54edo: Difference between revisions

Line 1:

== Theory ==

54edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a [[No-threes subgroup temperaments|no-fifth]] tuning system. Using the sharp fifth, it can be viewed as two [[ring number|rings]] of [[27edo]], which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]]. Using the flat fifth, it generates an ultrasoft [[diatonic scale]]. This 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]].

54edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a [[No-threes subgroup temperaments|no-fifth]] tuning system. Using the sharp fifth, it can be viewed as two [[ring number|rings]] of [[27edo]], which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]]. Using the flat fifth, it generates an ultrasoft [[diatonic scale]]. This scale is so [[soft]], with {{nowrap|L/s {{=}} 8/7}}, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]].

The [[patent val]] of this edo takes the same fifth as [[27edo]], but the [[mapping]] for harmonic 5 is different. It 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-limit using the [[pajarous]] mapping and to the 13-limit using the 54f val, falling neatly between the 7- and 13-limit [[Target tuning #Minimax tuning|minimax tunings]].

The [[patent val]] of this edo takes the same fifth as [[27edo]], but the [[mapping]] for harmonic 5 is different. It 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{{c}} 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-limit using the [[pajarous]] mapping and to the 13-limit using the 54f 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 [[hexe]] temperament, while the 54bdf val does higher limit [[muggles]] about as well as it can be tuned. However, even these best temperament interpretations of 54edo are quite high in [[badness]] compared to its immediate neighbours [[53edo|53-]] and [[55edo]], both of which are [[Historical temperaments|historically significant]] for different reasons, leaving it mostly unexplored so far.

Line 28:

{| class="wikitable"

|+Table of intervals

|+ style="font-size: 105%;" | Table of intervals in 54edo

! Degree

|-

! Cents

! rowspan="2" | Degree

! [[Ups and downs notation~~|Ups and Downs Notation~~]]~~<br>~~(~~Flat Fifth~~ 31\54)

! rowspan="2" | Cents

! ~~[[Ups and downs notation|Ups and Downs Notation]]<br>~~(~~Sharp Fifth~~ 16\27)

! colspan="2" | [[Ups and downs notation]]

|-

! Flat fifth (31\54)

! Sharp fifth (16\27)

|-

| 0

Line 311:

Line 314:

== Notation ==

===Sagittal notation===

=== Ups and downs notation ===

Using [[Helmholtz–Ellis]] accidentals, 54edo can also be notated using [[ups and downs notation]]:

Here, a sharp raises by eight steps, and a flat lowers by eight steps, so single, double, and triple arrows along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals can be used to fill in the gap.

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[61edo#Sagittal notation|61-EDO]], and is a superset of the notation for [[27edo#Sagittal notation|27-EDO]].

~~====Evo flavor====~~

==== Evo flavor ====

File:54-EDO_Evo_Sagittal.svg

Line 327:

Line 335:

</imagemap>

====Revo flavor====

==== Revo flavor ====

File:54-EDO_Revo_Sagittal.svg

Line 341:

Line 348:

</imagemap>

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

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

File:54-EDO_Evo-SZ_Sagittal.svg

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|54}}
+{{ED intro}}
 == Theory ==
-edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a [[No-threes subgroup temperaments|no-fifth]] tuning system. Using the sharp fifth, it can be viewed as two [[ring number|rings]] of [[27edo]], which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]]. Using the flat fifth, it generates an ultrasoft [[diatonic scale]]. This 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]].
+edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a [[No-threes subgroup temperaments|no-fifth]] tuning system. Using the sharp fifth, it can be viewed as two [[ring number|rings]] of [[27edo]], which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]]. Using the flat fifth, it generates an ultrasoft [[diatonic scale]]. This scale is so [[soft]], with {{nowrap|L/s {{=}} 8/7}}, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]].
-The [[patent val]] of this edo takes the same fifth as [[27edo]], but the [[mapping]] for harmonic 5 is different. It 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-limit using the [[pajarous]] mapping and to the 13-limit using the 54f val, falling neatly between the 7- and 13-limit [[Target tuning #Minimax tuning|minimax tunings]].
+The [[patent val]] of this edo takes the same fifth as [[27edo]], but the [[mapping]] for harmonic 5 is different. It 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{{c}} 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-limit using the [[pajarous]] mapping and to the 13-limit using the 54f 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 [[hexe]] temperament, while the 54bdf val does higher limit [[muggles]] about as well as it can be tuned. However, even these best temperament interpretations of 54edo are quite high in [[badness]] compared to its immediate neighbours [[53edo|53-]] and [[55edo]], both of which are [[Historical temperaments|historically significant]] for different reasons, leaving it mostly unexplored so far.
@@ Line 28: / Line 28: @@
 {| class="wikitable"
-|+Table of intervals
+|+ style="font-size: 105%;" | Table of intervals in 54edo
-! Degree
+|-
-! Cents
+! rowspan="2" | Degree
-! [[Ups and downs notation|Ups and Downs Notation]]<br>(Flat Fifth 31\54)
+! rowspan="2" | Cents
-! [[Ups and downs notation|Ups and Downs Notation]]<br>(Sharp Fifth 16\27)
+! colspan="2" | [[Ups and downs notation]]
+|-
+! Flat fifth (31\54)
+! Sharp fifth (16\27)
 |-
 | 0
@@ Line 311: / Line 314: @@
 == Notation ==
-===Sagittal notation===
+=== Ups and downs notation ===
+Using [[Helmholtz–Ellis]] accidentals, 54edo can also be notated using [[ups and downs notation]]:
+{{Sharpness-sharp8}}
+Here, a sharp raises by eight steps, and a flat lowers by eight steps, so single, double, and triple arrows along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals can be used to fill in the gap.
+=== Sagittal notation ===
 This notation uses the same sagittal sequence as [[61edo#Sagittal notation|61-EDO]], and is a superset of the notation for [[27edo#Sagittal notation|27-EDO]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:54-EDO_Evo_Sagittal.svg
@@ Line 327: / Line 335: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:54-EDO_Revo_Sagittal.svg
@@ Line 341: / Line 348: @@
 </imagemap>
-====Evo-SZ flavor====
+==== Evo-SZ flavor ====
 <imagemap>
 File:54-EDO_Evo-SZ_Sagittal.svg