62edo: Difference between revisions
→Regular temperament properties: +demivalentine |
→Instruments: Insert music section after this, starting with Bryan Deister's ''microtonal improvisation in 62edo'' (2025) |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
{{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 | {{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 [[ | 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]]. | 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]]. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 62 factors into | Since 62 factors into 2 × 31, 62edo does not contain nontrivial subset edos other than [[2edo]] and 31edo. [[186edo]] and [[248edo]] are notable supersets. | ||
=== | === 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. | 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. | ||
The 11 & 62 temperament | The 11 & 62 temperament is called mabon, named so because its associated year length corresponds to an autumnal equinoctial year. 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 | The 15 & 62 temperament, corresponding to the leap day cycle, is [[demivalentine]] in the 13-limit. | ||
== Intervals == | == Intervals == | ||
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== Notation == | == Notation == | ||
=== Ups and downs notation === | === Ups and downs notation === | ||
62edo can be notated with | 62edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat. | ||
{{Sharpness-sharp4a}} | |||
[[Alternative symbols for ups and downs notation]] uses sharps and flats and quarter-tone accidentals combined with arrows, borrowed from extended [[Helmholtz–Ellis notation]]: | |||
{{Sharpness-sharp4}} | {{Sharpness-sharp4}} | ||
=== Sagittal notation === | === 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]]. | 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]]. | ||
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== Approximation to JI == | == Approximation to JI == | ||
=== Zeta peak index === | === Zeta peak index === | ||
{ | {{ZPI | ||
| zpi = 314 | |||
| steps = 61.9380472360525 | |||
| step size = 19.3741981471691 | |||
| tempered height = 6.262952 | |||
| pure height = 4.11259 | |||
| integral = 0.952068 | |||
| gap = 15.026453 | |||
| octave = 1201.20028512448 | |||
| consistent = 8 | |||
| distinct = 8 | |||
}} | |||
| 19.3741981471691 | |||
| 6.262952 | |||
| 0.952068 | |||
| 15.026453 | |||
| | |||
| 8 | |||
| 8 | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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|} | |} | ||
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Instruments == | |||
=== Lumatone === | |||
* [[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) | |||