62edo: Difference between revisions

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{{Nowrap| 62 {{=}} 2 × 31 }} and the [[patent val]] of 62edo is a [[contorsion|contorted]] [[31edo]] through the [[11-limit]], but it makes for a good tuning in the higher limits. In the 13-limit it [[tempering out|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 size differences between consecutive harmonics are monotonically decreasing 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 [[gallium]], [[semivalentine]] and [[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]]. ~~(''See [[regular temperament]] for more about what all this means and how to use it.'')~~

It provides the [[optimal patent val]] for [[gallium]], [[semivalentine]] and [[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]].

=== Odd harmonics ===

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

=== ~~Miscellaneous properties~~ ===

=== Miscellany ===

62 years is the amount of years in a leap week calendar cycle which corresponds to a year of 365 days 5 hours 48 minutes 23 seconds, meaning it is both a simple cycle for a calendar, and 62 being a multiple of 31 makes it a harmonically useful and playable cycle. The corresponding maximal evenness scales are 15 & 62 and 11 & 62.

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== Instruments ==

~~; Fretted instruments~~

* [[Skip fretting system 62 6 11]]

=== Lumatone ===

* [[Skip fretting system 62 9 11]]

* [[Lumatone mapping for 62edo]]

=== Skip fretting ===

'''[[Skip fretting]] system 62 6 11''' has strings tuned 11\62 apart, while frets are 6\62.

On a 4-string bass, here are your open strings:

0 11 22 33

A good supraminor 3rd is found on the 2nd string, 1st fret. A supermajor third is found on the open 3rd string. The major 6th can be found on the 4th string, 2nd fret.

5-string bass

51 0 11 22 33

This adds an interval of a major 7th (minus an 8ve) at the first string, 1st fret.

6-string guitar

0 11 22 33 44 55

”Major” 020131

7-string guitar

0 11 22 33 44 55 4

'''Skip fretting system 62 9 11''' is another 62edo skip fretting system. The 5th is on the 5th string. The major 3rd is on the 2nd string, 1st fret.

{{todo|add illustration|text=Base it off of the diagram from [[User:MisterShafXen/Skip fretting system 62 9 11]]}}

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/UerD0NqBbng ''microtonal improvisation in 62edo''] (2025)

@@ Line 5: / Line 5: @@
 {{Nowrap| 62 {{=}} 2 × 31 }} and the [[patent val]] of 62edo is a [[contorsion|contorted]] [[31edo]] through the [[11-limit]], but it makes for a good tuning in the higher limits. In the 13-limit it [[tempering out|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 size differences between consecutive harmonics are monotonically decreasing 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 [[gallium]], [[semivalentine]] and [[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]]. (''See [[regular temperament]] for more about what all this means and how to use it.'')
+It provides the [[optimal patent val]] for [[gallium]], [[semivalentine]] and [[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]].
 === Odd harmonics ===
@@ Line 13: / Line 15: @@
 Since 62 factors into 2 × 31, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets.
-=== Miscellaneous properties ===
+=== Miscellany ===
 years is the amount of years in a leap week calendar cycle which corresponds to a year of 365 days 5 hours 48 minutes 23 seconds, meaning it is both a simple cycle for a calendar, and 62 being a multiple of 31 makes it a harmonically useful and playable cycle. The corresponding maximal evenness scales are 15 & 62 and 11 & 62.
@@ Line 921: / Line 923: @@
 == Instruments ==
-; Fretted instruments
-* [[Skip fretting system 62 6 11]]
+=== Lumatone ===
-* [[Skip fretting system 62 9 11]]
+* [[Lumatone mapping for 62edo]]
+=== Skip fretting ===
+'''[[Skip fretting]] system 62 6 11''' has strings tuned 11\62 apart, while frets are 6\62.
+On a 4-string bass, here are your open strings:
+11 22 33
+A good supraminor 3rd is found on the 2nd string, 1st fret. A supermajor third is found on the open 3rd string. The major 6th can be found on the 4th string, 2nd fret.
+-string bass
+0 11 22 33
+This adds an interval of a major 7th (minus an 8ve) at the first string, 1st fret.
+-string guitar
+11 22 33 44 55
+”Major” 020131
+-string guitar
+11 22 33 44 55 4
+'''Skip fretting system 62 9 11''' is another 62edo skip fretting system. The 5th is on the 5th string. The major 3rd is on the 2nd string, 1st fret.
+{{todo|add illustration|text=Base it off of the diagram from [[User:MisterShafXen/Skip fretting system 62 9 11]]}}
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/UerD0NqBbng ''microtonal improvisation in 62edo''] (2025)