Gentle tuning - Revision history

Vector at 02:06, 2 June 2026

2026-06-02T02:06:04Z

← Older revision		Revision as of 02:06, 2 June 2026
Line 1:		Line 1:
	[[File:Gentle12.png\|thumb\|The chromatic MOS of a gentle-fifth temperament.]]		[[File:Gentle12.png\|thumb\|The chromatic MOS of a gentle-fifth temperament. Note the two step sizes.]]
	'''Gentle tuning''' (aka the '''gentle region''' of fifths) is the regular tuning of the perfect fifth somewhere between the tunings given by 29edo and 17edo (~703.4 to 705.9 cents), producing MOS [[diatonic]] scales with step size ratios between 5:2 and 3:1 (and consequently [[chromatic]] scales between 3:2 and 2:1, including the [[Golden generator\|golden tuning]] of p-chromatic). As a result, the [[major third]] generated is between 413.8 and 423.5 cents, and approximates 14/11. It is so named because melodically it functions as a "gentle" version of [[Superpyth]]; conversely, it can instead be seen as exaggerating the characteristics of Pythagorean tuning as it compares with 12-tone equal temperament and is usable as a replacement for Pythagorean tuning. One notable characteristic of gentle tuning is that it narrows the diatonic semitone towards a size of 70 cents, described by some theorists as an ideal leading tone; Aura, however, considers it to be between a leading tone and a diesis or quarter tone in function, being able to either lead to the tonic or "skip over" it.		'''Gentle tuning''' (aka the '''gentle region''' of fifths) is the regular tuning of the perfect fifth somewhere between the tunings given by 29edo and 17edo (~703.4 to 705.9 cents), producing MOS [[diatonic]] scales with step size ratios between 5:2 and 3:1 (and consequently [[chromatic]] scales between 3:2 and 2:1, including the [[Golden generator\|golden tuning]] of p-chromatic). As a result, the [[major third]] generated is between 413.8 and 423.5 cents, and approximates 14/11. It is so named because melodically it functions as a "gentle" version of [[Superpyth]]; conversely, it can instead be seen as exaggerating the characteristics of Pythagorean tuning as it compares with 12-tone equal temperament and is usable as a replacement for Pythagorean tuning. One notable characteristic of gentle tuning is that it narrows the diatonic semitone towards a size of 70 cents, described by some theorists as an ideal leading tone; Aura, however, considers it to be between a leading tone and a diesis or quarter tone in function, being able to either lead to the tonic or "skip over" it.

Vector at 02:05, 2 June 2026

2026-06-02T02:05:42Z

← Older revision		Revision as of 02:05, 2 June 2026
Line 1:		Line 1:
			[[File:Gentle12.png\|thumb\|The chromatic MOS of a gentle-fifth temperament.]]
	'''Gentle tuning''' (aka the '''gentle region''' of fifths) is the regular tuning of the perfect fifth somewhere between the tunings given by 29edo and 17edo (~703.4 to 705.9 cents), producing MOS [[diatonic]] scales with step size ratios between 5:2 and 3:1 (and consequently [[chromatic]] scales between 3:2 and 2:1, including the [[Golden generator\|golden tuning]] of p-chromatic). As a result, the [[major third]] generated is between 413.8 and 423.5 cents, and approximates 14/11. It is so named because melodically it functions as a "gentle" version of [[Superpyth]]; conversely, it can instead be seen as exaggerating the characteristics of Pythagorean tuning as it compares with 12-tone equal temperament and is usable as a replacement for Pythagorean tuning. One notable characteristic of gentle tuning is that it narrows the diatonic semitone towards a size of 70 cents, described by some theorists as an ideal leading tone; Aura, however, considers it to be between a leading tone and a diesis or quarter tone in function, being able to either lead to the tonic or "skip over" it.		'''Gentle tuning''' (aka the '''gentle region''' of fifths) is the regular tuning of the perfect fifth somewhere between the tunings given by 29edo and 17edo (~703.4 to 705.9 cents), producing MOS [[diatonic]] scales with step size ratios between 5:2 and 3:1 (and consequently [[chromatic]] scales between 3:2 and 2:1, including the [[Golden generator\|golden tuning]] of p-chromatic). As a result, the [[major third]] generated is between 413.8 and 423.5 cents, and approximates 14/11. It is so named because melodically it functions as a "gentle" version of [[Superpyth]]; conversely, it can instead be seen as exaggerating the characteristics of Pythagorean tuning as it compares with 12-tone equal temperament and is usable as a replacement for Pythagorean tuning. One notable characteristic of gentle tuning is that it narrows the diatonic semitone towards a size of 70 cents, described by some theorists as an ideal leading tone; Aura, however, considers it to be between a leading tone and a diesis or quarter tone in function, being able to either lead to the tonic or "skip over" it.

Inthar: /* Names */

2026-04-05T07:19:46Z

Names

← Older revision		Revision as of 07:19, 5 April 2026
Line 14:		Line 14:

	==== Names ====		==== Names ====
	The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is '''Leapday.''' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a 2.3.7.11.13 temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''[[Parapyth]]'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''		The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is ''Leapday.'' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a 2.3.7.11.13 temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''[[Parapyth]]'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''

	== Interval chain ==		== Interval chain ==

Vector at 07:14, 5 April 2026

2026-04-05T07:14:17Z

← Older revision		Revision as of 07:14, 5 April 2026
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	The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is '''Leapday.''' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a 2.3.7.11.13 temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''[[Parapyth]]'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''		The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is '''Leapday.''' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a 2.3.7.11.13 temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''[[Parapyth]]'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''

			== Interval chain ==
			{\| class="wikitable"
			\|+
			! colspan="3" \|Generators up
			! colspan="3" \|Generators down
			\|-
			!#
			!Cents
			!Regular interpretation (2.3.7.11.13)
			!#
			!Cents
			!Regular interpretation (2.3.7.11.13)
			\|-
			\|0
			\|0.00
			\|1/1
			\|0
			\|1,200.00
			\|2/1
			\|-
			\|1
			\|704.35
			\|3/2
			\|1
			\|495.65
			\|4/3
			\|-
			\|2
			\|208.70
			\|9/8
			\|2
			\|991.30
			\|16/9
			\|-
			\|3
			\|913.04
			\|22/13
			\|3
			\|286.96
			\|13/11
			\|-
			\|4
			\|417.39
			\|14/11
			\|4
			\|782.61
			\|11/7
			\|-
			\|5
			\|1,121.74
			\|21/11
			\|5
			\|78.26
			\|22/21
			\|-
			\|6
			\|626.09
			\|13/9
			\|6
			\|573.91
			\|18/13
			\|-
			\|7
			\|130.43
			\|14/13
			\|7
			\|1,069.57
			\|13/7
			\|-
			\|8
			\|834.78
			\|13/8
			\|8
			\|365.22
			\|16/13
			\|-
			\|9
			\|339.13
			\|11/9
			\|9
			\|860.87
			\|18/11
			\|-
			\|10
			\|1,043.48
			\|11/6, 20/11
			\|10
			\|156.52
			\|11/10, 12/11
			\|-
			\|11
			\|547.83
			\|11/8, 15/11
			\|11
			\|652.17
			\|16/11, 22/15
			\|-
			\|12
			\|52.17
			\|33/32, 40/39
			\|12
			\|1,147.83
			\|64/33, 39/20
			\|-
			\|13
			\|756.52
			\|14/9, 20/13
			\|13
			\|443.48
			\|9/7, 13/10
			\|-
			\|14
			\|260.87
			\|7/6
			\|14
			\|939.13
			\|12/7
			\|-
			\|15
			\|965.22
			\|7/4
			\|15
			\|234.78
			\|8/7
			\|}
	{{Navbox regtemp}}		{{Navbox regtemp}}

Inthar at 23:47, 4 April 2026

2026-04-04T23:47:34Z

← Older revision		Revision as of 23:47, 4 April 2026
Line 2:		Line 2:

	== Regular temperament theory ==		== Regular temperament theory ==
	Gentle tuning is associated with the 2.3.(11/7).(13/7) temperament, 29 & 46, that finds 14/11 at the major third and 13/11 at the minor third, which has a number of names depending on the subgroup (see [[#Names]]). Therefore, the chromatic semitone is equated to 14/13 and the diatonic semitone is equated to 22/21. This implies the use of these thirds as elements of 13-limit harmony over the 11th and 13th harmonics themselves, similar to, but more intense and approach-specific than, the prioritization of 9/7 and 7/6 in Archy (and in septal harmony as a whole) as opposed to 7/4; both approaches are patterned off of [[Meantone]], which is often described outside of a regular temperament context as tempering the fifth to bring the major and minor third more in-tune with 5/4 and 6/5, where 5/4 is simply treated as "the major third" rather than a generator of just intonation in its own right.		Gentle tuning is associated with the 2.3.(11/7).(13/7) temperament, 29 & 46, that finds 14/11 at the major third and 13/11 at the minor third, which has a number of names depending on the subgroup (see [[#Names]]). Therefore, the chromatic semitone is equated to 14/13 and the diatonic semitone is equated to 22/21; the whole tone is equated to 44/39.

			This implies the use of these thirds as elements of 13-limit harmony over the 11th and 13th harmonics themselves, similar to, but more intense and approach-specific than, the prioritization of 9/7 and 7/6 in Archy (and in septal harmony as a whole) as opposed to 7/4; both approaches are patterned off of [[Meantone]], which is often described outside of a regular temperament context as tempering the fifth to bring the major and minor third more in-tune with 5/4 and 6/5, where 5/4 is simply treated as "the major third" rather than a generator of just intonation in its own right.

	To extend to the 2.3.7.11.13 subgroup, 7/6 is equated to two chromatic semitones (so that 7/4 is C-G##). As a result, 11/8 is placed at the augmented third and 13/8 at the augmented fifth (this is the same interval that becomes 8/5 in [[Schismic]]).		To extend to the 2.3.7.11.13 subgroup, 7/6 is equated to two chromatic semitones (so that 7/4 is C-G##). As a result, 11/8 is placed at the augmented third and 13/8 at the augmented fifth (this is the same interval that becomes 8/5 in [[Schismic]]).

Inthar: /* Regular temperament theory */

2026-04-04T23:43:25Z

Regular temperament theory

← Older revision		Revision as of 23:43, 4 April 2026
Line 2:		Line 2:

	== Regular temperament theory ==		== Regular temperament theory ==
	Gentle tuning is associated with the 2.3.(11/7).(13/7) temperament, 29 & 46, that finds 14/11 at the major third and 13/11 at the minor third, which has a number of names depending on the subgroup (see [[#Names]]). Therefore, the chromatic semitone is ~~equal~~ to 14/13 and the diatonic semitone is equated to 22/21. This implies the use of these thirds as elements of 13-limit harmony over the 11th and 13th harmonics themselves, similar to, but more intense and approach-specific than, the prioritization of 9/7 and 7/6 in Archy (and in septal harmony as a whole) as opposed to 7/4; both approaches are patterned off of [[Meantone]], which is often described outside of a regular temperament context as tempering the fifth to bring the major and minor third more in-tune with 5/4 and 6/5, where 5/4 is simply treated as "the major third" rather than a generator of just intonation in its own right.		Gentle tuning is associated with the 2.3.(11/7).(13/7) temperament, 29 & 46, that finds 14/11 at the major third and 13/11 at the minor third, which has a number of names depending on the subgroup (see [[#Names]]). Therefore, the chromatic semitone is equated to 14/13 and the diatonic semitone is equated to 22/21. This implies the use of these thirds as elements of 13-limit harmony over the 11th and 13th harmonics themselves, similar to, but more intense and approach-specific than, the prioritization of 9/7 and 7/6 in Archy (and in septal harmony as a whole) as opposed to 7/4; both approaches are patterned off of [[Meantone]], which is often described outside of a regular temperament context as tempering the fifth to bring the major and minor third more in-tune with 5/4 and 6/5, where 5/4 is simply treated as "the major third" rather than a generator of just intonation in its own right.

	To extend to the 2.3.7.11.13 subgroup, 7/6 is equated to two chromatic semitones (so that 7/4 is C-G##). As a result, 11/8 is placed at the augmented third and 13/8 at the augmented fifth (this is the same interval that becomes 8/5 in [[Schismic]]).		To extend to the 2.3.7.11.13 subgroup, 7/6 is equated to two chromatic semitones (so that 7/4 is C-G##). As a result, 11/8 is placed at the augmented third and 13/8 at the augmented fifth (this is the same interval that becomes 8/5 in [[Schismic]]).

Inthar at 23:42, 4 April 2026

2026-04-04T23:42:57Z

← Older revision		Revision as of 23:42, 4 April 2026
Line 1:		Line 1:
	'''Gentle tuning''' is the regular tuning of the perfect fifth somewhere between the tunings given by 29edo and 17edo (~703.4 to 705.9 cents), producing MOS [[diatonic]] scales with step size ratios between 5:2 and 3:1 (and consequently [[chromatic]] scales between 3:2 and 2:1, including the [[Golden generator\|golden tuning]] of p-chromatic). As a result, the [[major third]] generated is between 413.8 and 423.5 cents, and approximates 14/11. It is so named because melodically it functions as a "gentle" version of [[Superpyth]]; conversely, it can instead be seen as exaggerating the characteristics of Pythagorean tuning as it compares with 12-tone equal temperament and is usable as a replacement for Pythagorean tuning. One notable characteristic of gentle tuning is that it narrows the diatonic semitone towards a size of 70 cents, described by some theorists as an ideal leading tone; Aura, however, considers it to be between a leading tone and a diesis or quarter tone in function, being able to either lead to the tonic or "skip over" it.		'''Gentle tuning''' (aka the '''gentle region''' of fifths) is the regular tuning of the perfect fifth somewhere between the tunings given by 29edo and 17edo (~703.4 to 705.9 cents), producing MOS [[diatonic]] scales with step size ratios between 5:2 and 3:1 (and consequently [[chromatic]] scales between 3:2 and 2:1, including the [[Golden generator\|golden tuning]] of p-chromatic). As a result, the [[major third]] generated is between 413.8 and 423.5 cents, and approximates 14/11. It is so named because melodically it functions as a "gentle" version of [[Superpyth]]; conversely, it can instead be seen as exaggerating the characteristics of Pythagorean tuning as it compares with 12-tone equal temperament and is usable as a replacement for Pythagorean tuning. One notable characteristic of gentle tuning is that it narrows the diatonic semitone towards a size of 70 cents, described by some theorists as an ideal leading tone; Aura, however, considers it to be between a leading tone and a diesis or quarter tone in function, being able to either lead to the tonic or "skip over" it.

	== Regular temperament theory ==		== Regular temperament theory ==
	Gentle tuning is associated with the 2.3.(11/7).(13/7) temperament, 29 & 46, that finds 14/11 at the major third and 13/11 at the minor third, which has a number of names depending on the subgroup (see [[#Names]]). Therefore, the chromatic semitone is equal to 14/13. This implies the use of these thirds as elements of 13-limit harmony over the 11th and 13th harmonics themselves, similar to, but more intense and approach-specific than, the prioritization of 9/7 and 7/6 in Archy (and in septal harmony as a whole) as opposed to 7/4; both approaches are patterned off of [[Meantone]], which is often described outside of a regular temperament context as tempering the fifth to bring the major and minor third more in-tune with 5/4 and 6/5, where 5/4 is simply treated as "the major third" rather than a generator of just intonation in its own right.		Gentle tuning is associated with the 2.3.(11/7).(13/7) temperament, 29 & 46, that finds 14/11 at the major third and 13/11 at the minor third, which has a number of names depending on the subgroup (see [[#Names]]). Therefore, the chromatic semitone is equal to 14/13 and the diatonic semitone is equated to 22/21. This implies the use of these thirds as elements of 13-limit harmony over the 11th and 13th harmonics themselves, similar to, but more intense and approach-specific than, the prioritization of 9/7 and 7/6 in Archy (and in septal harmony as a whole) as opposed to 7/4; both approaches are patterned off of [[Meantone]], which is often described outside of a regular temperament context as tempering the fifth to bring the major and minor third more in-tune with 5/4 and 6/5, where 5/4 is simply treated as "the major third" rather than a generator of just intonation in its own right.

	To extend to the 2.3.7.11.13 subgroup, 7/6 is equated to two chromatic semitones (so that 7/4 is C-G##). As a result, 11/8 is placed at the augmented third and 13/8 at the augmented fifth (this is the same interval that becomes 8/5 in [[Schismic]]).		To extend to the 2.3.7.11.13 subgroup, 7/6 is equated to two chromatic semitones (so that 7/4 is C-G##). As a result, 11/8 is placed at the augmented third and 13/8 at the augmented fifth (this is the same interval that becomes 8/5 in [[Schismic]]).

Inthar: /* Names */

2026-04-04T16:20:37Z

Names

← Older revision		Revision as of 16:20, 4 April 2026
Line 12:		Line 12:

	==== Names ====		==== Names ====
	The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is '''Leapday.''' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a ~~no-5s~~ temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''[[Parapyth]]'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''		The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is '''Leapday.''' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a 2.3.7.11.13 temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''[[Parapyth]]'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''

	{{Navbox regtemp}}		{{Navbox regtemp}}

Inthar: /* Names */

2026-04-04T08:33:05Z

Names

← Older revision		Revision as of 08:33, 4 April 2026
Line 12:		Line 12:

	==== Names ====		==== Names ====
	The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is '''Leapday.''' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a no-5s temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''Parapyth'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''		The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is '''Leapday.''' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a no-5s temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''[[Parapyth]]'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''

	{{Navbox regtemp}}		{{Navbox regtemp}}

Inthar: /* Names */

2026-04-04T06:03:52Z

Names

← Older revision		Revision as of 06:03, 4 April 2026
Line 12:		Line 12:

	==== Names ====		==== Names ====
	The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is '''Leapday.''' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a no-5s temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''Parapyth'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni)'', and simply ''gentle.''		The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is '''Leapday.''' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a no-5s temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''Parapyth'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''

	{{Navbox regtemp}}		{{Navbox regtemp}}