Re: superparticular subdivision of 3^12:2^19 for Bachs 1723 WTC tuning instructi

--- In tuning-math@yahoogroups.com, "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?

--- In tuning-math@yahoogroups.com, "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

>> ! 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

>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

>>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

--- In tuning-math@yahoogroups.com, 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.