17-limit TOP damage

Successive improvements in 17-limit TOP damage:

3 5 7 11 13 17
----------------------------------
2 : 33.0 77.3 77.3 77.3 77.6 77.6
3 : 30.2 30.2 61.0 61.0 61.0 61.0
4 : 33.0 33.0 33.0 40.0 41.0 41.0
5 : 5.7 19.8 21.4 30.2 30.2 32.8
6 : 30.2 30.2 30.2 30.2 35.6 35.6
7 : 5.1 9.4 20.0 20.0 20.0 20.0
9 : 11.2 14.2 14.2 14.2 14.2 14.7
10 : 5.7 11.4 11.4 12.7 12.7 12.7
12 : 0.6 3.6 6.1 7.6 12.5 12.5
15 : 5.7 5.7 7.2 7.2 7.2 8.7
19 : 2.3 2.3 3.8 6.3 6.3 6.4
22 : 2.2 3.2 3.3 3.3 5.3 5.3
26 : 3.1 3.7 3.8 4.1 4.1 4.1
31 : 1.6 1.8 1.8 1.8 3.1 3.1
41 : 0.2 1.4 1.4 1.9 2.4 2.7
46 : 0.8 1.1 1.7 1.7 1.9 1.9
58 : 0.5 1.5 1.5 1.5 1.5 1.6
72 : 0.6 0.6 0.6 0.6 1.0 1.0

Successive improvements in 17-limit TOP badness
(damage * notes):

3 5 7 11 13 17
-------------------------------
2 : 33.0 77.3 77.3 77.3 77.6 77.6 = 155.2
7 : 5.1 9.4 20.0 20.0 20.0 20.0 = 140
9 : 11.2 14.2 14.2 14.2 14.2 14.7 = 132.3
10 : 5.7 11.4 11.4 12.7 12.7 12.7 = 127
19 : 2.3 2.3 3.8 6.3 6.3 6.4 = 121.6
22 : 2.2 3.2 3.3 3.3 5.3 5.3 = 116.6
26 : 3.1 3.7 3.8 4.1 4.1 4.1 = 106.6
31 : 1.6 1.8 1.8 1.8 3.1 3.1 = 96.1
46 : 0.8 1.1 1.7 1.7 1.9 1.9 = 87.4
72 : 0.6 0.6 0.6 0.6 1.0 1.0 = 72

My only gripe is that I thin 41 is being
underrated.

-Carl

Carl Lumma wrote:
> Successive improvements in 17-limit TOP damage:
> > 3 5 7 11 13 17
> ----------------------------------
> 2 : 33.0 77.3 77.3 77.3 77.6 77.6
> 3 : 30.2 30.2 61.0 61.0 61.0 61.0
> 4 : 33.0 33.0 33.0 40.0 41.0 41.0
> 5 : 5.7 19.8 21.4 30.2 30.2 32.8
> 6 : 30.2 30.2 30.2 30.2 35.6 35.6
> 7 : 5.1 9.4 20.0 20.0 20.0 20.0
> 9 : 11.2 14.2 14.2 14.2 14.2 14.7
> 10 : 5.7 11.4 11.4 12.7 12.7 12.7
> 12 : 0.6 3.6 6.1 7.6 12.5 12.5
> 15 : 5.7 5.7 7.2 7.2 7.2 8.7
> 19 : 2.3 2.3 3.8 6.3 6.3 6.4
> 22 : 2.2 3.2 3.3 3.3 5.3 5.3
> 26 : 3.1 3.7 3.8 4.1 4.1 4.1
> 31 : 1.6 1.8 1.8 1.8 3.1 3.1
> 41 : 0.2 1.4 1.4 1.9 2.4 2.7
> 46 : 0.8 1.1 1.7 1.7 1.9 1.9
> 58 : 0.5 1.5 1.5 1.5 1.5 1.6
> 72 : 0.6 0.6 0.6 0.6 1.0 1.0
> > Successive improvements in 17-limit TOP badness
> (damage * notes):
> > 3 5 7 11 13 17
> -------------------------------
> 2 : 33.0 77.3 77.3 77.3 77.6 77.6 = 155.2
> 7 : 5.1 9.4 20.0 20.0 20.0 20.0 = 140
> 9 : 11.2 14.2 14.2 14.2 14.2 14.7 = 132.3
> 10 : 5.7 11.4 11.4 12.7 12.7 12.7 = 127
> 19 : 2.3 2.3 3.8 6.3 6.3 6.4 = 121.6
> 22 : 2.2 3.2 3.3 3.3 5.3 5.3 = 116.6
> 26 : 3.1 3.7 3.8 4.1 4.1 4.1 = 106.6
> 31 : 1.6 1.8 1.8 1.8 3.1 3.1 = 96.1
> 46 : 0.8 1.1 1.7 1.7 1.9 1.9 = 87.4
> 72 : 0.6 0.6 0.6 0.6 1.0 1.0 = 72
> > My only gripe is that I thin 41 is being
> underrated.

Interesting that 9 and 10 come out so well, though. I guess that's because 7 isn't all that great for more than the 5-limit, but it's surprising that 9 and 10 are much better.

At 08:02 PM 5/29/2007, you wrote:
>Carl Lumma wrote:
>> Successive improvements in 17-limit TOP damage:
>>
>> 3 5 7 11 13 17
>> ----------------------------------
>> 2 : 33.0 77.3 77.3 77.3 77.6 77.6
>> 3 : 30.2 30.2 61.0 61.0 61.0 61.0
>> 4 : 33.0 33.0 33.0 40.0 41.0 41.0
>> 5 : 5.7 19.8 21.4 30.2 30.2 32.8
>> 6 : 30.2 30.2 30.2 30.2 35.6 35.6
>> 7 : 5.1 9.4 20.0 20.0 20.0 20.0
>> 9 : 11.2 14.2 14.2 14.2 14.2 14.7
>> 10 : 5.7 11.4 11.4 12.7 12.7 12.7
>> 12 : 0.6 3.6 6.1 7.6 12.5 12.5
>> 15 : 5.7 5.7 7.2 7.2 7.2 8.7
>> 19 : 2.3 2.3 3.8 6.3 6.3 6.4
>> 22 : 2.2 3.2 3.3 3.3 5.3 5.3
>> 26 : 3.1 3.7 3.8 4.1 4.1 4.1
>> 31 : 1.6 1.8 1.8 1.8 3.1 3.1
>> 41 : 0.2 1.4 1.4 1.9 2.4 2.7
>> 46 : 0.8 1.1 1.7 1.7 1.9 1.9
>> 58 : 0.5 1.5 1.5 1.5 1.5 1.6
>> 72 : 0.6 0.6 0.6 0.6 1.0 1.0
>>
>> Successive improvements in 17-limit TOP badness
>> (damage * notes):
>>
>> 3 5 7 11 13 17
>> -------------------------------
>> 2 : 33.0 77.3 77.3 77.3 77.6 77.6 = 155.2
>> 7 : 5.1 9.4 20.0 20.0 20.0 20.0 = 140
>> 9 : 11.2 14.2 14.2 14.2 14.2 14.7 = 132.3
>> 10 : 5.7 11.4 11.4 12.7 12.7 12.7 = 127
>> 19 : 2.3 2.3 3.8 6.3 6.3 6.4 = 121.6
>> 22 : 2.2 3.2 3.3 3.3 5.3 5.3 = 116.6
>> 26 : 3.1 3.7 3.8 4.1 4.1 4.1 = 106.6
>> 31 : 1.6 1.8 1.8 1.8 3.1 3.1 = 96.1
>> 46 : 0.8 1.1 1.7 1.7 1.9 1.9 = 87.4
>> 72 : 0.6 0.6 0.6 0.6 1.0 1.0 = 72
>>
>> My only gripe is that I thin 41 is being
>> underrated.
>
>Interesting that 9 and 10 come out so well, though. I guess that's
>because 7 isn't all that great for more than the 5-limit, but it's
>surprising that 9 and 10 are much better.

In my recent search of logflat-good ETs up to 1000 based
on rms tenny-weighted error of primes, 10 kept showing up
in the top 10. It's apparently a very good ET. Maybe
because it's a miracle MOS?

-Carl

Carl Lumma wrote:

> In my recent search of logflat-good ETs up to 1000 based
> on rms tenny-weighted error of primes, 10 kept showing up
> in the top 10. It's apparently a very good ET. Maybe
> because it's a miracle MOS?

The 10 note miracle MOS is close to being equally tempered: the ratio of large to small steps is around 7:5. Negri (19&29) gives a similar result. It comes down to 5 being a very good no-fives temperament and 10 filling in the gaps.

9 is involved with ennealimmal.

Graham

Carl Lumma wrote:

> Successive improvements in 17-limit TOP badness
> (damage * notes):
> > 3 5 7 11 13 17
> -------------------------------
> 2 : 33.0 77.3 77.3 77.3 77.6 77.6 = 155.2
> 7 : 5.1 9.4 20.0 20.0 20.0 20.0 = 140
> 9 : 11.2 14.2 14.2 14.2 14.2 14.7 = 132.3
> 10 : 5.7 11.4 11.4 12.7 12.7 12.7 = 127
> 19 : 2.3 2.3 3.8 6.3 6.3 6.4 = 121.6
> 22 : 2.2 3.2 3.3 3.3 5.3 5.3 = 116.6
> 26 : 3.1 3.7 3.8 4.1 4.1 4.1 = 106.6
> 31 : 1.6 1.8 1.8 1.8 3.1 3.1 = 96.1
> 46 : 0.8 1.1 1.7 1.7 1.9 1.9 = 87.4
> 72 : 0.6 0.6 0.6 0.6 1.0 1.0 = 72

I agree with this ordering for the mappings taking the nearest approximation to each prime interval. But my figures are different:

2 155.3
7 139.8
9 132.3
10 126.9
19 122.4
22 117.4
26 105.4
31 97.1
46 85.1
72 70.0

Maybe you have a rounding error. However, taking the best TOP-max mappings does make a difference. The only mapping in your list that changes is 22:

best TOP: <22, 35, 51, 62, 76, 82, 90]

nearest primes: <22, 35, 51, 62, 76, 81, 90]

The best TOP error is 116.7 cents/octave.

The list using the best mappings is:

2 155.3
3 135.9
5 127.3
8 119.7
12 103.2
27 102.9
29 102.2
31 97.1
46 85.1
72 70.0

and a comparison of the best (left) and nearest prime (right) mappings:

<2, 3, 5, 6, 7, 7, 8] <2, 3, 5, 6, 7, 7, 8]
<3, 5, 7, 9, 11, 11, 13] <3, 5, 7, 8, 10, 11, 12]
<5, 8, 12, 14, 18, 19, 21] <5, 8, 12, 14, 17, 19, 20]
<8, 13, 19, 23, 28, 30, 33] <8, 13, 19, 22, 28, 30, 33]
<12, 19, 28, 34, 42, 45, 49] <12, 19, 28, 34, 42, 44, 49]
<27, 43, 63, 76, 94, 100, 111] <27, 43, 63, 76, 93, 100, 110]
<29, 46, 67, 81, 100, 107, 118] <29, 46, 67, 81, 100, 107, 119]
<31, 49, 72, 87, 107, 115, 127] <31, 49, 72, 87, 107, 115, 127]
<46, 73, 107, 129, 159, 170, 188] <46, 73, 107, 129, 159, 170, 188]
<72, 114, 167, 202, 249, 266, 294] <72, 114, 167, 202, 249, 266, 294]

> My only gripe is that I thin 41 is being
> underrated.

No change there.

Graham