54edo: Difference between revisions

{{EDO intro|54}}

54edo is suitable for usage with [[dual-fifth tuning]] systems, or alternately, no-fifth tuning systems. 54edo has an ultrahard diatonic scale using the sharp fifth of [[27edo]] and an ultrasoft diatonic using the flat fifth. The soft diatonic scale is so soft, with L/s = 8/7, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]].

It's a rare temperament which adds better approximations of the 11th and 15th harmonics from [[27edo]], which it doubles. 54edo contains an alternate (flat) mapping of the fifth and an "extreme bayati" 6 6 10 10 2 10 10 diatonic scale.

Using the patent val, 54edo tempers out [[2048/2025]] in the 5-limit, making it a [[diaschismic]] system. It is the highest [[EDO]] in which the best mappings of the major 3rd ([[5/4]]) and harmonic 7th ([[7/4]]), 17\54 and 44\54, are exactly 600 cents apart, making them suitable for harmonies using tritone substitutions. In other words, this is the last EDO tempering out [[50/49]]. This means it extends quite simply to the 7 and 11 limits using the [[pajarous]] mapping and to the 13-limit using the f val, falling neatly between the 7 and 13 limit [[Target_tuning#Minimax_tuning|minimax]] tunings. The 54cd val makes for an excellent tuning of 7-limit [[Augmented_family#Hexe|hexe temperament]], while the bdf val does higher limit [[Magic family #Muggles|muggles]] about as well as it can be tuned. However, even these best temperament interpretations are quite high in badness compared to it's immediate neighbours 53 & 55edo, both of which are historically significant for different reasons, leaving it mostly unexplored so far.

===Odd harmonics===

Using the sharp fifth as a generator, 54edo require an incredibly large amount of ups and downs to notate, and using the flat fifth as a generator, 54edo requires an incredibly large amount of sharps and flats to notate. Because the flat fifth generates a diatonic scale with a chroma of 1 step, ups and downs are not needed in notation if the flat fifth is used.

{| class="wikitable mw-collapsible mw-collapsed"

|+Table of intervals

![[Ups and downs notation]] (flat fifth 31\54)

![[Ups and downs notation]] (sharp fifth 16\27)

|0

|0.000

|{{UDnote|fifth=31|step=0}}

|{{UDnote|fifth=31|step=1}}

|2

|44.444

|{{UDnote|fifth=31|step=2}}

|{{UDnote|step=2}}

|3

|66.667

|{{UDnote|fifth=31|step=3}}

|{{UDnote|step=3}}

|4

|88.889

|{{UDnote|fifth=31|step=4}}

|{{UDnote|step=4}}

|5

|111.111

|{{UDnote|fifth=31|step=5}}

|{{UDnote|step=5}}

|6

|133.333

|{{UDnote|fifth=31|step=6}}

|{{UDnote|step=6}}

|7

|155.556

|{{UDnote|fifth=31|step=7}}

|{{UDnote|step=7}}

|8

|177.778

|{{UDnote|fifth=31|step=8}}

|{{UDnote|step=8}}

|9

|200.000

|{{UDnote|fifth=31|step=9}}

|{{UDnote|step=9}}

|10

|222.222

|{{UDnote|fifth=31|step=10}}

|{{UDnote|step=10}}

|11

|244.444

|{{UDnote|fifth=31|step=11}}

|{{UDnote|step=11}}

|12

|266.667

|{{UDnote|fifth=31|step=12}}

|{{UDnote|step=12}}

|13

|288.889

|{{UDnote|fifth=31|step=13}}

|{{UDnote|step=13}}

|14

|311.111

|{{UDnote|fifth=31|step=14}}

|{{UDnote|step=14}}

|15

|333.333

|{{UDnote|fifth=31|step=15}}

|{{UDnote|step=15}}

|16

|355.556

|{{UDnote|fifth=31|step=16}}

|{{UDnote|step=16}}

|17

|377.778

|{{UDnote|fifth=31|step=17}}

|{{UDnote|step=17}}

|18

|400.000

|{{UDnote|fifth=31|step=18}}

|{{UDnote|step=18}}

|19

|422.222

|{{UDnote|fifth=31|step=19}}

|{{UDnote|step=19}}

|20

|444.444

|{{UDnote|fifth=31|step=20}}

|{{UDnote|step=20}}

|21

|466.667

|{{UDnote|fifth=31|step=21}}

|{{UDnote|step=21}}

|22

|488.889

|{{UDnote|fifth=31|step=22}}

|{{UDnote|step=22}}

|23

|511.111

|{{UDnote|fifth=31|step=23}}

|{{UDnote|step=23}}

|24

|533.333

|{{UDnote|fifth=31|step=24}}

|{{UDnote|step=24}}

|25

|555.556

|{{UDnote|fifth=31|step=25}}

|{{UDnote|step=25}}

|26

|577.778

|{{UDnote|fifth=31|step=26}}

|{{UDnote|step=26}}

|27

|600.000

|{{UDnote|fifth=31|step=27}}

|{{UDnote|step=27}}

|28

|622.222

|{{UDnote|fifth=31|step=28}}

|{{UDnote|step=28}}

|29

|644.444

|{{UDnote|fifth=31|step=29}}

|{{UDnote|step=29}}

|30

|666.667

|{{UDnote|fifth=31|step=30}}

|{{UDnote|step=30}}

|31

|688.889

|{{UDnote|fifth=31|step=31}}

|{{UDnote|step=31}}

|32

|711.111

|{{UDnote|fifth=31|step=32}}

|{{UDnote|step=32}}

|33

|733.333

|{{UDnote|fifth=31|step=33}}

|{{UDnote|step=33}}

|34

|755.556

|{{UDnote|fifth=31|step=34}}

|{{UDnote|step=34}}

|35

|777.778

|{{UDnote|fifth=31|step=35}}

|{{UDnote|step=35}}

|36

|800.000

|{{UDnote|fifth=31|step=36}}

|{{UDnote|step=36}}

|37

|822.222

|{{UDnote|fifth=31|step=37}}

|{{UDnote|step=37}}

|38

|844.444

|{{UDnote|fifth=31|step=38}}

|{{UDnote|step=38}}

|39

|866.667

|{{UDnote|fifth=31|step=39}}

|{{UDnote|step=39}}

|40

|888.889

|{{UDnote|fifth=31|step=40}}

|{{UDnote|step=40}}

|41

|911.111

|{{UDnote|fifth=31|step=41}}

|{{UDnote|step=41}}

|42

|933.333

|{{UDnote|fifth=31|step=42}}

|{{UDnote|step=42}}

|43

|955.556

|{{UDnote|fifth=31|step=43}}

|{{UDnote|step=43}}

|44

|977.778

|{{UDnote|fifth=31|step=44}}

|{{UDnote|step=44}}

|45

|1000.000

|{{UDnote|fifth=31|step=45}}

|{{UDnote|step=45}}

|46

|1022.222

|{{UDnote|fifth=31|step=46}}

|{{UDnote|step=46}}

|47

|1044.444

|{{UDnote|fifth=31|step=47}}

|{{UDnote|step=47}}

|48

|1066.667

|{{UDnote|fifth=31|step=48}}

|{{UDnote|step=48}}

|49

|1088.889

|{{UDnote|fifth=31|step=49}}

|{{UDnote|step=49}}

|50

|1111.111

|{{UDnote|fifth=31|step=50}}

|{{UDnote|step=50}}

|51

|1133.333

|{{UDnote|fifth=31|step=51}}

|{{UDnote|step=51}}

|52

|1155.556

|{{UDnote|fifth=31|step=52}}

|{{UDnote|step=52}}

|53

|1177.778

|{{UDnote|fifth=31|step=53}}

|{{UDnote|step=53}}

|54

|1200.000

|{{UDnote|fifth=31|step=54}}

|{{UDnote|step=54}}