Topic: Re: superparticular subdivision of 3^12:2^19 for Bachs 1723 WTC tuning instructi
1 scales
| File | Description | Notes | Period (ยข) | Limit |
|---|---|---|---|---|
| sparschuch | Modified Collatz sequence well-temperament | 12 | 1200.0 | 157 |
Thread (6 messages)
From: Gene Ward Smith (2005-08-19) Subject: Re: superparticular subdivision of 3^12:2^19 for Bachs 1723 WTC tuning instructi --- In [email protected], "a_sparschuh" <a_sparschuh@y...> wrote: > dear members, > here comes an tuning-algorithm, > in math known as "Collatz-sequence" Did Collatz by any chance have music on his mind when he came up with his famous conjecture?
From: Gene Ward Smith (2005-08-24) Subject: Re: superparticular subdivision of 3^12:2^19 for Bachs 1723 WTC tuning instructi --- In [email protected], "a_sparschuh" <a_sparschuh@y...> wrote: > A:=420cps,210,105 "divide by 2 as long as possible until odd > *3=:E (315)314,157 "multyply by 3, then decrement that etc. > *3=:B (471)470,235 "real operation 470:=471*(470:471) > *3=:F# (705)704,352,176,88,44,22,11 "11Hz are difficult to hear > *3=:C# 33 "an 5th dinmished 32:33 would be to harsh > *3=:G# 99 "also an 99:98 wolf 5th > *3=:Eb 297 "same single Bach-squiggle as for C# and G# > *3=:Bb (891)890,445 "temper again > *3=:F (1335)1334,667 > *3=:C (2001)1000,500,250,125"1kHz inofficial normal pitch of acustics > *3=:G (375)374,187 > *3=:D (561)560,280,140,70,35 > *3=A 105 again, same as on start: therfore circular It had escaped my notice that this was a proposal to get synchronized beating, and it isn't clear to me what you are doing. Using 3n-1 is the Collatz sequence for negative numbers, which may go around in cycles, but you don't stick with that, since 3*11-1 is 32, which should come after 11. Collatz sequences starting at positive odd integers eventually go (conjecturally, and true for all values of interest to us) to one. Sometimes this ends up giving us a scale of twelve notes, for instance starting from 39, 79, 153 etc. This looks to be a nice rational well-temperament, and of course the beat ratios are rational, but I don't see that they are exceptionally interesting. Here it is as a Scala file; it strikes me as a very practical well-temperament which is worth trying out. ! sparschuch.scl Modified Collatz sequence well-temperament 12 ! 89/84 47/42 25/21 44/35 4/3 99/70 157/105 667/420 176/105 187/105 66/35 2
From: Carl Lumma (2005-08-26) Subject: Re: [tuning-math] Re: superparticular subdivision of 3^12:2^19 for Bachs 1723 WTC tuning instructi >> ! sparschuch.scl >> Modified Collatz sequence well-temperament >> 12 >> ! >> 89/84 >> 47/42 >> 25/21 >> 44/35 >> 4/3 >> 99/70 >> 157/105 >> 667/420 >> 176/105 >> 187/105 >> 66/35 >> 2 > >Thanky you for that compilation into scl. >That's very kind of you, but it's already done and on the list: >http://www.xs4all.nl/~huygensf/doc/scalesdir.txt >as >sparschuh.scl 12 Andreas Sparschuh WTC temperament, >1/1=C=250 That's funny -- Gene gave a different mode. I'm not sure what you're up to with beat rates but I wonder if this affects them? >Also my friend Tom, suggestet an improvement: >sparschuh2.scl 12 Modified Sparschuh temperament with >A=419Hz by Tom Dent > >in accordance to the algorithm dviding @ A: >by splitting: >(315:314)=(420:419)*(1257:1256) >adding an further tuning step. Why is this an improvement? -Carl
From: Carl Lumma (2005-08-26) Subject: Re: [tuning-math] Re: superparticular subdivision of 3^12:2^19 for Bachs 1723 WTC tuning instructi >Scepsis is advisible on their hypothesises they assume. >c.f. >http://em.oxfordjournals.org/cgi/content/full/33/3/547-a This requires a subscription. -Carl
From: Carl Lumma (2005-08-26) Subject: Re: [tuning-math] Re: superparticular subdivision of 3^12:2^19 for Bachs 1723 WTC tuning instructi >>Thanky you for that compilation into scl. >>That's very kind of you, but it's already done and on the list: >>http://www.xs4all.nl/~huygensf/doc/scalesdir.txt >>as >>sparschuh.scl 12 Andreas Sparschuh WTC temperament, >>1/1=C=250 > >That's funny -- Gene gave a different mode. I'm not sure what >you're up to with beat rates but I wonder if this affects them? It looks like you're trying to get the 5ths to beat a whole-number of times/second? Gene's file doesn't seem to do that, anyway... sparschuh.scl from the Scale Archive (ver. 50)... Base frequency : 250.0000 Hertz Beat frequencies of 5/4 3/2 0: 0.000: 6.0000 -2.0000 1: 94.332: 14.0000 -0.0000 2: 196.198: 8.0000 -0.0000 3: 298.242: 11.0000 -1.0000 4: 394.604: 14.0000 -2.0000 5: 498.910: 12.5000 -0.5000 6: 592.377: 20.0000 -0.0000 7: 697.332: 10.0000 -2.0000 8: 796.287: 20.0000 -0.0000 9: 898.153: 12.0000 -4.0000 10: 998.253: 15.0000 -1.0000 11: 1092.879: 26.0000 -2.0000 12: 1200.000: 12.0000 -4.0000 Total abs. beats : 168.5000 14.5000 Average abs. beats: 14.0417 1.2083 Highest abs. beats: 26.0000 4.0000 Modified Sparschuh temperament with A=419Hz by Tom Dent... Base frequency : 250.0000 Hertz Beat frequencies of 5/4 3/2 0: 0.000: 6.0000 -2.0000 1: 94.332: 14.0000 -0.0000 2: 196.198: 8.0000 -2.0000 3: 298.242: 11.0000 -1.0000 4: 394.604: 14.0000 -2.0000 5: 498.910: 8.5000 -0.5000 6: 592.377: 20.0000 -0.0000 7: 697.332: 10.0000 -2.0000 8: 796.287: 20.0000 -0.0000 9: 894.027: 17.0000 -1.0000 10: 998.253: 15.0000 -1.0000 11: 1092.879: 26.0000 -2.0000 12: 1200.000: 12.0000 -4.0000 Total abs. beats : 169.5000 13.5000 Average abs. beats: 14.1250 1.1250 Highest abs. beats: 26.0000 2.0000 Gene's .scl file... Base frequency : 250.0000 Hertz Beat frequencies of 5/4 3/2 0: 0.000: 7.1586 -2.3595 1: 100.100: 8.8933 -0.5758 2: 194.700: 15.5065 -1.1893 3: 301.800: 7.2258 -2.3460 4: 396.200: 16.6865 -0.0300 5: 498.000: 9.5446 0.0260 6: 600.100: 13.1052 -1.1971 7: 696.500: 16.5767 -2.4298 8: 800.800: 14.8316 -0.6569 9: 894.200: 23.8387 0.0327 10: 999.200: 11.8442 -2.4320 11: 1098.100: 23.7909 0.0368 12: 1200.000: 14.3172 -4.7190 Total abs. beats : 169.0025 13.3118 Average abs. beats: 14.0835 1.1093 Highest abs. beats: 23.8387 2.4320 | Base frequency : 261.9048 Hertz Beat frequencies of 5/4 3/2 0: 0.000: 7.4995 -2.4718 1: 100.100: 9.3168 -0.6033 2: 194.700: 16.2449 -1.2459 3: 301.800: 7.5699 -2.4577 4: 396.200: 17.4811 -0.0314 5: 498.000: 9.9991 0.0272 6: 600.100: 13.7293 -1.2542 7: 696.500: 17.3661 -2.5455 8: 800.800: 15.5378 -0.6881 9: 894.200: 24.9739 0.0342 10: 999.200: 12.4082 -2.5478 11: 1098.100: 24.9238 0.0385 12: 1200.000: 14.9990 -4.9437 Total abs. beats : 177.0502 13.9457 Average abs. beats: 14.7542 1.1621 Highest abs. beats: 24.9739 2.5478 -Carl
From: Gene Ward Smith (2005-08-26) Subject: Re: superparticular subdivision of 3^12:2^19 for Bachs 1723 WTC tuning instructi --- In [email protected], Carl Lumma <ekin@l...> wrote: > That's funny -- Gene gave a different mode. I'm not sure what > you're up to with beat rates but I wonder if this affects them? My mode was simply the one the original posting, starting from 105, suggested. In practice you might very well want a different one.