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add top 20 up to 1280edo and add a top 10 up to 311edo as i deemed that the best cutoff
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→Top 10: move list to top
 
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== Top 9 ==
== Top 10 ==
Below is a list of [[equal temperaments]] (that is, [[edo]]s with octave-tempering) that the [[zeta]] function considers to be tuned well relative to their size, up to [[311edo|311et]].
Below is a list of [[equal temperaments]] (that is, [[edo]]s with octave-tempering) that the [[zeta]] function considers to be tuned well relative to their size, up to [[311edo|311et]].


We can be confident in their notability because they appear in a wide range of odd-limits (19 thru 123) according to [[User:Godtone/optimal edo sequences|the <code>optimal_edo_sequence</code>s for those odd-limits]]. (You can calculate them yourself by using my [[User:Godtone#My_Python_3_code|copyleft Python 3 code]] (which needs no dependencies).)
We can be confident in their notability because the larger entries appear in a wide range of odd-limits (19 thru 123) according to [[User:Godtone/optimal edo sequences|the <code>optimal_edo_sequence</code>s for those odd-limits]]. (You can calculate them yourself by using my [[User:Godtone#My_Python_3_code|copyleft Python 3 code]] (which needs no dependencies).)


Specifically, an equal temperament is included if it does better than the 9th-best-scoring equal temperament so far*, and the below details both what is added to the sequence and the top 9 at any given point. These can mostly be thought of as corresponding to [[edo]]s with the exception of 39 equal temperament, because 39edo is close to a zeta valley so that it isn't meaningfully considered in-tune by zeta without the 3.8{{cent}} flat [[2/1|octave]] that zeta recommends.
Specifically, an equal temperament is included if it does better than the 10th-best-scoring equal temperament so far*, and the below details both what is added to the sequence and the top 9 at any given point. These can mostly be thought of as corresponding to [[edo]]s with the exception of 39 equal temperament, because 39edo is close to a zeta valley so that it isn't meaningfully considered in-tune by zeta without the 3.8{{cent}} flat [[2/1|octave]] that zeta recommends.


<nowiki>*</nowiki> For this reason, we start the list at 10et (corresponding to 10edo) because all previous equal temperaments' inclusion is unsurprising. (The first to be excluded is 18et, corresponding to [[18edo]] being close to a zeta valley.)
<nowiki>*</nowiki> For this reason, we start the list at 10et (corresponding to 10edo) because all previous equal temperaments' inclusion is unsurprising. (The first to be excluded is 18et, corresponding to [[18edo]] being close to a zeta valley.)


10 [10, 7, 9, 5, 8, 4, 6, 3, 2]
The resulting sequence is: {{EDOs| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 34, 36, 37, 38, 39, 41, 43, 46, 48, 50, 53, 56, 58, 60, 62, 63, 65, 68, 72, 77, 80, 84, 87, 94, 99, 103, 106, 111, 113, 118, 121, 125, 130, 140, 149, 152, 159, 161, 171, 183, 190, 193, 198, 212, 217, 224, 229, 239, 243, 255, 270, 282, 289, 301, 311 }}.


11 [10, 7, 9, 5, 8, 4, 11, 6, 3]
=== Top 10 placement data ===
[[10], 7, 9, 5, 8, 4, 6, 3, 2, 6]


12 [12, 10, 7, 9, 5, 8, 4, 11, 6]
[10, 7, 9, 5, 8, 4, [11], 6, 3, 2]


13 [12, 10, 7, 9, 5, 8, 13, 4, 11]
[10, 7, 9, 5, 8, 4, [11], 6, 3, [11]]


14 [12, 14, 10, 7, 9, 5, 8, 13, 4]
[[12], 10, 7, 9, 5, 8, 4, 11, 6, 3]


15 [12, 15, 14, 10, 7, 9, 5, 8, 13]
[12, 10, 7, 9, 5, 8, [13], 4, 11, 6]


16 [12, 15, 14, 10, 16, 7, 9, 5, 8]
[12, [14], 10, 7, 9, 5, 8, 13, 4, 11]


17 [12, 17, 15, 14, 10, 16, 7, 9, 5]
[12, [15], 14, 10, 7, 9, 5, 8, 13, 4]


19 [19, 12, 17, 15, 14, 10, 16, 7, 9]
[12, 15, 14, 10, [16], 7, 9, 5, 8, 13]


21 [19, 12, 17, 15, 14, 10, 16, 7, 21]
[12, [17], 15, 14, 10, 16, 7, 9, 5, 8]


22 [22, 19, 12, 17, 15, 14, 10, 16, 7]
[[19], 12, 17, 15, 14, 10, 16, 7, 9, 5]


24 [22, 19, 24, 12, 17, 15, 14, 10, 16]
[19, 12, 17, 15, 14, 10, 16, 7, [21], 9]


26 [22, 19, 24, 26, 12, 17, 15, 14, 10]
[[22], 19, 12, 17, 15, 14, 10, 16, 7, 21]


27 [27, 22, 19, 24, 26, 12, 17, 15, 14]
[22, 19, [24], 12, 17, 15, 14, 10, 16, 7]


29 [27, 22, 19, 24, 29, 26, 12, 17, 15]
[22, 19, 24, [26], 12, 17, 15, 14, 10, 16]


31 [31, 27, 22, 19, 24, 29, 26, 12, 17]
[[27], 22, 19, 24, 26, 12, 17, 15, 14, 10]


34 [31, 34, 27, 22, 19, 24, 29, 26, 12]
[27, 22, 19, 24, [29], 26, 12, 17, 15, 14]


36 [31, 34, 27, 22, 36, 19, 24, 29, 26]
[[31], 27, 22, 19, 24, 29, 26, 12, 17, 15]


38 [31, 34, 27, 22, 36, 19, 38, 24, 29]
[31, [34], 27, 22, 19, 24, 29, 26, 12, 17]


39 [31, 34, 27, 22, 36, 19, 38, 24, 39]
[31, 34, 27, 22, [36], 19, 24, 29, 26, 12]


41 [41, 31, 34, 27, 22, 36, 19, 38, 24]
[31, 34, 27, 22, 36, 19, 24, 29, 26, [37]]


43 [41, 31, 34, 43, 27, 22, 36, 19, 38]
[31, 34, 27, 22, 36, 19, [38], 24, 29, 26]


46 [41, 46, 31, 34, 43, 27, 22, 36, 19]
[31, 34, 27, 22, 36, 19, 38, 24, [39], 29]


50 [41, 46, 31, 34, 50, 43, 27, 22, 36]
[[41], 31, 34, 27, 22, 36, 19, 38, 24, 39]


53 [53, 41, 46, 31, 34, 50, 43, 27, 22]
[41, 31, 34, [43], 27, 22, 36, 19, 38, 24]


58 [53, 58, 41, 46, 31, 34, 50, 43, 27]
[41, [46], 31, 34, 43, 27, 22, 36, 19, 38]


60 [53, 58, 41, 46, 60, 31, 34, 50, 43]
[41, 46, 31, 34, 43, 27, 22, 36, 19, [48]]


62 [53, 58, 41, 46, 60, 31, 34, 50, 62]
[41, 46, 31, 34, [50], 43, 27, 22, 36, 19]


63 [53, 58, 41, 46, 60, 31, 63, 34, 50]
[[53], 41, 46, 31, 34, 50, 43, 27, 22, 36]


65 [53, 58, 65, 41, 46, 60, 31, 63, 34]
[53, 41, 46, 31, 34, 50, 43, 27, [56], 22]


68 [53, 58, 65, 68, 41, 46, 60, 31, 63]
[53, [58], 41, 46, 31, 34, 50, 43, 27, 56]


72 [72, 53, 58, 65, 68, 41, 46, 60, 31]
[53, 58, 41, 46, [60], 31, 34, 50, 43, 27]


77 [72, 53, 77, 58, 65, 68, 41, 46, 60]
[53, 58, 41, 46, 60, 31, 34, 50, [62], 43]


80 [72, 53, 77, 80, 58, 65, 68, 41, 46]
[53, 58, 41, 46, 60, 31, [63], 34, 50, 62]


84 [72, 53, 77, 84, 80, 58, 65, 68, 41]
[53, 58, [65], 41, 46, 60, 31, 63, 34, 50]


87 [72, 87, 53, 77, 84, 80, 58, 65, 68]
[53, 58, 65, [68], 41, 46, 60, 31, 63, 34]


94 [72, 87, 94, 53, 77, 84, 80, 58, 65]
[[72], 53, 58, 65, 68, 41, 46, 60, 31, 63]


99 [99, 72, 87, 94, 53, 77, 84, 80, 58]
[72, 53, [77], 58, 65, 68, 41, 46, 60, 31]


103 [99, 72, 87, 94, 103, 53, 77, 84, 80]
[72, 53, 77, [80], 58, 65, 68, 41, 46, 60]


106 [99, 72, 87, 94, 103, 53, 77, 106, 84]
[72, 53, 77, [84], 80, 58, 65, 68, 41, 46]


111 [99, 72, 111, 87, 94, 103, 53, 77, 106]
[72, [87], 53, 77, 84, 80, 58, 65, 68, 41]


113 [99, 72, 111, 87, 94, 103, 113, 53, 77]
[72, 87, [94], 53, 77, 84, 80, 58, 65, 68]


118 [118, 99, 72, 111, 87, 94, 103, 113, 53]
[[99], 72, 87, 94, 53, 77, 84, 80, 58, 65]


121 [118, 99, 72, 111, 87, 94, 121, 103, 113]
[99, 72, 87, 94, [103], 53, 77, 84, 80, 58]


125 [118, 99, 72, 111, 87, 94, 121, 125, 103]
[99, 72, 87, 94, 103, 53, 77, [106], 84, 80]


130 [130, 118, 99, 72, 111, 87, 94, 121, 125]
[99, 72, [111], 87, 94, 103, 53, 77, 106, 84]


140 [130, 140, 118, 99, 72, 111, 87, 94, 121]
[99, 72, 111, 87, 94, 103, [113], 53, 77, 106]


149 [130, 140, 118, 99, 72, 149, 111, 87, 94]
[[118], 99, 72, 111, 87, 94, 103, 113, 53, 77]


152 [152, 130, 140, 118, 99, 72, 149, 111, 87]
[118, 99, 72, 111, 87, 94, [121], 103, 113, 53]


159 [152, 130, 140, 118, 159, 99, 72, 149, 111]
[118, 99, 72, 111, 87, 94, 121, [125], 103, 113]


171 [171, 152, 130, 140, 118, 159, 99, 72, 149]
[[130], 118, 99, 72, 111, 87, 94, 121, 125, 103]


183 [171, 183, 152, 130, 140, 118, 159, 99, 72]
[130, [140], 118, 99, 72, 111, 87, 94, 121, 125]


190 [171, 183, 152, 190, 130, 140, 118, 159, 99]
[130, 140, 118, 99, 72, 111, [149], 87, 94, 121]


193 [171, 183, 152, 190, 130, 193, 140, 118, 159]
[[152], 130, 140, 118, 99, 72, 111, 149, 87, 94]


198 [171, 183, 152, 190, 130, 198, 193, 140, 118]
[152, 130, 140, 118, [159], 99, 72, 111, 149, 87]


217 [217, 171, 183, 152, 190, 130, 198, 193, 140]
[152, 130, 140, 118, 159, 99, 72, 111, 149, [161]]


224 [224, 217, 171, 183, 152, 190, 130, 198, 193]
[[171], 152, 130, 140, 118, 159, 99, 72, 111, 149]


239 [224, 217, 171, 183, 152, 190, 130, 198, 239]
[171, [183], 152, 130, 140, 118, 159, 99, 72, 111]


243 [224, 243, 217, 171, 183, 152, 190, 130, 198]
[171, 183, [190], 152, 130, 140, 118, 159, 99, 72]


255 [224, 243, 217, 171, 183, 152, 190, 130, 255]
[171, 183, 190, 152, 130, [193], 140, 118, 159, 99]


270 [270, 224, 243, 217, 171, 183, 152, 190, 130]
[171, 183, 190, 152, 130, [198], 193, 140, 118, 159]


282 [270, 224, 243, 217, 171, 282, 183, 152, 190]
[171, 183, 190, 152, 130, 198, 193, 140, 118, [212]]


289 [270, 224, 243, 217, 171, 282, 183, 289, 152]
[[217], 171, 183, 190, 152, 130, 198, 193, 140, 118]


301 [270, 224, 301, 243, 217, 171, 282, 183, 289]
[[224], 217, 171, 183, 190, 152, 130, 198, 193, 140]


311 [270, 311, 224, 301, 243, 217, 171, 282, 183]
[224, 217, 171, 183, 190, 152, 130, 198, 193, [229]]


The resulting sequence is: {{EDOs| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 34, 36, 38, 39, 41, 43, 46, 50, 53, 58, 60, 62, 63, 65, 68, 72, 77, 80, 84, 87, 94, 99, 103, 106, 111, 113, 118, 121, 125, 130, 140, 149, 152, 159, 171, 183, 190, 193, 198, 217, 224, 239, 243, 255, 270, 282, 289, 301, 311 }}.
[224, 217, 171, 183, 190, 152, 130, 198, 193, [239]]
 
[224, [243], 217, 171, 183, 190, 152, 130, 198, 193]
 
[224, 243, 217, 171, 183, 190, 152, 130, [255], 198]
 
[[270], 224, 243, 217, 171, 183, 190, 152, 130, 255]
 
[270, 224, 243, 217, 171, 183, [282], 190, 152, 130]
 
[270, 224, 243, 217, 171, 183, 282, [289], 190, 152]
 
[270, 224, [301], 243, 217, 171, 183, 282, 289, 190]
 
[270, [311], 224, 301, 243, 217, 171, 183, 282, 289]
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