54edo - Revision history

CompactStar: /* Theory */

2026-10-22T02:16:58Z

Theory

← Older revision		Revision as of 02:16, 22 October 2026
Line 9:		Line 9:
	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.		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. 54edo is still definitely usable to approximate low-complexity JI, since at this step size, every interval's direct approximation is within a maximum of about 11.1{{c}} [[error]], it's just ~~not very good~~ for its size.		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. 54edo is still definitely usable to approximate low-complexity JI, since at this step size, every interval's direct approximation is within a maximum of about 11.1{{c}} [[error]], it's just bad for its size.

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

CompactStar: /* Theory */

2026-10-22T02:16:50Z

Theory

← Older revision		Revision as of 02:16, 22 October 2026
Line 9:		Line 9:
	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.		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. 54edo is still definitely usable to approximate JI, since at this step size, every interval is ~~approximated~~ within a maximum of about 11.1{{c}} [[error]], it's just not very good for its size.		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. 54edo is still definitely usable to approximate low-complexity JI, since at this step size, every interval's direct approximation is within a maximum of about 11.1{{c}} [[error]], it's just not very good for its size.

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

CompactStar: /* Theory */

2026-10-22T02:16:09Z

Theory

← Older revision		Revision as of 02:16, 22 October 2026
Line 9:		Line 9:
	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.		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. 54edo is still definitely usable to approximate JI, since at this step size, every interval is approximated within a maximum of about 11.1{{c}} [[error]], it's just not very good for its size.

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

CompactStar: /* Octave stretch or compression */

2026-10-22T01:13:46Z

Octave stretch or compression

← Older revision		Revision as of 01:13, 22 October 2026
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	== Octave stretch or compression ==		== Octave stretch or compression ==
			If you use the patent val (27edo) fifth, then 54edo is a good candidate for octave-compression, since it has a consistently sharp tendency except for harmonic 5. If you use the flatter fifth, then octave-stretching will work better for 5-limit (but will have mixed results on higher limits).

	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.		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.

CompactStar: /* Theory */

2026-10-22T01:13:31Z

Theory

← Older revision		Revision as of 01:13, 22 October 2026
Line 10:		Line 10:

	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.

	If you use the patent val (27edo) fifth, then 54edo is a good candidate for octave-compression, since it has a consistently sharp tendency except for harmonic 5. If you use the flatter fifth, then octave-stretching will work better for 5-limit (but will have mixed results on higher limits).

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

CompactStar: /* Theory */

2026-10-22T01:13:11Z

Theory

← Older revision		Revision as of 01:13, 22 October 2026
Line 3:		Line 3:

	== 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]]. The flat fifth can be used [[deeptone]] generator, although it is perhaps a bit too flat even for deeptone, with the ~5/4 reached by 11 fifths up being only 378 cents.		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 flat fifth can be used as a [[deeptone]] generator, although it is perhaps a bit too flat even for deeptone, with the ~5/4 reached by 11 fifths up being only 378 cents.

	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]].

CompactStar: /* Theory */

2026-10-22T01:12:49Z

Theory

← Older revision		Revision as of 01:12, 22 October 2026
Line 3:		Line 3:

	== 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]]. The flat fifth can be used [[deeptone]] generator, although it is perhaps a bit too flat even for deeptone, with the 5/4 reached by 11 fifths up being only 378 cents.		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 flat fifth can be used [[deeptone]] generator, although it is perhaps a bit too flat even for deeptone, with the ~5/4 reached by 11 fifths up being only 378 cents.

	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]].

CompactStar: /* Theory */

2026-10-22T01:12:39Z

Theory

← Older revision		Revision as of 01:12, 22 October 2026
Line 3:		Line 3:

	== 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]]. The flat fifth can be used [[deeptone]] generator, although it is perhaps a bit too flat even for deeptone, with the 5/4 reached by 11 fifths up being only 378 cents.

	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]].

Lucius Chiaraviglio: /* Theory */ Fix missing words

2026-10-15T04:06:10Z

Theory: Fix missing words

← Older revision		Revision as of 04:06, 15 October 2026
Line 11:		Line 11:
	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.

	If you use the patent val (27edo) fifth, then 54edo is a good candidate for octave-compression, since it has a consistently sharp tendency except for harmonic 5. If you use the flatter fifth, then octave-stretching will work better for 5-limit (but mixed results on higher limits).		If you use the patent val (27edo) fifth, then 54edo is a good candidate for octave-compression, since it has a consistently sharp tendency except for harmonic 5. If you use the flatter fifth, then octave-stretching will work better for 5-limit (but will have mixed results on higher limits).

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

CompactStar: /* Theory */

2026-10-14T22:26:19Z

Theory

← Older revision		Revision as of 22:26, 14 October 2026
Line 11:		Line 11:
	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.

	If ~~using~~ the 27edo fifth, then 54edo is a good candidate for octave-~~stretching~~, since it has a consistently sharp tendency except for harmonic 5.		If you use the patent val (27edo) fifth, then 54edo is a good candidate for octave-compression, since it has a consistently sharp tendency except for harmonic 5. If you use the flatter fifth, then octave-stretching will work better for 5-limit (but mixed results on higher limits).

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