54edo - Revision history

Overthink: /* Theory */ fixed val

2025-11-21T16:54:12Z

Theory: fixed val

← Older revision		Revision as of 16:54, 21 November 2025
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	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 [[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. The overall best val of 54edo in the 17-limit is probably 54c, which preserves the 2.3.5.7.13 mapping of 27edo and corrects the 11th and 17th harmonics with a consistently sharp tendency.		The 54cdd 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. The overall best val of 54edo in the 17-limit is probably 54c, which preserves the 2.3.5.7.13 mapping of 27edo and corrects the 11th and 17th harmonics with a consistently sharp tendency.

	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.		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.

ArrowHead294: /* Ups and downs notation */ Functionality added to template

2025-09-25T16:16:34Z

Ups and downs notation: Functionality added to template

← Older revision		Revision as of 16:16, 25 September 2025
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	=== Ups and downs notation ===		=== Ups and downs notation ===
	54edo can be notated with [[ups and downs]], spoken as up, dup, trup, quup (or downquip), dudsharp, downsharp, sharp, upsharp, etc. and down, dud, trud, quud (or upquid), dupflat, etc. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and quupflat (quadruple-up flat) is equivalent to quid (quintuple-down).		54edo can be notated with [[ups and downs]], spoken as up, dup, trup, quup (or downquip), dudsharp, downsharp, sharp, upsharp, etc. and down, dud, trud, quud (or upquid), dupflat, etc. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and quupflat (quadruple-up flat) is equivalent to quid (quintuple-down).
	{{Ups and downs sharpness~~\|54~~}}		{{Ups and downs sharpness}}

	It can also be notated by borrowing [[Helmholtz–Ellis]] accidentals:		It can also be notated by borrowing [[Helmholtz–Ellis]] accidentals:

Tristanbay: /* Ups and downs notation */ Added non-HEJI/SZ ups and downs

2025-09-24T18:40:05Z

Ups and downs notation: Added non-HEJI/SZ ups and downs

← Older revision		Revision as of 18:40, 24 September 2025
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	== Notation ==		== Notation ==
	=== Ups and downs notation ===		=== Ups and downs notation ===
	~~Using~~ [[~~Helmholtz–Ellis~~]] ~~accidentals~~, ~~54edo~~ can also be notated ~~using~~ [[~~ups and downs notation~~]]:		54edo can be notated with [[ups and downs]], spoken as up, dup, trup, quup (or downquip), dudsharp, downsharp, sharp, upsharp, etc. and down, dud, trud, quud (or upquid), dupflat, etc. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and quupflat (quadruple-up flat) is equivalent to quid (quintuple-down).
			{{Ups and downs sharpness\|54}}

			It can also be notated by borrowing [[Helmholtz–Ellis]] accidentals:
	{{Sharpness-sharp8}}		{{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.		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.

BudjarnLambeth: /* Octave stretch or compression */

2025-09-23T03:22:45Z

Octave stretch or compression

@@ Line 359: / Line 359: @@
 edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[ed6|139ed6]], a [[Octave stretch|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1 and 19/1.
-If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[86edt]], [[ed5|126ed5]] and [[ed7|152ed7]] are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.
+If one prefers a ''[[Octave shrinking|compressed-octave]]'' tuning instead, [[zpi|264zpi]] is a good choice, improving upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.
 == Scales ==

Lériendil: /* Theory */

2025-09-11T07:35:42Z

Theory

← Older revision		Revision as of 07:35, 11 September 2025
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	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 [[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.		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. The overall best val of 54edo in the 17-limit is probably 54c, which preserves the 2.3.5.7.13 mapping of 27edo and corrects the 11th and 17th harmonics with a consistently sharp tendency.

			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.

	=== Odd harmonics ===		=== Odd harmonics ===

BudjarnLambeth: /* Scales */ Stretch compression

2025-09-02T03:10:51Z

Scales: Stretch compression

← Older revision		Revision as of 03:10, 2 September 2025
Line 353:		Line 353:

	In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma\|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.		In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma\|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

			== Octave stretch or compression ==
			54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[ed6\|139ed6]], a [[Octave stretch\|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1 and 19/1.

			If one prefers a ''[[Octave shrinking\|compressed-octave]]'' tuning instead, [[86edt]], [[ed5\|126ed5]] and [[ed7\|152ed7]] are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.

			What follows is a comparison of stretched- and compressed-octave 54edo tunings.

			; [[ed6\|139ed6]]
			* Octave size: 1205.08{{c}}
			Stretching the octave of 54edo by around 5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 10.15{{c}}. The tuning 139ed6 does this. So does the tuning 262zpi whose octave is identical to 139ed6 within 0.2{{c}}.
			{{Harmonics in equal\|139\|6\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 139ed6}}
			{{Harmonics in equal\|139\|6\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 139ed6 (continued)}}

			; [[ed7\|151ed7]]
			* Octave size: 1204.75{{c}}
			Stretching the octave of 54edo by around 4.5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 11.12{{c}}. The tuning 151ed7 does this.
			{{Harmonics in equal\|151\|7\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 151ed7}}
			{{Harmonics in equal\|151\|7\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 151ed7 (continued)}}

			; [[ed12\|193ed12]]
			* Octave size: 1203.66{{c}}
			Stretching the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 10.97{{c}}. The tuning 193ed12 does this.
			{{Harmonics in equal\|193\|12\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 193ed12}}
			{{Harmonics in equal\|193\|12\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 193ed12 (continued)}}

			; [[zpi\|263zpi]]
			* Step size: 22.243{{c}}, octave size: 1201.12{{c}}
			Stretching the octave of 54edo by around 1{{c}} results in an improved prime 5, but worse primes 2, 3, 7, 11 and 13. This approximates all harmonics up to 16 within 10.94{{c}}. The tuning 263zpi does this.
			{{Harmonics in cet\|22.243\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 263zpi}}
			{{Harmonics in cet\|22.243\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 263zpi (continued)}}

			; 54edo
			* Step size: 22.222{{c}}, octave size: 1200.00{{c}}
			Pure-octaves 54edo approximates all harmonics up to 16 within 9.16{{c}}.
			{{Harmonics in equal\|54\|2\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 54edo}}
			{{Harmonics in equal\|54\|2\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 54edo (continued)}}

			; [[WE\|54et, 13-limit WE tuning]]
			* Step size: 22.198{{c}}, octave size: 1198.69{{c}}
			Compressing the octave of 54edo by around 1.5{{c}} results in improved primes 3, 7, 11, 13, 17 and 19, but worse primes 2 and 5. This approximates all harmonics up to 16 within 10.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning\|187ed11]] whose octave is identical to 13lim WE within 0.1{{c}}.
			{{Harmonics in cet\|22.198\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 54et, 13-limit WE tuning}}
			{{Harmonics in cet\|22.198\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 54et, 13-limit WE tuning (continued)}}

			; [[zpi\|264zpi]]
			* Step size: 22.175{{c}}, octave size: 1197.45{{c}}
			Compressing the octave of 54edo by around 2.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.19{{c}}. The tuning 264zpi does this. So does the tuning 194ed12 whose octave is identical to 264zpi within 0.01{{c}}.
			{{Harmonics in cet\|22.175\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 264zpi}}
			{{Harmonics in cet\|22.175\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 264zpi (continued)}}

			; [[ed7\|152ed7]]
			* Octave size: 1196.82{{c}}
			Compressing the octave of 54edo by around 3{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.36{{c}}. The tuning 152ed7 does this.
			{{Harmonics in equal\|152\|7\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 152ed7}}
			{{Harmonics in equal\|152\|7\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 152ed7 (continued)}}

			; [[ed6\|140ed6]]
			* Octave size: 1196.47{{c}}
			Compressing the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.59{{c}}. The tuning 140ed6 does this.
			{{Harmonics in equal\|140\|6\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 140ed6}}
			{{Harmonics in equal\|140\|6\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 140ed6 (continued)}}

			; [[ed5\|126ed5]]
			* Octave size: 1194.13{{c}}
			Compressing the octave of 54edo by around 6{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20{{c}}. The tuning 126ed5 does this. So does the tuning [[86edt]] whose octave is identical to 126ed5 within 0.1{{c}}.
			{{Harmonics in equal\|126\|5\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 126ed5}}
			{{Harmonics in equal\|126\|5\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 126ed5 (continued)}}

	== Scales ==		== Scales ==

BudjarnLambeth: /* Octave stretch */

2025-09-02T03:10:20Z

Octave stretch

← Older revision		Revision as of 03:10, 2 September 2025
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	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|54}}		{{Harmonics in equal\|54}}

	~~=== Octave stretch ===~~
	54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[38ed5/3]], a [[Octave stretch\|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1.

	If one prefers a ''[[Octave shrinking\|compressed-octave]]'' tuning instead, [[86edt]], [[126ed5]] and [[152ed7]] are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.

	[[40ed5/3]] is another compressed octave option. It improves upon 54edo’s 3/1, 5/1, 11/1, 13/1, 17/1 and 19/1, at slight cost to the 2/1 and 7/1. Its 2/1 is the least accurate of all the tunings mentioned in this section, though still accurate enough that it has low [[harmonic entropy]].

	There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve 54edo’s approximation of JI: 262zpi, 263zpi, 264zpi and 265zpi. The main Zeta peak index page details all four tunings.

	=== Subsets and supersets ===		=== Subsets and supersets ===

BudjarnLambeth: /* Octave stretch */ temporary improvement until the roll out

2025-08-22T05:19:50Z

Octave stretch: temporary improvement until the roll out

← Older revision		Revision as of 05:19, 22 August 2025
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	=== Octave stretch ===		=== Octave stretch ===
	54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[~~Gallery of arithmetic pitch sequences#APS of mérides\|APS4~~/~~5méride~~]], a [[Octave stretch\|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1.		54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[38ed5/3]], a [[Octave stretch\|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1.

	If one prefers a ''[[Octave shrinking\|compressed-octave]]'' tuning instead, [[86edt]], [[126ed5]] and [[152ed7]] are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.		If one prefers a ''[[Octave shrinking\|compressed-octave]]'' tuning instead, [[86edt]], [[126ed5]] and [[152ed7]] are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.

	[[~~ed255~~/~~128#54ed255/128\|54ed255/128~~]] is another compressed octave option. It improves upon 54edo’s 3/1, 5/1, 11/1, 13/1, 17/1 and 19/1, at slight cost to the 2/1 and 7/1. Its 2/1 is the least accurate of all the tunings mentioned in this section, though still accurate enough that it has low [[harmonic entropy]].		[[40ed5/3]] is another compressed octave option. It improves upon 54edo’s 3/1, 5/1, 11/1, 13/1, 17/1 and 19/1, at slight cost to the 2/1 and 7/1. Its 2/1 is the least accurate of all the tunings mentioned in this section, though still accurate enough that it has low [[harmonic entropy]].

	There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve 54edo’s approximation of JI: 262zpi, 263zpi, 264zpi and 265zpi. The main Zeta peak index page details all four tunings.		There are also some nearby [[Zeta peak index]] (ZPI) tunings which can be used to improve 54edo’s approximation of JI: 262zpi, 263zpi, 264zpi and 265zpi. The main Zeta peak index page details all four tunings.

Lucius Chiaraviglio: /* Instruments */ Insert music section after this, starting with Bryan Deister's ''microtonal improvisation in 54edo'' (2025)

2025-05-08T08:28:46Z

Instruments: Insert music section after this, starting with Bryan Deister's ''microtonal improvisation in 54edo'' (2025)

← Older revision		Revision as of 08:28, 8 May 2025
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	[[Category:Todo:add rank 2 temperaments table]]		[[Category:Todo:add rank 2 temperaments table]]

			== Music ==

			; [[Bryan Deister]]
			* [https://www.youtube.com/shorts/Bi5-YQUQHek ''microtonal improvisation in 54edo''] (2025)

FloraC: No need to remind readers of what a regular temperament is everywhere

2025-04-11T11:16:19Z

No need to remind readers of what a regular temperament is everywhere

← Older revision		Revision as of 11:16, 11 April 2025
Line 4:		Line 4:
	== Theory ==		== 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 {{nowrap\|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]].

	~~=== RTT ===~~
	~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

	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 [[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]].

← Older revision		Revision as of 03:10, 2 September 2025
Line 353:		Line 353:

	In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma\|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.		In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma\|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

			== Octave stretch or compression ==
			54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by [[ed6\|139ed6]], a [[Octave stretch\|stretched-octave]] version of 54edo. The trade-off is a slightly worse 2/1 and 19/1.

			If one prefers a ''[[Octave shrinking\|compressed-octave]]'' tuning instead, [[86edt]], [[ed5\|126ed5]] and [[ed7\|152ed7]] are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.

			What follows is a comparison of stretched- and compressed-octave 54edo tunings.

			; [[ed6\|139ed6]]
			* Octave size: 1205.08{{c}}
			Stretching the octave of 54edo by around 5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 10.15{{c}}. The tuning 139ed6 does this. So does the tuning 262zpi whose octave is identical to 139ed6 within 0.2{{c}}.
			{{Harmonics in equal\|139\|6\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 139ed6}}
			{{Harmonics in equal\|139\|6\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 139ed6 (continued)}}

			; [[ed7\|151ed7]]
			* Octave size: 1204.75{{c}}
			Stretching the octave of 54edo by around 4.5{{c}} results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 11.12{{c}}. The tuning 151ed7 does this.
			{{Harmonics in equal\|151\|7\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 151ed7}}
			{{Harmonics in equal\|151\|7\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 151ed7 (continued)}}

			; [[ed12\|193ed12]]
			* Octave size: 1203.66{{c}}
			Stretching the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 10.97{{c}}. The tuning 193ed12 does this.
			{{Harmonics in equal\|193\|12\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 193ed12}}
			{{Harmonics in equal\|193\|12\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 193ed12 (continued)}}

			; [[zpi\|263zpi]]
			* Step size: 22.243{{c}}, octave size: 1201.12{{c}}
			Stretching the octave of 54edo by around 1{{c}} results in an improved prime 5, but worse primes 2, 3, 7, 11 and 13. This approximates all harmonics up to 16 within 10.94{{c}}. The tuning 263zpi does this.
			{{Harmonics in cet\|22.243\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 263zpi}}
			{{Harmonics in cet\|22.243\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 263zpi (continued)}}

			; 54edo
			* Step size: 22.222{{c}}, octave size: 1200.00{{c}}
			Pure-octaves 54edo approximates all harmonics up to 16 within 9.16{{c}}.
			{{Harmonics in equal\|54\|2\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 54edo}}
			{{Harmonics in equal\|54\|2\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 54edo (continued)}}

			; [[WE\|54et, 13-limit WE tuning]]
			* Step size: 22.198{{c}}, octave size: 1198.69{{c}}
			Compressing the octave of 54edo by around 1.5{{c}} results in improved primes 3, 7, 11, 13, 17 and 19, but worse primes 2 and 5. This approximates all harmonics up to 16 within 10.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does the tuning [[equal tuning\|187ed11]] whose octave is identical to 13lim WE within 0.1{{c}}.
			{{Harmonics in cet\|22.198\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 54et, 13-limit WE tuning}}
			{{Harmonics in cet\|22.198\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 54et, 13-limit WE tuning (continued)}}

			; [[zpi\|264zpi]]
			* Step size: 22.175{{c}}, octave size: 1197.45{{c}}
			Compressing the octave of 54edo by around 2.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.19{{c}}. The tuning 264zpi does this. So does the tuning 194ed12 whose octave is identical to 264zpi within 0.01{{c}}.
			{{Harmonics in cet\|22.175\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 264zpi}}
			{{Harmonics in cet\|22.175\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 264zpi (continued)}}

			; [[ed7\|152ed7]]
			* Octave size: 1196.82{{c}}
			Compressing the octave of 54edo by around 3{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.36{{c}}. The tuning 152ed7 does this.
			{{Harmonics in equal\|152\|7\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 152ed7}}
			{{Harmonics in equal\|152\|7\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 152ed7 (continued)}}

			; [[ed6\|140ed6]]
			* Octave size: 1196.47{{c}}
			Compressing the octave of 54edo by around 3.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.59{{c}}. The tuning 140ed6 does this.
			{{Harmonics in equal\|140\|6\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 140ed6}}
			{{Harmonics in equal\|140\|6\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 140ed6 (continued)}}

			; [[ed5\|126ed5]]
			* Octave size: 1194.13{{c}}
			Compressing the octave of 54edo by around 6{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20{{c}}. The tuning 126ed5 does this. So does the tuning [[86edt]] whose octave is identical to 126ed5 within 0.1{{c}}.
			{{Harmonics in equal\|126\|5\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 126ed5}}
			{{Harmonics in equal\|126\|5\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 126ed5 (continued)}}

	== Scales ==		== Scales ==