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Grade A temperaments, 5-limit to 13-limit

🔗Herman Miller <hmiller@...>

1/1/2012 4:09:29 PM

The best temperaments with error < 20 and complexity < 100.

5-limit
1&2 father [<1 2 2|, <0 -1 1|]
3&4 dicot [<1 1 2|, <0 2 1|]
5&7 meantone [<1 2 4|, <0 -1 -4|]
2&10 srutal [<2 3 5|, <0 1 -2|]
4&15 hanson [<1 0 1|, <0 6 5|]
12&17 helmholtz [<1 2 -1|, <0 -1 8|]
16&18 vishnu [<2 4 5|, <0 -7 -3|]
25&31 luna [<1 4 2|, <0 -15 2|]
27&45 ennealimmal [<9 15 22|, <0 -2 -3|]
22&52 kwazy [<2 1 6|, <0 8 -5|]
53&188 monzismic [<1 2 10|, <0 -2 -37|]
26&145 senior [<1 11 19|, <0 -35 -62|]
71&84 pirate [<1 -6 0|, <0 49 15|]
7&381 raider [<1 -9 -26|, <0 37 99|]
12&576 atomic [<12 19 28|, <0 1 -7|]

7-limit
4&7 dicot [<1 1 2 2|, <0 2 1 3|]
4&5 beep [<1 2 3 3|, <0 -2 -3 -1|]
2d&5 father [<1 2 2 4|, <0 -1 1 -3|]
5&10 blacksmith [<5 8 12 14|, <0 0 -1 0|]
4&12 diminished [<4 6 9 11|, <0 1 1 1|]
5&7 dominant [<1 2 4 2|, <0 -1 -4 2|]
2&10 pajara [<2 3 5 6|, <0 1 -2 -2|]
12&19 meantone [<1 2 4 7|, <0 -1 -4 -10|]
19&27 sensi [<1 -1 -1 -2|, <0 7 9 13|]
19&22 magic [<1 0 2 -1|, <0 5 1 12|]
9&22 orwell [<1 0 3 1|, <0 7 -3 8|]
12&29 garibaldi [<1 2 -1 -3|, <0 -1 8 14|]
10&11 miracle [<1 1 3 3|, <0 6 -7 -2|]
19&53 catakleismic [<1 0 1 -3|, <0 6 5 22|]
6&31 hemiw�rschmidt [<1 -1 2 2|, <0 16 2 5|]
41&58 hemififths [<1 1 -5 -1|, <0 2 25 13|]
7&39 amity [<1 3 6 -2|, <0 -5 -13 17|]
19&80 parakleismic [<1 5 6 12|, <0 -13 -14 -35|]
27&45 ennealimmal [<9 15 22 26|, <0 -2 -3 -2|]
80&91 supermajor [<1 15 19 30|, <0 -37 -46 -75|]

11-limit
4e&7 (dicot) [<1 1 2 2 2|, <0 2 1 3 5|]
5&10e (blacksmith) [<5 8 12 14 17|, <0 0 -1 0 0|]
5&8d (father) [<1 2 2 4 5|, <0 -1 1 -3 -4|]
7&14c jamesbond [<7 11 16 19 24|, <0 0 0 1 0|]
9&12 august [<3 5 7 9 11|, <0 -1 0 -2 -2|]
7&12 domineering [<1 2 4 2 6|, <0 -1 -4 2 -6|]
12&24d duodecim [<12 19 28 34 42|, <0 0 0 0 -1|]
12&19 meanenneadecal [<1 2 4 7 6|, <0 -1 -4 -10 -6|]
5&12 dominant [<1 2 4 2 1|, <0 -1 -4 2 6|]
12&26 injera [<2 3 4 5 6|, <0 1 4 4 6|]
3&12 augene [<3 5 7 8 10|, <0 -1 0 2 2|]
8d&22 hedgehog [<2 4 6 7 8|, <0 -3 -5 -5 -4|]
7&8 porcupine [<1 2 3 2 4|, <0 -3 -5 6 -4|]
12&22 pajara [<2 3 5 6 8|, <0 1 -2 -2 -6|]
12&31 meantone [<1 2 4 7 11|, <0 -1 -4 -10 -18|]
9&22 orwell [<1 0 3 1 3|, <0 7 -3 8 2|]
31&58 myna [<1 -1 0 1 -3|, <0 10 9 7 25|]
10&31 miracle [<1 1 3 3 2|, <0 6 -7 -2 15|]
60e&240e compton*5 [<60 95 139 168 207|, <0 0 1 2 3|]
31&56 luna, hemithirds [<1 4 2 2 7|, <0 -15 2 5 -22|]
58&72 harry [<2 4 7 7 9|, <0 -6 -17 -10 -15|]
26&46 unidec [<2 5 8 5 6|, <0 -6 -11 2 3|]
72&80 octoid [<8 13 19 23 28|, <0 -3 -4 -5 -3|]

13-limit
7&14c septimal [<7 11 16 20 24 26|, <0 0 0 -1 0 0|]
10&20 [<10 16 23 28 35 37|, <0 0 0 0 -1 0|]
8d&12e (diminished) [<4 6 9 11 13 14|, <0 1 1 1 2 2|]
9&11 (mavila) [<1 2 1 1 3 1|, <0 -1 3 4 1 6|]
2&9 (mavila) [<1 2 1 1 3 5|, <0 -1 3 4 1 -3|]
5e&7 domineering [<1 2 4 2 6 5|, <0 -1 -4 2 -6 -3|]
9&21 august [<3 5 7 9 11 12|, <0 -1 0 -2 -2 -3|]
8d&9 progression [<1 1 2 2 3 3|, <0 5 3 7 4 6|]
15&30 [<15 24 35 42 52 56|, <0 0 0 0 0 -1|]
2f&10 (pajaric) [<2 3 5 6 7 8|, <0 1 -2 -2 0 -3|]
7d&15 opossum [<1 2 3 4 4 4|, <0 -3 -5 -9 -4 -2|]
9&10 negri [<1 2 2 3 4 4|, <0 -4 3 -2 -5 -3|]
1&9 negric [<1 2 2 3 3 4|, <0 -4 3 -2 4 -3|]
8d&22 hedgehog [<2 4 6 7 8 9|, <0 -3 -5 -5 -4 -6|]
7&8 porcupine [<1 2 3 2 4 4|, <0 -3 -5 6 -4 -2|]
30f&34e (keemun)*2 [<2 6 7 7 13 14|, <0 -6 -5 -3 -13 -14|]
22&44 [<22 35 51 62 76 82|, <0 0 0 0 0 -1|]
10&29 (negri) [<1 2 2 3 5 4|, <0 -4 3 -2 -15 -3|]
12f&31 meantone [<1 2 4 7 11 10|, <0 -1 -4 -10 -18 -15|]
31&62 gallium [<31 49 72 87 107 115|, <0 0 0 0 0 -1|]
46&73 sensus [<1 -1 -1 -2 -8 0|, <0 7 9 13 31 10|]
29&31 tritonic [<1 4 -3 -3 2 -5|, <0 -5 11 12 3 18|]
29&46 leapday [<1 2 11 9 8 7|, <0 -1 -21 -15 -11 -8|]
9&41 [<1 4 1 5 5 7|, <0 -11 6 -10 -7 -15|]
19e&41 witchcraft [<1 0 2 -1 -7 -2|, <0 5 1 12 33 18|]
31&58 myna [<1 -1 0 1 -3 5|, <0 10 9 7 25 -5|]
41&58 hemififths [<1 1 -5 -1 2 4|, <0 2 25 13 5 -1|]
58&77 alicorn [<1 2 3 4 6 5|, <0 -8 -13 -23 -49 -25|]
19&53 catakleismic [<1 0 1 -3 9 0|, <0 6 5 22 -21 14|]
41&72 manna [<1 1 3 3 2 0|, <0 6 -7 -2 15 38|]
29&58 mystery [<29 46 67 81 100 107|, <0 0 1 1 1 1|]
58&72 harry [<2 4 7 7 9 11|, <0 -6 -17 -10 -15 -26|]

🔗genewardsmith <genewardsmith@...>

1/1/2012 4:28:30 PM

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

You're still missing names. The first I checked was 9&41, which is septimin. If we could find out which ones haven't been cataloged already, it would be boon.

> The best temperaments with error < 20 and complexity < 100.
>
> 5-limit
> 1&2 father [<1 2 2|, <0 -1 1|]
> 3&4 dicot [<1 1 2|, <0 2 1|]
> 5&7 meantone [<1 2 4|, <0 -1 -4|]
> 2&10 srutal [<2 3 5|, <0 1 -2|]
> 4&15 hanson [<1 0 1|, <0 6 5|]
> 12&17 helmholtz [<1 2 -1|, <0 -1 8|]
> 16&18 vishnu [<2 4 5|, <0 -7 -3|]
> 25&31 luna [<1 4 2|, <0 -15 2|]
> 27&45 ennealimmal [<9 15 22|, <0 -2 -3|]
> 22&52 kwazy [<2 1 6|, <0 8 -5|]
> 53&188 monzismic [<1 2 10|, <0 -2 -37|]
> 26&145 senior [<1 11 19|, <0 -35 -62|]
> 71&84 pirate [<1 -6 0|, <0 49 15|]
> 7&381 raider [<1 -9 -26|, <0 37 99|]
> 12&576 atomic [<12 19 28|, <0 1 -7|]
>
> 7-limit
> 4&7 dicot [<1 1 2 2|, <0 2 1 3|]
> 4&5 beep [<1 2 3 3|, <0 -2 -3 -1|]
> 2d&5 father [<1 2 2 4|, <0 -1 1 -3|]
> 5&10 blacksmith [<5 8 12 14|, <0 0 -1 0|]
> 4&12 diminished [<4 6 9 11|, <0 1 1 1|]
> 5&7 dominant [<1 2 4 2|, <0 -1 -4 2|]
> 2&10 pajara [<2 3 5 6|, <0 1 -2 -2|]
> 12&19 meantone [<1 2 4 7|, <0 -1 -4 -10|]
> 19&27 sensi [<1 -1 -1 -2|, <0 7 9 13|]
> 19&22 magic [<1 0 2 -1|, <0 5 1 12|]
> 9&22 orwell [<1 0 3 1|, <0 7 -3 8|]
> 12&29 garibaldi [<1 2 -1 -3|, <0 -1 8 14|]
> 10&11 miracle [<1 1 3 3|, <0 6 -7 -2|]
> 19&53 catakleismic [<1 0 1 -3|, <0 6 5 22|]
> 6&31 hemiwürschmidt [<1 -1 2 2|, <0 16 2 5|]
> 41&58 hemififths [<1 1 -5 -1|, <0 2 25 13|]
> 7&39 amity [<1 3 6 -2|, <0 -5 -13 17|]
> 19&80 parakleismic [<1 5 6 12|, <0 -13 -14 -35|]
> 27&45 ennealimmal [<9 15 22 26|, <0 -2 -3 -2|]
> 80&91 supermajor [<1 15 19 30|, <0 -37 -46 -75|]
>
> 11-limit
> 4e&7 (dicot) [<1 1 2 2 2|, <0 2 1 3 5|]
> 5&10e (blacksmith) [<5 8 12 14 17|, <0 0 -1 0 0|]
> 5&8d (father) [<1 2 2 4 5|, <0 -1 1 -3 -4|]
> 7&14c jamesbond [<7 11 16 19 24|, <0 0 0 1 0|]
> 9&12 august [<3 5 7 9 11|, <0 -1 0 -2 -2|]
> 7&12 domineering [<1 2 4 2 6|, <0 -1 -4 2 -6|]
> 12&24d duodecim [<12 19 28 34 42|, <0 0 0 0 -1|]
> 12&19 meanenneadecal [<1 2 4 7 6|, <0 -1 -4 -10 -6|]
> 5&12 dominant [<1 2 4 2 1|, <0 -1 -4 2 6|]
> 12&26 injera [<2 3 4 5 6|, <0 1 4 4 6|]
> 3&12 augene [<3 5 7 8 10|, <0 -1 0 2 2|]
> 8d&22 hedgehog [<2 4 6 7 8|, <0 -3 -5 -5 -4|]
> 7&8 porcupine [<1 2 3 2 4|, <0 -3 -5 6 -4|]
> 12&22 pajara [<2 3 5 6 8|, <0 1 -2 -2 -6|]
> 12&31 meantone [<1 2 4 7 11|, <0 -1 -4 -10 -18|]
> 9&22 orwell [<1 0 3 1 3|, <0 7 -3 8 2|]
> 31&58 myna [<1 -1 0 1 -3|, <0 10 9 7 25|]
> 10&31 miracle [<1 1 3 3 2|, <0 6 -7 -2 15|]
> 60e&240e compton*5 [<60 95 139 168 207|, <0 0 1 2 3|]
> 31&56 luna, hemithirds [<1 4 2 2 7|, <0 -15 2 5 -22|]
> 58&72 harry [<2 4 7 7 9|, <0 -6 -17 -10 -15|]
> 26&46 unidec [<2 5 8 5 6|, <0 -6 -11 2 3|]
> 72&80 octoid [<8 13 19 23 28|, <0 -3 -4 -5 -3|]
>
> 13-limit
> 7&14c septimal [<7 11 16 20 24 26|, <0 0 0 -1 0 0|]
> 10&20 [<10 16 23 28 35 37|, <0 0 0 0 -1 0|]
> 8d&12e (diminished) [<4 6 9 11 13 14|, <0 1 1 1 2 2|]
> 9&11 (mavila) [<1 2 1 1 3 1|, <0 -1 3 4 1 6|]
> 2&9 (mavila) [<1 2 1 1 3 5|, <0 -1 3 4 1 -3|]
> 5e&7 domineering [<1 2 4 2 6 5|, <0 -1 -4 2 -6 -3|]
> 9&21 august [<3 5 7 9 11 12|, <0 -1 0 -2 -2 -3|]
> 8d&9 progression [<1 1 2 2 3 3|, <0 5 3 7 4 6|]
> 15&30 [<15 24 35 42 52 56|, <0 0 0 0 0 -1|]
> 2f&10 (pajaric) [<2 3 5 6 7 8|, <0 1 -2 -2 0 -3|]
> 7d&15 opossum [<1 2 3 4 4 4|, <0 -3 -5 -9 -4 -2|]
> 9&10 negri [<1 2 2 3 4 4|, <0 -4 3 -2 -5 -3|]
> 1&9 negric [<1 2 2 3 3 4|, <0 -4 3 -2 4 -3|]
> 8d&22 hedgehog [<2 4 6 7 8 9|, <0 -3 -5 -5 -4 -6|]
> 7&8 porcupine [<1 2 3 2 4 4|, <0 -3 -5 6 -4 -2|]
> 30f&34e (keemun)*2 [<2 6 7 7 13 14|, <0 -6 -5 -3 -13 -14|]
> 22&44 [<22 35 51 62 76 82|, <0 0 0 0 0 -1|]
> 10&29 (negri) [<1 2 2 3 5 4|, <0 -4 3 -2 -15 -3|]
> 12f&31 meantone [<1 2 4 7 11 10|, <0 -1 -4 -10 -18 -15|]
> 31&62 gallium [<31 49 72 87 107 115|, <0 0 0 0 0 -1|]
> 46&73 sensus [<1 -1 -1 -2 -8 0|, <0 7 9 13 31 10|]
> 29&31 tritonic [<1 4 -3 -3 2 -5|, <0 -5 11 12 3 18|]
> 29&46 leapday [<1 2 11 9 8 7|, <0 -1 -21 -15 -11 -8|]
> 9&41 [<1 4 1 5 5 7|, <0 -11 6 -10 -7 -15|]
> 19e&41 witchcraft [<1 0 2 -1 -7 -2|, <0 5 1 12 33 18|]
> 31&58 myna [<1 -1 0 1 -3 5|, <0 10 9 7 25 -5|]
> 41&58 hemififths [<1 1 -5 -1 2 4|, <0 2 25 13 5 -1|]
> 58&77 alicorn [<1 2 3 4 6 5|, <0 -8 -13 -23 -49 -25|]
> 19&53 catakleismic [<1 0 1 -3 9 0|, <0 6 5 22 -21 14|]
> 41&72 manna [<1 1 3 3 2 0|, <0 6 -7 -2 15 38|]
> 29&58 mystery [<29 46 67 81 100 107|, <0 0 1 1 1 1|]
> 58&72 harry [<2 4 7 7 9 11|, <0 -6 -17 -10 -15 -26|]
>

🔗genewardsmith <genewardsmith@...>

1/1/2012 4:29:44 PM

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> The best temperaments with error < 20 and complexity < 100.

I accidentially posted an article before I had finished the edit. I was going to ask, what error and complexity measure you were using?

🔗Herman Miller <hmiller@...>

1/1/2012 6:01:40 PM

On 1/1/2012 7:29 PM, genewardsmith wrote:
>
>
> --- In tuning@yahoogroups.com, Herman Miller<hmiller@...> wrote:
>>
>> The best temperaments with error< 20 and complexity< 100.
>
> I accidentially posted an article before I had finished the edit. I
> was going to ask, what error and complexity measure you were using?

For this list I used the same error and complexity measure as in Paul's Middle Path paper.