Neutral second - Revision history

Vector: Redirected page to Second

2026-03-06T09:59:57Z

Redirected page to Second

← Older revision		Revision as of 09:59, 6 March 2026
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			#REDIRECT [[Second]]

	~~A '''neutral second''', as a concrete interval region, is typically near 150 cents in size, distinct from~~ [[Semitone\|semitones]] of roughly 100 cents and [[Major second\|major seconds]] of roughly 200 cents. A rough tuning range of the neutral second is 130 to 170 cents, and they characteristically are half of some kind of [[minor third]]. Neutral seconds are not found in diatonic, but they do exist in a diatonic functional context as "neutral" diatonic intervals.

	~~They are one of the most distinctive-sounding yet versatile xenharmonic intervals, which makes them highly valuable.~~

	~~==Function==~~

	~~The large step of armotonic is always some sort of neutral second, which functions similar to both a whole tone and semitone.~~

	~~A step of 13ed3 is a middle neutral second. It is notable for having an unusually good approximation of LCJI for a system generated by neutral seconds.~~

	~~Supraminor/submajor scales, such as 2.3.17/7 blackdye, have a large number of neutral seconds.~~

	'''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (~10/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.

	In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.
	~~== In just intonation ==~~
	~~=== By prime-limit ===~~
	~~The [[Pythagorean tuning\|3-limit]] does not have a simple neutral second, so we start with the 5-limit:~~

	* The 5-limit '''acute minor second''' or '''large limma''' is a ratio of [[27/25]], and is about 133{{c}}.
	* The 7-limit neutral second is a ratio of [[35/32]], and is about 155{{c}}.
	** There is also a 7-limit '''swetismic neutral second''', which is a ratio of [[49/45]], and is about 147{{c}}.
	* The 11-limit neutral/submajor seconds are the ratios of [[12/11]] and [[11/10]]~~, which are about 151{{c}} and 165{{c}}, respectively; 11/10 in particular can also be analyzed as a [[major second]]. Despite that, it is also here for completeness.~~
	* The 13-limit neutral/supraminor seconds are the ratios of [[14/13]] and [[13/12]], which are about 128{{c}} and 139{{c}}, respectively; 14/13 in particular can also be analyzed as a [[semitone]]. Despite that, it is also here for completeness.

	~~== In mos scales ==~~
	~~Intervals between 120 and 171{{c}} generate the following [[MOS\|mos]] scales:~~

	~~These tables start from the last monolarge mos generated by the interval range.~~

	~~Scales with more than 12 notes are not included.~~
	~~{\| class="wikitable"~~
	~~!Range~~
	~~! colspan="2" \|Mos~~
	\|-
	~~\|120–133{{c}}~~
	~~\|[[1L 8s]]~~
	~~\|[[9L 1s]]~~
	\|-
	~~\|133–150{{c}}~~
	~~\|[[1L 7s]]~~
	~~\|[[8L 1s]]~~
	\|-
	~~\|150–171{{c}}~~
	~~\|[[1L 6s]]~~
	~~\|[[7L 1s]]~~
	\|}

	~~==Categorization==~~

	~~===Proposal: Ground's Neutral Second Categorization System===~~
	~~As neutral intervals, neutral seconds can be named based on interval splitting.~~
	~~{\| class="wikitable"~~
	~~\|+ Names Based on Interval Splitting (with Nearby Edo Intervals)~~
	\|-
	~~! ¢ !! Definition !! Name (accepted names are bold) !! Edo !! ¢ !! Error ¢~~
	\|-
	~~\| 119.443 \|\| '''15/14''' \|\| '''Septimal Major Semitone''' \|\| 1\10 \|\| 120.000 \|\| 0.557~~
	\|-
	~~\| 121.243 \|\| 7/6 / <sup>4</sup>√(7/5) \|\| Quadranseptimal Supraminor Second \|\| 1\10 \|\| 120.000 \|\| -1.243~~
	\|-
	~~\| 124.511 \|\| <sup>4</sup>√(4/3) \|\| Quadranpyth Supraminor Second \|\| 3\29 \|\| 124.138 \|\| -0.373~~
	\|-
	~~\| 128.298 \|\| '''14/13''' \|\| '''Tridecimal Supraminor Second''' \|\| 3\28 \|\| 128.571 \|\| 0.273~~
	\|-
	~~\| 133.435 \|\| √(7/6) \|\| Semiseptal Neutral Second \|\| 1\9 \|\| 133.333 \|\| -0.102~~
	\|-
	~~\| 138.573 \|\| '''13/12''' \|\| '''Tridecimal Neutral Second''' \|\| 3\26 \|\| 138.462 \|\| -0.111~~
	\|-
	~~\| 140.391 \|\| <sup>5</sup>√(3/2) \|\| Quintanpyth Neutral Second \|\| 2\17 \|\| 141.176 \|\| 0.785~~
	\|-
	~~\| 145.628 \|\| <sup>4</sup>√(7/5) \|\| Quadranseptimal Neutral Second \|\| 4\33 \|\| 145.455 \|\| -0.174~~
	\|-
	~~\| 150.637 \|\| '''12/11''' \|\| '''Undecimal Neutral Second''' \|\| 1\8 \|\| 150.000 \|\| -0.637~~
	\|-
	~~\| 157.821 \|\| √(6/5) \|\| Semipental Neutral Second \|\| 5\38 \|\| 157.895 \|\| 0.074~~
	\|-
	~~\| 165.004 \|\| '''11/10''' \|\| '''Undecimal Submajor Second''' \|\| 4\29 \|\| 165.517 \|\| 0.513~~
	\|-
	~~\| 166.015 \|\| <sup>3</sup>√(4/3) \|\| Trienpyth Submajor Second \|\| 4\29 \|\| 165.517 \|\| -0.498~~
	\|-
	~~\| 170.013 \|\| 6/5 / <sup>4</sup>√(7/5) \|\| Quadranseptimal Submajor Second \|\| 1\7 \|\| 171.429 \|\| 1.415~~
	\|-
	~~\| 175.489 \|\| <sup>4</sup>√(3/2) \|\| Quadranpyth Major Second \|\| 6\41 \|\| 175.610 \|\| 0.121~~
	\|}

	~~{{Interval regions}}~~

Inthar: /* Proposal: Ground's Neutral Second Categorization System */

2026-02-13T04:27:29Z

Proposal: Ground's Neutral Second Categorization System

← Older revision		Revision as of 04:27, 13 February 2026
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	\| 175.489 \|\| <sup>4</sup>√(3/2) \|\| Quadranpyth Major Second \|\| 6\41 \|\| 175.610 \|\| 0.121		\| 175.489 \|\| <sup>4</sup>√(3/2) \|\| Quadranpyth Major Second \|\| 6\41 \|\| 175.610 \|\| 0.121
	\|}		\|}

			{{Interval regions}}

Vector at 19:01, 26 December 2025

2025-12-26T19:01:15Z

← Older revision		Revision as of 19:01, 26 December 2025
Line 17:		Line 17:
	In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.		In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.
	== In just intonation ==		== In just intonation ==
	=== By ~~primelimit~~ ===		=== By prime-limit ===
	The [[Pythagorean tuning\|3-limit]] does not have a simple neutral second, so we start with the 5-limit:		The [[Pythagorean tuning\|3-limit]] does not have a simple neutral second, so we start with the 5-limit:

Inthar at 17:47, 26 December 2025

2025-12-26T17:47:28Z

← Older revision		Revision as of 17:47, 26 December 2025
Line 17:		Line 17:
	In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.		In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.
	== In just intonation ==		== In just intonation ==
	=== By ~~prime limit~~ ===		=== By primelimit ===
	The [[Pythagorean tuning\|3-limit]] does not have a simple neutral second, so we start with the 5-limit:		The [[Pythagorean tuning\|3-limit]] does not have a simple neutral second, so we start with the 5-limit:

Vector at 09:37, 11 December 2025

2025-12-11T09:37:49Z

← Older revision		Revision as of 09:37, 11 December 2025
Line 13:		Line 13:
	Supraminor/submajor scales, such as 2.3.17/7 blackdye, have a large number of neutral seconds.		Supraminor/submajor scales, such as 2.3.17/7 blackdye, have a large number of neutral seconds.

	'''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (~~T10~~/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.		'''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (~10/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.

	In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.		In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.

Vector at 09:37, 11 December 2025

2025-12-11T09:37:25Z

← Older revision		Revision as of 09:37, 11 December 2025
Line 15:		Line 15:
	'''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (T10/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.		'''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (T10/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.

	In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the ~~T10~~/9 is a slightly larger 168¢ in exchange for making the ~~T81~~/80 48¢, a much more usable aberrisma than Porcupine typically offers.		In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.
	== In just intonation ==		== In just intonation ==
	=== By prime limit ===		=== By prime limit ===

Vector at 09:36, 11 December 2025

2025-12-11T09:36:10Z

@@ Line 1: / Line 1: @@
-'''Neutral seconds''' are intervals with a size in between a whole tone and a semitone. They are one of the most distinctive-sounding yet versatile xenharmonic intervals, which makes them highly valuable.
 ==Function==
@@ Line 12: / Line 16: @@
 In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the T10/9 is a slightly larger 168¢ in exchange for making the T81/80 48¢, a much more usable aberrisma than Porcupine typically offers.
-==Categorization==
+== In mos scales ==
-Neutral seconds are generally considered to be between 1\9 and 1\7, being half of some sort of minor third. If semifourths are included, they can be as narrow as 1\10.
+These tables start from the last monolarge mos generated by the interval range.
-===Proposal: Ground's Neutral Second Categorization System===
+Scales with more than 12 notes are not included.
-Import splitting table here.
+==Categorization==
 {| class="wikitable"
 |+ Names Based on Interval Splitting (with Nearby Edo Intervals)
 |-
 ! ¢ !! Definition !! Name (accepted names are bold) !! Edo !! ¢ !! Error ¢
 |-
 | 119.443 || '''15/14''' || '''Septimal Major Semitone''' || 1\10 || 120.000 || 0.557
@@ Line 57: / Line 85: @@
 |-
 | 175.489 || <sup>4</sup>√(3/2) || Quadranpyth Major Second || 6\41 || 175.610 || 0.121
 |}

Ground at 02:34, 26 October 2025

2025-10-26T02:34:53Z

← Older revision		Revision as of 02:34, 26 October 2025
Line 6:		Line 6:

	A step of 13ed3 is a middle neutral second. It is notable for having an unusually good approximation of LCJI for a system generated by neutral seconds.		A step of 13ed3 is a middle neutral second. It is notable for having an unusually good approximation of LCJI for a system generated by neutral seconds.

			Supraminor/submajor scales, such as 2.3.17/7 blackdye, have a large number of neutral seconds.

	'''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (T10/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.		'''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (T10/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.

Ground at 00:34, 24 October 2025

2025-10-24T00:34:22Z

@@ Line 15: / Line 15: @@
 Neutral seconds are generally considered to be between 1\9 and 1\7, being half of some sort of minor third. If semifourths are included, they can be as narrow as 1\10.
-===Ground's Neutral Second Categorization System===
+===Proposal: Ground's Neutral Second Categorization System===
-This is going to need a huge table import.
+Import splitting table here.

Ground at 03:16, 22 October 2025

2025-10-22T03:16:15Z

← Older revision		Revision as of 03:16, 22 October 2025
Line 3:		Line 3:
	==Function==		==Function==

	The ~~small~~ step of ~~antidiatonic~~ is a neutral second, which functions similar to both a whole tone and semitone.		The large step of armotonic is always some sort of neutral second, which functions similar to both a whole tone and semitone.

	'''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone, and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.		A step of 13ed3 is a middle neutral second. It is notable for having an unusually good approximation of LCJI for a system generated by neutral seconds.

			'''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (T10/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.

			In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the T10/9 is a slightly larger 168¢ in exchange for making the T81/80 48¢, a much more usable aberrisma than Porcupine typically offers.

	==Categorization==		==Categorization==