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trained-pianist
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« Reply #1 on: 06:27:11, 09-05-2007 » |
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I don't know what message you mean SusanDoris,
Do you mean this message?
Critical Thinker
Join Date: Jul 2006
Posts: 384
My Profile
complex of indices of 6 ^ n mod 709 has triads throughout
Audiofreak, I trust you don't phryg like your aunt lucy, either.
Aggle, I'd like to understand:
"A combination of 12 notes taken 4 at a time is 495."
Here's the complete list of 29 4-note sets:
(From Straus Intro to Post-Tonal Theory)
0123
0124
0125
0126
0127
0134
0135
0136
0137 (all-interval)
0145
0146 (all-interval and 5m-related to 0137)
0147
0148
0156
0157
0158
0167
0235
0236
0237
0246
0247
0248
0257 stack of 4ths/5ths
0258 minor-7th flat five, or err, "half-diminished"
0268
0347
0358
0369 diminished
I like this ordering more than Forte's, because it is more intuitive.
Anyway, another example of an interesting power-residue sequence:
indices of 6^n mod 709:
708 221 488 442 362 1 164 663 268 583 98 222 249 385 142 176 197 489 368 96 652 319 359 443 16 470 48 606 160 363 83 397 586 418 526 2 695 589 29 317 379 165 122 540 630 580 654 664 328 237 685 691 201 269 460 119 148 381 132 584 137 304 432 618 611 99 508 639 139 39 125 223 69 208 504 102 262 250 661 538 536 600 321 386 559 343 648 53 533 143 413 93 571 167 22 177 553 549 366 458 411 198 58 204 306 422 153 490 105 681 475 340 436 369 13 602 517 353 361 97 196 358 159 525 378 653 200 131 610 124 261 320 532 21 410 152 435 360 377 260 434 346 347 444 522 290 108 429 348 17 218 323 465 483 445 471 448 174 689 51 523 49 60 113 240 542 291 607 498 72 636 564 109 161 180 274 620 46 430 364 408 634 625 314 349 84 295 388 212 243 18 398 698 66 391 62 219 587 206 679 288 632 324 419 33 279 627 425 466 527 676 643 613 374 484 3 247 326 557 194 446 696 183 561 284 657 472 590 185 234 42 115 449 30 308 574 441 582 175 318 605 417 316 579 690 380 617 38 101 599 52 166 457 421 339 352 524 123 151 345 428 482 50 541 563 45 313 242 61 631 424 373 193 656 114 581 578 598 351 481 241 655 480 567 510 568 543 665 394 35 333 511 292 329 494 650 146 569 608 238 286 439 191 544 499 686 155 704 546 666 73 692 454 669 86 395 637 202 258 272 641 36 565 270 265 281 593 334 110 461 79 55 255 512 162 120 659 11 216 293 181 149 492 77 501 330 275 382 231 401 297 495 621 133 487 267 141 651 47 585 28 629 684 147 431 138 503 535 647 570 365 305 474 516 158 609 409 433 107 464 688 239 635 619 624 211 390 287 626 612 556 283 41 440 315 100 338 427 312 192 350 509 332 145 190 545 85 640 592 254 215 500 296 140 683 646 157 687 389 40 311 189 214 156 213 126 301 595 127 705 244 224 227 468 302 547 19 70 277 415 596 667 399 209 187 404 128 74 699 505 477 170 706 693 67 103 520 406 245 455 392 263 26 336 225 670 63 251 672 529 228 87 220 662 384 95 469 396 588 539 236 118 303 638 207 537 342 92 548 203 680 601 357 130 20 259 289 322 173 112 71 273 633 387 65 678 278 642 325 560 233 573 416 37 420 344 44 372 597 566 34 649 438 703 668 271 280 54 10 76 400 266 628 534 515 463 210 282 426 144 253 645 188 594 467 414 403 169 405 335 528 94 117 91 129 111 677 572 371 702 75 462 644 168 90 701 700 80 614 23 6 81 506 56 375 178 674 615 478 256 485 554 299 24 171 513 4 550 451 7 707 163 248 367 15 82 694 121 327 459 136 507 68 660 558 412 552 57 104 12 195 199 531 376 521 217 447 59 497 179 407 294 697 205 32 675 246 182 184 307 604 616 456 150 562 423 577 479 393 493 285 154 453 257 264 78 658 491 230 486 27 502 473 106 623 555 337 331 591 682 310 300 226 276 186 476 519 25 671 383 235 341 356 172 64 232 43 437 9 514 252 402 116 370 89 5 673 298 450 14 135 551 530 496 31 603 576 452 229 622 309 518 355 8 88 134 575 354
mod 12: 0 5 8 10 2 1 8 3 4 7 2 6 9 1 10 8 5 9 8 0 4 7 11 11 4 2 0 6 4 3 11 1 10 10 10 2 11 1 5 5 7 9 2 0 6 4 6 4 4 9 1 7 9 5 4 11 4 9 0 8 5 4 0 6 11 3 4 3 7 3 5 7 9 4 0 6 10 10 1 10 8 0 9 2 7 7 0 5 5 11 5 9 7 11 10 9 1 9 6 2 3 6 10 0 6 2 9 10 9 9 7 4 4 9 1 2 1 5 1 1 4 10 3 9 6 5 8 11 10 4 9 8 4 9 2 8 3 0 5 8 2 10 11 0 6 2 0 9 0 5 2 11 9 3 1 3 4 6 5 3 7 1 0 5 0 2 3 7 6 0 0 0 1 5 0 10 8 10 10 4 0 10 1 2 1 0 7 4 8 3 6 2 2 6 7 2 3 11 2 7 0 8 0 11 9 3 3 5 10 11 4 7 1 2 4 3 7 2 5 2 2 0 3 9 8 9 4 2 5 6 6 7 5 6 8 10 9 6 7 6 5 9 4 3 6 8 5 2 5 11 4 10 1 1 3 4 8 3 7 9 8 2 2 1 11 9 1 2 1 7 4 1 1 8 6 5 2 10 3 1 1 7 0 3 6 4 3 5 10 11 9 7 4 5 2 2 2 5 8 10 10 7 11 4 7 2 11 8 6 6 1 8 10 9 2 11 1 10 6 8 5 0 1 6 1 5 5 10 2 5 7 7 3 8 6 0 11 11 0 5 1 5 0 5 9 6 11 10 3 5 9 3 9 1 7 3 9 3 11 9 4 5 0 3 11 6 11 7 11 6 5 5 6 0 2 9 1 1 11 8 4 11 11 7 0 7 6 11 2 0 4 7 5 8 3 4 2 7 0 0 2 5 8 1 10 5 1 4 4 2 11 8 8 8 11 10 1 3 5 4 11 9 10 0 9 6 1 7 7 9 4 8 11 0 2 7 7 10 1 7 8 7 3 5 7 8 8 2 3 1 9 2 10 9 7 7 4 10 5 11 8 11 2 0 9 10 3 11 0 1 0 3 4 2 0 11 1 0 0 11 8 10 3 2 3 9 6 8 8 11 8 1 9 10 8 7 1 10 5 4 11 9 9 3 5 6 2 6 1 8 5 9 8 1 0 8 8 0 9 2 10 1 6 7 8 7 4 6 10 4 4 2 4 6 11 7 6 6 6 0 1 9 8 6 11 6 7 1 9 11 0 10 9 7 9 3 5 8 11 6 3 6 8 0 6 5 4 8 2 11 6 9 2 8 3 10 2 3 10 4 5 2 11 0 3 9 4 10 7 7 11 7 8 7 3 10 10 1 3 3 4 3 8 0 6 4 0 9 8 0 3 7 3 4 5 1 3 11 5 11 11 6 1 1 8 3 6 2 4 7 4 4 0 6 10 3 1 11 9 1 9 10 9 5 0 6 10 11 2 6 3 10 5 10 11 3 1 7 3 10 10 0 10 0 6 8 3 1 11 11 7 5 8 4 4 4 7 5 9 10 0 6 8 10 5 5 1 10 6 2 3 11 2 4 7 3 0 8 1 10 9 2 7 8 4 2 11 6
It seems that there are more 037 cells (including all transpositions, inversions, etc.) than there ought to be by chance in this set of numbers.
Derived 12-tone row: C,F,Ab,Bb,D,Db,Eb,E,G,F#,A,B
Below is a schematic of combined canons with this series. The second voice comes in on second beat and changes notes every 2 beats. The third voice comes in on the third beat and changes every 3 beats, and so on--forever. Obviously, in the actual piece, voices would be selected from this array. The important thing here is showing how different transpositions of this series at different speeds all add up to triadic combinations of voices.
I'm hoping, fervently, that this aligns on your screen as it does on mine: you should be seeing many aligned unisons. This should make sense to the eye. If it doesn't, oh well.
2:F-Bb-Db-Eb-G-Gb-Db
3: Ab--Db--E---Gb---Bb--A---e---B---C---Eb---bb--D---F--------G
-4: Bb----Eb---Gb----Ab--C------B-----f#---C#----D---F------E-G-A
--5: D-------G------Bb---C--E-----------Eb---bb------F---F#-----A-----e---Ab---B---(C#)
----6: Db------Gb-------A-------B-------Eb-------D-----a----E-------F-----Ab----eb----G--Bb---C
-----7: Ab--------Db--------E---------F#-----Bb--------A----e----------B----C---Eb-------bb----D---F---G
-------8: Eb---------Ab---------B----------C#--------F------E----------B-----F#-------G--Bb--------F-----A--------C--D
--------9: E------------A-----------C------------D-------F#---------F-------c---------G----Ab-------B-----F#--Bb----C#-Eb
---------10: G-----------C--------------Eb-----------F----------A---------Ab----eb-------Bb---------B-D--A----------C#--E-F#
Just after glancing it I can not understand it too. May be if I stay and look at it longer I may find out what is being said here.
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smittims
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« Reply #2 on: 08:59:31, 09-05-2007 » |
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While music has always been the most important thing in my life I've always been allergic to maths,possibly because I have no aptitude for it. Even simple sums like £5.00 minus 3 x £1.17 make no sense to me. So I've always been dismayed by claims that music is essentially mathematical. To me music is about human feelings, and composing is no more about maths than writing a novel is.
I suppse this post will irritate some, in which case apologies and please resist the temptation to post a sarky counterblast,as I shan't read it. Sensible replies will be welcome.. .
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Ian Pace
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« Reply #3 on: 09:19:38, 09-05-2007 » |
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Well, aside from considerations of the abstract beauty of maths, it is an essential tool for understanding scientific phenomena. Human feelings at least in part are one such thing.
There are of course branches of music theory that make a fetish out of numerical relationships in a work of music for their own sake, in part as an attempt to lend music theory a level of prestige comparable to that of various types of science (though not that much science is actually like that). But various such relationships, distributions, or whatever, are means for producing certain types of expressive effects. Xenakis attempts to capture something of complexity of natural phenomena as well as one's sense of awe produced by watching such spectacles - to do that he sometimes tries to re-create some of these natural phenomena's scientific properties in musical terms. That requires an understanding of such things in mathematical terms.
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« Last Edit: 09:26:38, 09-05-2007 by Ian Pace »
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'These acts of keeping politics out of music, however, do not prevent musicology from being a political act . . .they assure that every apolitical act assumes a greater political immediacy' - Philip Bohlman, 'Musicology as a Political Act'
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time_is_now
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« Reply #4 on: 09:41:26, 09-05-2007 » |
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While music has always been the most important thing in my life I've always been allergic to maths,possibly because I have no aptitude for it. Even simple sums like £5.00 minus 3 x £1.17 make no sense to me. So I've always been dismayed by claims that music is essentially mathematical. To me music is about human feelings, and composing is no more about maths than writing a novel is.
I suppse this post will irritate some, in which case apologies and please resist the temptation to post a sarky counterblast,as I shan't read it. Sensible replies will be welcome.. .
I don't think music is essentially mathematical or essentially anything else, or at least, I don't know what it would mean to claim so. Some music is about human feelings, some isn't. As Ian says, maths is a useful tool. If you find it difficult to 'think mathematically', then inevitably it's going to be a less useful tool for you than for someone else. That's fine, isn't it? There's some music I don't get on with at all but I don't expect to understand everything, in music or in maths or, indeed, in life. |
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The city is a process which always veers away from the form envisaged and desired, ... whose revenge upon its architects and planners undoes every dream of mastery. It is [also] one of the sites where Dasein is assigned the impossible task of putting right what can never be put right. - Rob Lapsley
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martle
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« Reply #5 on: 10:02:28, 09-05-2007 » |
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It's surely undeniable that music has inherent mathematical structures and properties (acoustical ones, for a start). Whether one chooses to take advantage of them in writing music is another matter. Composers have been doing so, more or less explicitly, however, since the ancient Greeks.
Personally, I find some sorts of mathematical thinking very useful indeed in composing, particularly when exploring issues of structural proportion, or permuations of basic pitch and rhythmic materials. I'm never conscious of the resulting music as 'experssing' maths in any way, though. It's a tool, for me.
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Green. Always green.
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George Garnett
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« Reply #6 on: 10:13:59, 09-05-2007 » |
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A bit of translation is occasionally necessary though.
It was only very recently that a kindly Babel Fish took me to one side and explained that when musicians talk of 'irrational rhythms' they don't really mean 'irrational' in the mathematical sense. Scales fell from my eyes, or ears (I think it must have been the Babel moulting season, either that or I'm particularly tight-eared).
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« Last Edit: 10:16:45, 09-05-2007 by George Garnett »
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time_is_now
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« Reply #7 on: 10:29:28, 09-05-2007 » |
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(though not that much science is actually like that) Indeed! The interesting thing about the kinds of music that are often described as being closer to maths is that, as far as I can understand (and unlike Ian I'm not a trained mathematician), they're actually not very close at all to maths as practised by mathematicians. |
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The city is a process which always veers away from the form envisaged and desired, ... whose revenge upon its architects and planners undoes every dream of mastery. It is [also] one of the sites where Dasein is assigned the impossible task of putting right what can never be put right. - Rob Lapsley
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richard barrett
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« Reply #8 on: 10:30:33, 09-05-2007 » |
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when musicians talk of 'irrational rhythms' they don't really mean 'irrational' in the mathematical sense Indeed they would more properly be called "rational rhythms". Unless the use of the word "irrational" here is trying to say something else entirely, as in "irrational behaviour"... |
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George Garnett
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« Reply #9 on: 10:40:21, 09-05-2007 » |
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they're actually not very close at all to maths as practised by mathematicians.
Only to a very, very tiny and simple subset of 'mathematics'. |
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Bryn
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« Reply #10 on: 10:52:26, 09-05-2007 » |
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A bit of translation is occasionally necessary though.
It was only very recently that a kindly Babel Fish took me to one side and explained that when musicians talk of 'irrational rhythms' they don't really mean 'irrational' in the mathematical sense. Scales fell from my eyes, or ears (I think it must have been the Babel moulting season, either that or I'm particularly tight-eared).
http://en.wikipedia.org/wiki/Irrational_rhythm is to be rationalized in the near future, it seems. [Rationalized in the sense of "let's rationalize the steel industry", back in the '70s, and again, and again, ... , that is.  ] |
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richard barrett
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« Reply #11 on: 11:19:04, 09-05-2007 » |
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[flash=200,200][flash=200,200] [Rationalized in the sense of "let's rationalize the steel industry", back in the '70s, and again, and again, ... , that is. ] So I see. I really don't like this word "tuplet" though, for some reason. |
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time_is_now
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« Reply #12 on: 11:22:16, 09-05-2007 » |
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I really don't like this word "tuplet" though Me neither. In fact, I was talking to another composer about this a couple of weeks ago, mainly because I wanted to know where it came from. It seems it was popularised by Sibelius software, which would explain why I started hearing composers use it a lot around 5 years ago but never adopted it myself (never having seen Sibelius in action in my life). |
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The city is a process which always veers away from the form envisaged and desired, ... whose revenge upon its architects and planners undoes every dream of mastery. It is [also] one of the sites where Dasein is assigned the impossible task of putting right what can never be put right. - Rob Lapsley
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Ian Pace
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« Reply #13 on: 11:24:56, 09-05-2007 » |
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How would simply 'rhythmic ratio' be? Oh, no, that could apply to lots of other things. Hmmmm - can't think of a better one than 'tuplet'....
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'These acts of keeping politics out of music, however, do not prevent musicology from being a political act . . .they assure that every apolitical act assumes a greater political immediacy' - Philip Bohlman, 'Musicology as a Political Act'
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time_is_now
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« Reply #14 on: 11:27:01, 09-05-2007 » |
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How would simply 'rhythmic ratio' be? Oh, no, that could apply to lots of other things. Hmmmm - can't think of a better one than 'tuplet'....
Well, I don't really have a problem with 'irrational rhythm' myself, since everyone always seemed to know what it meant, but if we really want to replace it how about something like 'embedded value'? |
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The city is a process which always veers away from the form envisaged and desired, ... whose revenge upon its architects and planners undoes every dream of mastery. It is [also] one of the sites where Dasein is assigned the impossible task of putting right what can never be put right. - Rob Lapsley
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