written as row vectors of integers (always possible for free groups) then the dimension of these vectors is the rank.

Message: 7390 Date: Thu, 04 Sep 2003 09:23:23 Subject: Re: uniqueness From: Graham Breed Me:

> >I don't think it makes sense for a temperament to be an injective > >mapping. Meaning that at least one interval in the tempered scale > >should represent more than one interval in the ideal scale.

Carl:

> What about things like dicot?

I don't know, what's dicot? It's not in the catalog. A search shows it's something to do with neutral thirds. Well, I don't see what stops it being injective. If it's only 5-limit then it isn't surjective, and so not strictly a temperament by the definition below. Me:

> >Otherwise, all you have is a subset. Like the 5-limit is a subset > >of the 7-limit, but not a temperament of it (unless you map the 7:4 > >in some way).

Carl:

> I don't think there's necessarily a homomorphism from a group > to all its subsets. You have to keep closure. For example, > the integers are closed under addition but the subset (1 2 3) > is not. Am I getting this right?

Yes, I should have said "subgroup". And I got it the wrong way round. The injective homomorphism would have to be from 5-limit to 7-limit, so 7-limit would be the temperament. That sounds bizarre which supports the idea that temperaments should be surjective but not injective. Me:

> >Furthermore, I suggest a temperament should strictly be a > >surjective homomorphism, or an epimorphism. That means each > >interval in the tempered scale represents at least one ideal > >interval.

Carl:

> I don't think that's possible. Unless you allow compound > intervals to be ideal. But then you're always going to be > injective, since any comma will confound two compound > intervals (wheras only commas like 25:24 confound two > primary intervals).

When I say "scale" I mean the whole infinity of notes in the tuning system. What you describe is "surjective" not "injective". Another term for "injective" is "one to one" which I find easier to understand, as long as I remember it isn't the same as "bijective" or "one to one onto". Mathworld has a diagram that explains it: Injection -- from MathWorld * [with cont.]

> I wasn't sure if card. was defined on groups. Turns out that > groups are special types of sets.

Yes, a group is a set with a binary operation that fulfils a number of criteria.

> >Only octave-specific equal temperaments have a finite number > >of elements. Every other regular scale is a countable infinity, > >which I think means the cardinality is the same (aleph 1).

> > Wait, are temperaments groups of intervals or pitches? I assumed > intervals.

Ah! I meant "octave-equivalent" there. If they're groups, they have to be sets of intervals, because an operation on elements of the group has to give another element of the same group. More specifically they have to be directed intervals. An equal temperament with n notes the octave can be thought of as a set of n notes in octave equivalent terms. But it also has n distinct directed intervals, making it a cyclic group of order n. So for 12-equal, the 12 ascending intervals are: 0 unison 1 minor second 2 major second 3 minor third 4 major third 5 perfect fourth 6 tritone/augmented fourth/diminished fifth 7 perfect fifth 8 minor sixth 9 major sixth 10 minor seventh 11 major seventh And octave is the same as a unison in octave-equivalent terms. If you add an ascending perfect fifth to an ascending minor sixth, you get in interval of +15 steps. That's the same modulo 12 to a 3 steps, so the result is an ascending minor third. The inverse operation relates each ascending interval to a descending one. A descending minor second has -2 steps, and in octave equivalent terms that's the same as 10 steps, and so a descending minor second and an ascending minor seventh are the same octave equivalent interval. Graham

Message: 7391 Date: Thu, 04 Sep 2003 23:04:12 Subject: Re: uniqueness From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> Anyway, what about uniqueness...

uniqueness, unlike consistency, is well-defined for temperaments -- it is quite possible for temperaments to violate uniqueness, for example dicot in the 5-limit . . .

Message: 7392 Date: Fri, 05 Sep 2003 04:55:58 Subject: Re: Classificiation of musical scales From: Dave Keenan

> > --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> wrote:

> > > Motivated by a recent dicussion in the group I have tried to

> > provide a

> > > comprehensive and clear classification of musical scales.

... Carlos, Your seem to be classifying methods of scale _construction_, rather than how the scales sound or can be used. And that's fine, so long as you recognise it for what it is, and realise that two scales may be in different categories but be audibly (and even immeasurably) indistingushable from each other, in part because these categories rely on the purely mathematical distinction between ratios and their neighbouring irrational numbers that may only be infinitesimally different. I prefer to use the term "justly intoned" for an audible distinction (with fuzzy boundaries), and "rationally constructed" otherwise. Even so, your scheme seems to have no place for tunings that have a single generator but no particular interval of equivalence, such as Gary Morrison's 88-cET. This is where successive pitches are simply placed 88 cents apart for as far as you like in either direction from a starting pitch. From a purely mathematical point of view. 88-cET is a one-dimensional system, while meantone and Pythagorean (3-limit rational) are two dimensional. We call the latter "linear" temperaments because we are taking the octave for granted as the other generator. ... Or are we? We are certainly taking _something_ for granted as the other generator. Let's look at another example. Paul Erlich's pajara is a mathematically 2D system which is an octave-equivalent linear temperament. One generator is a slightly wide fifth, but the other generator is not the octave. It is the exact half-octave. There are other octave-equivalent linear temperaments that are mathematically 2D and that use 1/3 octave or 1/4 octave or 1/5 octave etc. Graham Breed even found an apparently useful one that used 1/29 octave. When a generator is an exact submultiple of the interval of equivalence like this, we call it a "period". But it's unclear how important that distinction is (between a period and an ordinary generator) for classification. Perhaps the so-called interval-of-equivalence (IoE) should be considered as just another just interval to be approximated. What's special about it? .. I think that what's special about it is that when defining a scale, the number of times the IoE is to be iterated is left undefined. It is considered to depend on the physical properties of a particular instrument that the scale might be realised on. How wide its compass. Octave-equivalent 5-limit rational tunings are clearly mathematically 3D, although we call them "planar". And clearly planar temperaments Now what about N equal divisions of the octave (N-tETs or N-EDOs)? Are they (mathematically) closed 2D systems, or are they 1D? Do they have a period of an octave and a generator that just happens to close on the octave after a certain number of iterations, or do they have a period of 1/N octave and no other generator. The latter seems more meaningful to me, i.e. they are mathematically 1D. Since otherwise the generator can be arbitrarily chosen as any interval of the ET. If one generator "closes" on another generator, then it doesn't really belong to a different dimension, does it? So I'd put -tETs and -cETs in the same mathematically 1D category called "equal tunings". Then we have "linear tunings", "planar tunings", etc. Then orthogonal to those categories we have the question of what the IoE is if any, then the size of the period which, for tunings that have an IoE, is expressed as how many periods there are in the IoE, and for tunings that don't have an IoE, is simply given in cents or whatever. Then we have the sizes of any subsequent generators and how many times they are iterated. All the best, -- Dave Keenan

Message: 7395 Date: Sat, 06 Sep 2003 14:20:24 Subject: Re: Classificiation of musical scales From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> wrote:

> > you recognise it for what it is, and realise that two scales may be in > > different categories but be audibly (and even immeasurably) > > indistingushable from each other, in part because these categories > > rely on the purely mathematical distinction between ratios and their > > neighbouring irrational numbers that may only be infinitesimally > > different.

> > Are you referring to the "fuzzy neigbourghoods" of Lindley and

Tuner-Smith. I don't think so. But I haven't read about them.

> >I prefer to use the term "justly intoned" for an audible > > distinction (with fuzzy boundaries), and "rationally constructed" > > otherwise. > >

> > I am not sure I am understanding you here. Do you use the term

"justly" in a

> reference to the just ratios of prime numbers? and the term

rationally, I

> guess you do not mean it in the sense of using rational numbers but

rather to

> having a rational appraoch to the construction. But if this is the

case, well

> all case are rational, arent't they?

No. By "rationally constructed" I did mean "constructed using ratios" and in future I will use the latter phrase to avoid confusion. Many take "justly intoned" to be a synonym for "constructed using ratios". This is not necessarily so. One can easily construct a scale using ratios, that no one in their right mind would call justly intoned if they _listened_ to it, in a typical harmonic timbre. I understand that a tuning indistinguishable from 12-tET can be constructed using only ratios of numbers less than 100 - the famous Hammond organ.

Message: 7396 Date: Sat, 06 Sep 2003 23:52:35 Subject: Re: Classificiation of musical scales From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> wrote:

> But justly intoned by most usual definitions (see for example Randell > Dictionary of Music) is a scale that tries to replicate a portion of

the

> natural harmonics, hence you have ratios of the numbers 1,2,3,4, ..

and so

> forth, which are normally expressed in terms of their prime components.

There's no doubt that justly intoned scales include many intervals whose frequency ratio are very close to simple ratios. But one can't really _define_ just intonation that way, because such mathematical models of the perceptual quality called "justness" are not (yet) sufficiently accurate. It's currently too difficult to say, with any hope of wide agreement among experts, what counts as a simple enough ratio and what counts as close enough, in what musical contexts. And note that a complex ratio may or may not be justly intoned. For example, in most contexts, 20001:30001 is just, while 34:55 is not.

> On the other hand, not all scales based in ratios have to be JI.

I'm very glad you realise that. The equal tempered, but rational, Hammond Organ is very useful for making this point. The other side of the coin is that not all just intonation involves ratios. The preeminent example here is Barbershop singing. There need be no intention on the part of the singers to produce exact ratios of frequencies, and no measurement could, even in principle, determine whether they are producing ratios or merely irrational relationships that happen to be very close to simple ratios. And yet no one denies that they are producing justly intoned harmonies. Fortunately human genetics and environment are sufficiently stable to make it possible to define just intonation _injunctively_. That is, by giving a series of instructions, which if correctly followed will lead a person to experience that salient perceptual quality of some harmonies that we call "justness", or "purity". We usually do so by first defining (injunctively) another perceptual quality called "beating", etc. In my opinion the Oxford English Dictionary has a far better definion of "just" than most musical dictionaries and other publications that attempt to define it purely in terms of ratios. "Just ... /Mus/. in /just interval/, etc.: Harmonically pure; sounding perfectly in tune 1811." I think that equating justness with numerical rationality (rather than merely associating it) is a recent abberation and I hope it will soon fade. Of course, such simplifications will always be understandable in casual discussions, but not in dictionaries or other texts which claim to be authoritative.

Message: 7398 Date: Sun, 07 Sep 2003 23:29:53 Subject: Re: Classificiation of musical scales From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> wrote:

> Your > definition is more difficult to tackle because it involves a subjective > perception by the observer: "sounding perfectly in tune". > > But a classification is just a means to organize your views, so as

long as we

> know we are talking about, there is no disagreement here.

Agreed. --- In tuning-math@xxxxxxxxxxx.xxxx "pitchcolor" <pitchcolor@a...> wrote:

> Dave wrote: > > << I think that equating justness with numerical rationality (rather > than merely associating it) is a recent abberation and I hope it > will soon fade.>> > > I think it's fairly standard that 'just' means constructed by ratios of > (usually small) whole numbers.

Yes unfortunately, it has become fairly standard. "A recent abberation" as I said. When it's pointed out that the "just" = "numerically rational" definition would lead to the 12-tET Hammond organ and top-octave-divider electronic organs being called "justly intoned", while no one could ever say for sure whether Barbershop singing is justly intoned or not, then people usually realise that what they really mean by "just" is a perceptual quality, not a mathematical one, and they have been oversimplifying.

> On the other hand, the term > 'pure' is often used to describe perceptions of such intervals.

An interesting idea, but I'm afraid it's unsupportable. A quick search on "just pure intonation tuning" (without the quotes) in Google showed that the terms "pure" and "just" are still used as synonyms, even by those who define "just" purely in terms of ratios. As just one example, the very first sentence of Kyle Gann's "Just Intonation Explained" contains the phrase "just-intonation (pure) tuning".

> 'Beatless' is strictly a matter of physics.

Really? What about when the beats get faster and we no longer perceive them as beats, only as roughness, and then eventually we don't perceive them at all? Perhaps there are two meaning of "beatless" that we should be careful to distinguish.

> Note also that many > European language music theories classify intervals as 'just' > that we in the US call 'perfect' - that is, 'fifths' and 'fourths'.

These

> terms are not 'perceptual.' They are a product of music theory.

We are of course discussing the English meaning of "just", however, even if you are right about this usage in other languages, it doesn't help the case for "just" = "rational" any more than it does the perceptual definition.

> In > my opinion, the dictionary entry you cite which defines 'just' as > 'sounding pure' is a poor one because it conflates the theoretical > term 'just' with the perceptual term 'pure'. These terms should be > kept in their seperate spheres.

Can you provide any evidence that they have ever been in separate spheres?

> It sounds like you want to say that intervals which are just do not > necessarily sound pure.

No, I want to say that intervals which are rationally constructed do not necessarily sound just or pure. And that intervals which sound just or pure are not necessarily rationally constructed.

> I find this to be true in my experience, > but this is of course subjective, because statements about > perceptions always depend on the listener.

Of course. But we needn't run in terror from this subjectivity and try to bury our heads in an objective but oversimplified mathematical definition. The fact is that most people can recognise what is meant by justly intoned when you demonstrate it to them by, for example, playing two notes simultaneously, in a sustained timbre, rich in harmonics, and slowly varying the pitch of one of them while asking the listener to tell you when they hear something unusual. I raised this some years ago on the main tuning list, and I found that some people had a strong emotional attachment to ratios, whether they resulted in any audible quality or not. And the term "JI" had become something like a badge of cult status for them, so they felt very threatened by what I was saying. So much so that some went off and started new Yahoo groups. And others who could see the logic of what I was saying, nevertheless decided not to support it, for the sake of peace. Some folks agreed to use the term RI (for rational intonation) instead of JI in those cases where the ratios did not imply (in your terms) sounding pure. I never felt quite comfortable about that term (RI) but I couldn't say why, and let it go (for the sake of peace). In this present discussion, thankfully far less heated, I have realised why. I think it is because the word "intonation" also refers to perception, not mathematics. One cannot rationally _intone_ something, because one cannot _hear_ rationality. "intonation" and "tuning" seem to be almost synonyms, but not quite. Otherwise Kyle Gann would be guilty of redundancy when he wrote ""just-intonation (pure) tuning". So I think "RT" for "rational tuning" would be better, or possibly we could get away with "RCI" for "rationally constructed intonation".

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