62edo: Difference between revisions

{{EDO intro|62}}

{{nowrap|62 {{=}} 2 × 31}} and the [[patent val]] of 62edo is a contorted [[31edo]] through the 11-limit, but it makes for a good tuning in the higher limits. In the 13-limit it tempers out [[169/168]], [[1188/1183]], [[847/845]] and [[676/675]]; in the 17-limit [[221/220]], [[273/272]], and [[289/288]]; in the 19-limit [[153/152]], [[171/170]], [[209/208]], [[286/285]], and [[361/360]]. Unlike 31edo, which has a sharp profile for primes [[13/1|13]], [[17/1|17]], [[19/1|19]] and [[23/1|23]], 62edo has a flat profile for these, as it removes the distinction of otonal and utonal [[superparticular]] pairs of the primes (e.g. 13/12 vs 14/13 for prime 13) by tempering out the corresponding [[square-particular]]s. This flat tendency extends to higher primes too, as the first prime harmonic that is tuned sharper than its [[5/4]] is its [[59/32]]. Interestingly, the relative size differences of consecutive harmonics are well preserved for all first 24 harmonics, and 62edo is one of the few meantone edos that achieve this, great for those who seek higher-limit [[meantone]] harmony.  

It provides the [[optimal patent val]] for [[31st-octave temperaments#Gallium|gallium]], [[Starling temperaments #Valentine|semivalentine]] and [[Meantone family #Hemimeantone|hemimeantone]] temperaments.  

Using the 35\62 generator, which leads to the {{val| 62 97 143 173 }} val, 62edo is also an excellent tuning for septimal [[mavila]] temperament; alternatively {{val| 62 97 143 172 }} [[support]]s [[hornbostel]].

Since 62 factors into {{factorization|62}}, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets.  

=== Miscellaneous properties ===

The 11 & 62 temperament in the 2.9.7 subgroup tempers out 44957696/43046721, and the three generators of 17\62 correspond to [[16/9]]. It is possible to extend this to the 11-limit with comma basis {896/891, 1331/1296}, where 17\62 is mapped to [[11/9]] and two of them make [[16/11]]. In addition, three generators make the patent val 9/8, which is also created by combining the flat patent val fifth from 31edo with the sharp 37\62 fifth.

The 15 & 62 temperament, corresponding to the leap day cycle, is an unnamed extension to [[valentine]] in the 13-limit.

| {{UDnote|step=10}}

=== Ups and downs notation ===

62edo can be notated with quarter-tone accidentals and [[Alternative symbols for ups and downs notation #Sharp-4|ups and downs]]. This can be done by combining sharps and flats with arrows borrowed from extended [[Helmholtz-Ellis notation]]:

{{Sharpness-sharp4}}

===Sagittal notation===

This notation uses the same sagittal sequence as EDOs [[69edo#Sagittal notation|69]] and [[76edo#Sagittal notation|76]], and is a superset of the notation for [[31edo#Sagittal notation|31-EDO]].

====Revo flavor====

 

====Evo-SZ flavor====

 

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

! colspan="2" | #

| 58.06

| 135.48

| 251.61

| 329.03

| 58.06

| 77.42

| 116.13

| 503.22

| 561.29<br>(19.35)

| 367.74<br>(19.35)

<nowiki>*</nowiki> [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

==Zeta properties==

===Zeta peak index===

{| class="wikitable"

|-

!Step size (cents)

|-

|[[314zpi]]

|19.3741981471691

|6.262952

|0.952068

|15.026453

|62edo

|1201.20028512448