Minimal consistent EDOs: Difference between revisions

{{Idiosyncratic terms|The term ‘purely consistent’ was coined by [[User:ArrowHead294]] and does not so far see widespread use.}}

An [[edo]] ''N'' is ''[[consistent]]'' with respect to the [[Odd limit|''q''-odd-limit]] if the closest approximations of the odd harmonics of the q-odd-limit in that edo also give the closest approximations of all the differences between these odd harmonics. It is ''[[distinctly consistent]]'' if every one of those closest approximations is a distinct value, and ''purely consistent'' if its [[relative interval error|relative errors]] on odd harmonics up to and including ''q'' never exceed 25%. Below is a table of the smallest consistent, and the smallest distinctly consistent, edo for every odd number up to 135. Odd limits of {{nowrap|2^''n'' − 1}} are '''highlighted'''.

|+ style="font-size: 105%;" | Smallest consistent EDOs per odd limit

|-

! Odd<br>limit !! Smallest<br>consistent edo* !! Smallest distinctly<br>consistent edo !! Smallest purely<br>consistent edo* !! Smallest edo<br>consistent to<br>[[Consistency #Generalization|distance 2]]* !! Smallest edo<br>distinctly consistent<br>to distance 2

| 1 || [[1edo|1]]|| 1 || 1 || 1 || 1

|- style="font-weight: bold; background-color: #dddddd;"

| 3 || 1 || [[3edo|3]]|| [[2edo|2]]|| 2 || 3

|-

| 5 || [[3edo|3]]|| [[9edo|9]]|| 3 || 3 || [[12edo|12]]

|- style="font-weight: bold; background-color: #dddddd;"

| 7 || [[4edo|4]]|| [[27edo|27]]|| [[10edo|10]]|| [[31edo|31]]|| 31

|-

| 9 || [[5edo|5]]|| [[41edo|41]]|| 41 || 41 || 41

|-

| 11 || [[22edo|22]]|| [[58edo|58]]|| 41 || [[72edo|72]]|| 72

|-

| 13 || [[26edo|26]]|| [[87edo|87]]|| [[46edo|46]]|| [[270edo|270]]|| 270

|- style="font-weight: bold; background-color: #dddddd;"

| 15 || [[29edo|29]]|| [[111edo|111]]|| [[87edo|87]]|| [[494edo|494]]|| 494

|-

| 17 || [[58edo|58]]|| [[149edo|149]]|| [[311edo|311]]|| [[3395edo|3395]]|| 3395

|-

| 19 || [[80edo|80]]|| [[217edo|217]]|| 311 || [[8539edo|8539]]|| 8539

|-

| 21 || [[94edo|94]]|| [[282edo|282]]|| 311 || 8539 || 8539

|-

| 23 || 94 || 282 || 311 || [[16808edo|16808]]|| 16808

|-

| 25 || [[282edo|282]]|| [[388edo|388]]|| 311 || 16808 || 16808

|-

| 27 || 282 || 388 || 311 || 16808 || 16808

|-

| 29 || 282 || [[1323edo|1323]]|| 311 || 16808 || 16808

|- style="font-weight: bold; background-color: #dddddd;"

| 31 || [[311edo|311]]|| [[1600edo|1600]]|| 311 || 16808 || 16808

|-

| 33 || 311 || 1600 || 311 || 16808 || 16808

|-

| 35 || 311 || 1600 || 311 || 16808 || 16808

|-

| 37 || 311 || 1600 || 311 || [[324296edo|324296]]|| 324296

|-

| 39 || 311 || [[2554edo|2554]]|| 311 || [[2398629edo|2398629]]|| 2398629

|-

| 41 || 311 || 2554 || 311 || [[19164767edo|19164767]]|| 19164767

|-

| 43 || [[17461edo|17461]]|| 17461 || [[20567edo|20567]]|| [[19735901edo|19735901]]|| 19735901

|-

| 45 || 17461 || 17461 || 20567 || 19735901 || 19735901

|-

| 47 || [[20567edo|20567]]|| 20567 || 20567 || [[152797015edo|152797015]]|| 152797015

|-

| 49 || 20567 || 20567 || [[459944edo|459944]]|| ||  

|-

| 51 || 20567 || 20567 || 459944 ||  ||  

|-

| 53 || 20567 || 20567 || [[1705229edo|1705229]]|| ||  

|-

| 55 || 20567 || 20567 || 1705229 ||  ||  

|-

| 57 || 20567 || 20567 || 1705229 ||  ||  

|-

| 59 || [[253389edo|253389]]|| 253389 || [[3159811edo|3159811]]|| ||  

|-

| 61 || [[625534edo|625534]]|| 625534 || 3159811 ||  ||  

|- style="font-weight: bold; background-color: #dddddd;"

| 63 || 625534 || 625534 || 3159811 ||  ||  

|-

| 65 || 625534 || 625534 || 3159811 ||  ||  

|-

| 67 || 625534 || 625534 || [[7317929edo|7317929]]|| ||  

|-

| 69 || [[759630edo|759630]]|| 759630 || [[8595351edo|8595351]]|| ||  

|-

| 71 || 759630 || 759630 || 8595351 ||  ||  

|-

| 73 || 759630 || 759630 || [[27783092edo|27783092]]|| ||  

|-

| 75 || [[2157429edo|2157429]]|| 2157429 || [[34531581edo|34531581]]|| ||  

|-

| 77 || 2157429 || 2157429 || 34531581 ||  ||  

|-

| 79 || [[2901533edo|2901533]]|| 2901533 || [[50203972edo|50203972]]|| ||  

|-

| 81 || 2901533 || 2901533 || 50203972 ||  ||  

|-

| 83 || 2901533 || 2901533 || 50203972 ||  ||  

|-

| 85 || 2901533 || 2901533 || 50203972 ||  ||  

|-

| 87 || 2901533 || 2901533 || 50203972 ||  ||

| 89 || 2901533 || 2901533 || 50203972 ||  ||  

|-

| 91 || 2901533 || 2901533 || 50203972 ||  ||  

|-

| 93 || 2901533 || 2901533 || 50203972 ||  ||  

|-

| 95 || 2901533 || 2901533 || 50203972 ||  ||  

|-

| 97 || 2901533 || 2901533 || [[1297643131edo|1297643131]]|| ||  

|-

| 99 || 2901533 || 2901533 || 1297643131 ||  ||  

|-

| 101 || 2901533 || 2901533 || [[3888109922edo|3888109922]]|| ||  

|-

| 103 || 2901533 || 2901533 || 3888109922 ||  ||  

|-

| 105 || 2901533 || 2901533 || 3888109922 ||  ||  

|-

| 107 || 2901533 || 2901533 || [[13805152233edo|13805152233]]|| ||  

|-

| 109 || 2901533 || 2901533 || [[27218556026edo|27218556026]]|| ||  

|-

| 111 || 2901533 || 2901533 || 27218556026 ||  ||  

|-

| 113 || 2901533 || 2901533 || 27218556026 ||  ||  

|-

| 115 || 2901533 || 2901533 || 27218556026 ||  ||  

|-

| 117 || 2901533 || 2901533 || 27218556026 ||  ||  

|-

| 119 || 2901533 || 2901533 || [[42586208631edo|42586208631]]|| ||  

|-

| 121 || 2901533 || 2901533 || 42586208631 ||  ||  

|-

| 123 || 2901533 || 2901533 || 42586208631 ||  ||  

|-

| 125 || 2901533 || 2901533 || 42586208631 ||  ||  

|- style="font-weight: bold; background-color: #dddddd;"

| 127 || 2901533 || 2901533 || 42586208631 ||  ||  

|-

| 129 || 2901533 || 2901533 || 42586208631 ||  ||  

|-

| 131 || 2901533 || 2901533 || [[93678217813edo|93678217813]]** || ||  

|-

| 133 || [[70910024edo|70910024]]|| 70910024 || 93678217813 ||  ||  

|-

| 135 || 70910024 || 70910024 || 93678217813 ||  ||

<nowiki />* Apart from 0edo

The last entry, 70910024edo, is consistent up to the 135-odd-limit. The next edo is [[5407372813edo|5407372813]], reported to be consistent to the 155-odd-limit.