m wtf is the "long hundred" thing doing here, pretty sure that's MMTM's doing
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|120}}
{{ED intro}}
== Theory ==
== Theory ==
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).
120edo shares the [[perfect fifth]] with 12edo, [[tempering out]] the [[Pythagorean comma]]. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]]. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.
120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].
The 120bdd val is a tuning for [[superpyth]] where 3/2 is tuned to exactly 710{{c}}. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament.
120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.
=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|120}}
{{Harmonics in equal|120}}
== Interval list ==
=== Subsets and supersets ===
{|class="wikitable"
120edo is the 10th highly composite edo and the 5th factorial edo (since {{nowrap|120 {{=}} 5!}} {{nowrap|{{=}} 1 × 2 × 3 × 4 × 5}}). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}.
|-
!#
!Cents
![[Ups and downs notation]] (fifth 7\12)
![[Ups and downs notation]] (fifth 71\120)
|-
|0
|0
|{{UDnote|step=0}}
|{{UDnote|fifth=71|step=0}}
|-
|1
|10
|{{UDnote|step=1}}
|{{UDnote|fifth=71|step=1}}
|-
|2
|20
|{{UDnote|step=2}}
|{{UDnote|fifth=71|step=2}}
|-
|3
|30
|{{UDnote|step=3}}
|{{UDnote|fifth=71|step=3}}
|-
|4
|40
|{{UDnote|step=4}}
|{{UDnote|fifth=71|step=4}}
|-
|5
|50
|{{UDnote|step=5}}
|{{UDnote|fifth=71|step=5}}
|-
|6
|60
|{{UDnote|step=6}}
|{{UDnote|fifth=71|step=6}}
|-
|7
|70
|{{UDnote|step=7}}
|{{UDnote|fifth=71|step=7}}
|-
|8
|80
|{{UDnote|step=8}}
|{{UDnote|fifth=71|step=8}}
|-
|9
|90
|{{UDnote|step=9}}
|{{UDnote|fifth=71|step=9}}
|-
|10
|100
|{{UDnote|step=10}}
|{{UDnote|fifth=71|step=10}}
|-
|11
|110
|{{UDnote|step=11}}
|{{UDnote|fifth=71|step=11}}
|-
|12
|120
|{{UDnote|step=12}}
|{{UDnote|fifth=71|step=12}}
|-
|13
|130
|{{UDnote|step=13}}
|{{UDnote|fifth=71|step=13}}
|-
|14
|140
|{{UDnote|step=14}}
|{{UDnote|fifth=71|step=14}}
|-
|15
|150
|{{UDnote|step=15}}
|{{UDnote|fifth=71|step=15}}
|-
|16
|160
|{{UDnote|step=16}}
|{{UDnote|fifth=71|step=16}}
|-
|17
|170
|{{UDnote|step=17}}
|{{UDnote|fifth=71|step=17}}
|-
|18
|180
|{{UDnote|step=18}}
|{{UDnote|fifth=71|step=18}}
|-
|19
|190
|{{UDnote|step=19}}
|{{UDnote|fifth=71|step=19}}
|-
|20
|200
|{{UDnote|step=20}}
|{{UDnote|fifth=71|step=20}}
|-
|21
|210
|{{UDnote|step=21}}
|{{UDnote|fifth=71|step=21}}
|-
|22
|220
|{{UDnote|step=22}}
|{{UDnote|fifth=71|step=22}}
|-
|23
|230
|{{UDnote|step=23}}
|{{UDnote|fifth=71|step=23}}
|-
|24
|240
|{{UDnote|step=24}}
|{{UDnote|fifth=71|step=24}}
|-
|25
|250
|{{UDnote|step=25}}
|{{UDnote|fifth=71|step=25}}
|-
|26
|260
|{{UDnote|step=26}}
|{{UDnote|fifth=71|step=26}}
|-
|27
|270
|{{UDnote|step=27}}
|{{UDnote|fifth=71|step=27}}
|-
|28
|280
|{{UDnote|step=28}}
|{{UDnote|fifth=71|step=28}}
|-
|29
|290
|{{UDnote|step=29}}
|{{UDnote|fifth=71|step=29}}
|-
|30
|300
|{{UDnote|step=30}}
|{{UDnote|fifth=71|step=30}}
|-
|31
|310
|{{UDnote|step=31}}
|{{UDnote|fifth=71|step=31}}
|-
|32
|320
|{{UDnote|step=32}}
|{{UDnote|fifth=71|step=32}}
|-
|33
|330
|{{UDnote|step=33}}
|{{UDnote|fifth=71|step=33}}
|-
|34
|340
|{{UDnote|step=34}}
|{{UDnote|fifth=71|step=34}}
|-
|35
|350
|{{UDnote|step=35}}
|{{UDnote|fifth=71|step=35}}
|-
|36
|360
|{{UDnote|step=36}}
|{{UDnote|fifth=71|step=36}}
|-
|37
|370
|{{UDnote|step=37}}
|{{UDnote|fifth=71|step=37}}
|-
|38
|380
|{{UDnote|step=38}}
|{{UDnote|fifth=71|step=38}}
|-
|39
|390
|{{UDnote|step=39}}
|{{UDnote|fifth=71|step=39}}
|-
|40
|400
|{{UDnote|step=40}}
|{{UDnote|fifth=71|step=40}}
|-
|41
|410
|{{UDnote|step=41}}
|{{UDnote|fifth=71|step=41}}
|-
|42
|420
|{{UDnote|step=42}}
|{{UDnote|fifth=71|step=42}}
|-
|43
|430
|{{UDnote|step=43}}
|{{UDnote|fifth=71|step=43}}
|-
|44
|440
|{{UDnote|step=44}}
|{{UDnote|fifth=71|step=44}}
|-
|45
|450
|{{UDnote|step=45}}
|{{UDnote|fifth=71|step=45}}
|-
|46
|460
|{{UDnote|step=46}}
|{{UDnote|fifth=71|step=46}}
|-
|47
|470
|{{UDnote|step=47}}
|{{UDnote|fifth=71|step=47}}
|-
|48
|480
|{{UDnote|step=48}}
|{{UDnote|fifth=71|step=48}}
|-
|49
|490
|{{UDnote|step=49}}
|{{UDnote|fifth=71|step=49}}
|-
|50
|500
|{{UDnote|step=50}}
|{{UDnote|fifth=71|step=50}}
|-
|51
|510
|{{UDnote|step=51}}
|{{UDnote|fifth=71|step=51}}
|-
|52
|520
|{{UDnote|step=52}}
|{{UDnote|fifth=71|step=52}}
|-
|53
|530
|{{UDnote|step=53}}
|{{UDnote|fifth=71|step=53}}
|-
|54
|540
|{{UDnote|step=54}}
|{{UDnote|fifth=71|step=54}}
|-
|55
|550
|{{UDnote|step=55}}
|{{UDnote|fifth=71|step=55}}
|-
|56
|560
|{{UDnote|step=56}}
|{{UDnote|fifth=71|step=56}}
|-
|57
|570
|{{UDnote|step=57}}
|{{UDnote|fifth=71|step=57}}
|-
|58
|580
|{{UDnote|step=58}}
|{{UDnote|fifth=71|step=58}}
|-
|59
|590
|{{UDnote|step=59}}
|{{UDnote|fifth=71|step=59}}
|-
|60
|600
|{{UDnote|step=60}}
|{{UDnote|fifth=71|step=60}}
|-
|61
|610
|{{UDnote|step=61}}
|{{UDnote|fifth=71|step=61}}
|-
|62
|620
|{{UDnote|step=62}}
|{{UDnote|fifth=71|step=62}}
|-
|63
|630
|{{UDnote|step=63}}
|{{UDnote|fifth=71|step=63}}
|-
|64
|640
|{{UDnote|step=64}}
|{{UDnote|fifth=71|step=64}}
|-
|65
|650
|{{UDnote|step=65}}
|{{UDnote|fifth=71|step=65}}
|-
|66
|660
|{{UDnote|step=66}}
|{{UDnote|fifth=71|step=66}}
|-
|67
|670
|{{UDnote|step=67}}
|{{UDnote|fifth=71|step=67}}
|-
|68
|680
|{{UDnote|step=68}}
|{{UDnote|fifth=71|step=68}}
|-
|69
|690
|{{UDnote|step=69}}
|{{UDnote|fifth=71|step=69}}
|-
|70
|700
|{{UDnote|step=70}}
|{{UDnote|fifth=71|step=70}}
|-
|71
|710
|{{UDnote|step=71}}
|{{UDnote|fifth=71|step=71}}
|-
|72
|720
|{{UDnote|step=72}}
|{{UDnote|fifth=71|step=72}}
|-
|73
|730
|{{UDnote|step=73}}
|{{UDnote|fifth=71|step=73}}
|-
|74
|740
|{{UDnote|step=74}}
|{{UDnote|fifth=71|step=74}}
|-
|75
|750
|{{UDnote|step=75}}
|{{UDnote|fifth=71|step=75}}
|-
|76
|760
|{{UDnote|step=76}}
|{{UDnote|fifth=71|step=76}}
|-
|77
|770
|{{UDnote|step=77}}
|{{UDnote|fifth=71|step=77}}
|-
|78
|780
|{{UDnote|step=78}}
|{{UDnote|fifth=71|step=78}}
|-
|79
|790
|{{UDnote|step=79}}
|{{UDnote|fifth=71|step=79}}
|-
|80
|800
|{{UDnote|step=80}}
|{{UDnote|fifth=71|step=80}}
|-
|81
|810
|{{UDnote|step=81}}
|{{UDnote|fifth=71|step=81}}
|-
|82
|820
|{{UDnote|step=82}}
|{{UDnote|fifth=71|step=82}}
|-
|83
|830
|{{UDnote|step=83}}
|{{UDnote|fifth=71|step=83}}
|-
|84
|840
|{{UDnote|step=84}}
|{{UDnote|fifth=71|step=84}}
|-
|85
|850
|{{UDnote|step=85}}
|{{UDnote|fifth=71|step=85}}
|-
|86
|860
|{{UDnote|step=86}}
|{{UDnote|fifth=71|step=86}}
|-
|87
|870
|{{UDnote|step=87}}
|{{UDnote|fifth=71|step=87}}
|-
|88
|880
|{{UDnote|step=88}}
|{{UDnote|fifth=71|step=88}}
|-
|89
|890
|{{UDnote|step=89}}
|{{UDnote|fifth=71|step=89}}
|-
|90
|900
|{{UDnote|step=90}}
|{{UDnote|fifth=71|step=90}}
|-
|91
|910
|{{UDnote|step=91}}
|{{UDnote|fifth=71|step=91}}
|-
|92
|920
|{{UDnote|step=92}}
|{{UDnote|fifth=71|step=92}}
|-
|93
|930
|{{UDnote|step=93}}
|{{UDnote|fifth=71|step=93}}
|-
|94
|940
|{{UDnote|step=94}}
|{{UDnote|fifth=71|step=94}}
|-
|95
|950
|{{UDnote|step=95}}
|{{UDnote|fifth=71|step=95}}
|-
|96
|960
|{{UDnote|step=96}}
|{{UDnote|fifth=71|step=96}}
|-
|97
|970
|{{UDnote|step=97}}
|{{UDnote|fifth=71|step=97}}
|-
|98
|980
|{{UDnote|step=98}}
|{{UDnote|fifth=71|step=98}}
|-
|99
|990
|{{UDnote|step=99}}
|{{UDnote|fifth=71|step=99}}
|-
|100
|1000
|{{UDnote|step=100}}
|{{UDnote|fifth=71|step=100}}
|-
|101
|1010
|{{UDnote|step=101}}
|{{UDnote|fifth=71|step=101}}
|-
|102
|1020
|{{UDnote|step=102}}
|{{UDnote|fifth=71|step=102}}
|-
|103
|1030
|{{UDnote|step=103}}
|{{UDnote|fifth=71|step=103}}
|-
|104
|1040
|{{UDnote|step=104}}
|{{UDnote|fifth=71|step=104}}
|-
|105
|1050
|{{UDnote|step=105}}
|{{UDnote|fifth=71|step=105}}
|-
|106
|1060
|{{UDnote|step=106}}
|{{UDnote|fifth=71|step=106}}
|-
|107
|1070
|{{UDnote|step=107}}
|{{UDnote|fifth=71|step=107}}
|-
|108
|1080
|{{UDnote|step=108}}
|{{UDnote|fifth=71|step=108}}
|-
|109
|1090
|{{UDnote|step=109}}
|{{UDnote|fifth=71|step=109}}
|-
|110
|1100
|{{UDnote|step=110}}
|{{UDnote|fifth=71|step=110}}
|-
|111
|1110
|{{UDnote|step=111}}
|{{UDnote|fifth=71|step=111}}
|-
|112
|1120
|{{UDnote|step=112}}
|{{UDnote|fifth=71|step=112}}
|-
|113
|1130
|{{UDnote|step=113}}
|{{UDnote|fifth=71|step=113}}
|-
|114
|1140
|{{UDnote|step=114}}
|{{UDnote|fifth=71|step=114}}
|-
|115
|1150
|{{UDnote|step=115}}
|{{UDnote|fifth=71|step=115}}
|-
|116
|1160
|{{UDnote|step=116}}
|{{UDnote|fifth=71|step=116}}
|-
|117
|1170
|{{UDnote|step=117}}
|{{UDnote|fifth=71|step=117}}
|-
|118
|1180
|{{UDnote|step=118}}
|{{UDnote|fifth=71|step=118}}
|-
|119
|1190
|{{UDnote|step=119}}
|{{UDnote|fifth=71|step=119}}
|}
=== Miscellaneous properties ===
=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].
120edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.
== JI approximation ==
{{Q-odd-limit intervals|120}}
== Intervals ==
{{Interval table}}
120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.
== Notation ==
=== Ups and downs notation ===
120edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals and Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
120 equal divisions of the octave (abbreviated 120edo or 120ed2), also called 120-tone equal temperament (120tet) or 120 equal temperament (120et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 120 equal parts of exactly 10 ¢ each. Each step represents a frequency ratio of 21/120, or the 120th root of 2.
120edo shares the perfect fifth with 12edo, tempering out the Pythagorean comma. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5's 11-limit, it tempers out 243/242. In the patent val 120edo is also a tuning for the 7-limit decoid temperament.
The 120bdd val is a tuning for superpyth where 3/2 is tuned to exactly 710 ¢. It may be used as a de facto dual fifth in newcome temperament.
120edo is the 10th highly composite edo and the 5th factorial edo (since 120 = 5!= 1 × 2 × 3 × 4 × 5). It has many subsets: 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60.
Miscellaneous properties
120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.
JI approximation
The following tables show how 15-odd-limit intervals are represented in 120edo. Prime harmonics are in bold; inconsistent intervals are in italics.
15-odd-limit intervals in 120edo (direct approximation, even if inconsistent)
