Atomic temperament

family name: atomic
period: 100 cents
generator: schisma (32805/32768) 0.00144 to 0.00145 cents flat in the
5-limit

5-limit

name: atomic
comma: Kirnberger atom |161 -84 -12>
mapping: [<12 19 28|, <0 1 -7|]
poptimal generator: 75/46032
TOP period: 99.9999960286 generator: 1.9551521919
MOS: 612, 624, 1236, 1848, 2460, 3072, 4296, 7980, 12276, 16572,
20868, 25164, 46032

7-limit

name: atomic
wedgie: <<12 -84 -192 -161 -338 -210||
mapping: [<12 19 28 34|, <0 1 -7 -16|]
copoptimal generator: 13/8004
TOP period: 99.9993850271 generator: 1.9490287082
TM basis: |-4 6 -6 3>, |-55 30 2 1>
MOS: 612, 624, 1236, 1848, 2460, 4308, 6156, 8004

11-limit

name: atomic
wedgie: <<12 -84 -192 -300 -161 -338 -517 -210 -406 -178||
mapping: [<12 19 28 34 42|, <0 1 -7 -16 -25|]
poptimal generator: 5/3084
TOP period: 99.9992719361 generator: 1.9478879037
TM basis: {9801/9800, 151263/151250, 184549376/184528125}
MOS: 612, 624, 1236, 1848, 3084

note: atomic is compatible with 12-et, and ultra-accurate in the
5-limit, making it useful for analyzing 12-note temperaments. The
46032 division divides the Pythagorean comma into 900 parts and the
Didymus comma (81/80) into 825 parts, giving therefore 75 parts to the
schisma. It assigns the same number to the flatness of the 12-et
fifth, and is the temperament which equates the schisma with the
flattening of the 12-et fifth; whose near-identity was noted by Ellis
and implicitly used by Kirnberger. In higher limits, atomic is not
nearly so accurate but useful in connection with devising notation
schemes compatible with 12-et.

Gene Ward Smith wrote:

> 11-limit > > name: atomic
> wedgie: <<12 -84 -192 -300 -161 -338 -517 -210 -406 -178||
> mapping: [<12 19 28 34 42|, <0 1 -7 -16 -25|]
> poptimal generator: 5/3084
> TOP period: 99.9992719361 generator: 1.9478879037
> TM basis: {9801/9800, 151263/151250, 184549376/184528125}
> MOS: 612, 624, 1236, 1848, 3084

Oh, that looks like Hans Aberg's:

http://tinyurl.com/5ouj4

"""
So one might look for a two-dimensional keyboard having pitches 2^{k/12}
3^l. Looking for such pairs (k, l), I found:
Partial (k, l) Offset in cents
5 ~ (161, -7) 0.0013
7 ~ (338, -16) -0.11
11 ~ (517, -25) -0.19
13 ~ (-355, 21) 0.53
(615, -30) 0.82
17 ~ (981, -49) -0.75
(-8, 3) 0.91
Thus, on such a keyboard, one can play all the partials very accurately,
as well as all 12-equal music exactly. The question remains to find a good
keyboard layout, so that the keys one is interested in are grouped
together.
"""

Graham

>Gene Ward Smith wrote:
>
>> 11-limit
>>
>> name: atomic
>> wedgie: <<12 -84 -192 -300 -161 -338 -517 -210 -406 -178||
>> mapping: [<12 19 28 34 42|, <0 1 -7 -16 -25|]
>> poptimal generator: 5/3084
>> TOP period: 99.9992719361 generator: 1.9478879037
>> TM basis: {9801/9800, 151263/151250, 184549376/184528125}
>> MOS: 612, 624, 1236, 1848, 3084
>
>Oh, that looks like Hans Aberg's:
>
>http://tinyurl.com/5ouj4
>
>"""
>So one might look for a two-dimensional keyboard having pitches 2^{k/12}
>3^l. Looking for such pairs (k, l), I found:
>Partial (k, l) Offset in cents
> 5 ~ (161, -7) 0.0013
> 7 ~ (338, -16) -0.11
> 11 ~ (517, -25) -0.19
> 13 ~ (-355, 21) 0.53
> (615, -30) 0.82
> 17 ~ (981, -49) -0.75
> (-8, 3) 0.91
>Thus, on such a keyboard, one can play all the partials very accurately,
>as well as all 12-equal music exactly. The question remains to find a good
>keyboard layout, so that the keys one is interested in are grouped
>together.
>"""

Dude, how'd you find this?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

>
> family name: atomic
> period: 100 cents
> generator: schisma (32805/32768) 0.00144 to 0.00145 cents flat in the
> 5-limit
>
> <etc. -- snip>

awesome, Gene! thanks!

http://tonalsoft.com/enc/index2.htm?atomic.htm

please, keep these coming! please fill out these
data sheets for as many temperament families as you can,
and i'll include them all in the Encyclopaedia.

-monz