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			{{Interval regions}}

Vector: Created page with "''For just intervals with a denominator of 2, see 3/2.'' A '''second''' is a step of the 7-form, or an interval that could reasonably span 1 step of the 7-form. They are alongside smaller intervals heard as "steps" rather than "skips" to Western listeners. Seconds comprise whole tones, neutral seconds, and semitones, which may be considered subtypes of second or fully independent interval size/function categories in their own right. This..."

2026-03-06T09:59:34Z

Created page with "''For just intervals with a denominator of 2, see 3/2.'' A '''second''' is a step of the 7-form, or an interval that could reasonably span 1 step of the 7-form. They are alongside smaller intervals heard as "steps" rather than "skips" to Western listeners. Seconds comprise whole tones, neutral seconds, and semitones, which may be considered subtypes of second or fully independent interval size/function categories in their own right. This..."

New page

''For just intervals with a denominator of 2, see [[3/2]].''

A '''second''' is a step of the 7-form, or an interval that could reasonably span 1 step of the 7-form. They are alongside smaller intervals heard as "steps" rather than "skips" to Western listeners. Seconds comprise [[Whole tone|whole tones]], neutral seconds, and [[Semitone|semitones]], which may be considered subtypes of second or fully independent interval size/function categories in their own right. This conceptualization is precisely due to the second's role as the heptatonic scale step.

It is important to note that in most systems of interval arithmetic, the difference between two qualities of the same heptatonic interval ordinal, while it may be second-like in distance, is not a second, but rather a [[chromatic semitone]] (or a more general heptatonic [[chroma]]). It may regardless be useful to call it something like a "minor second" when discussing JI interpretations or size.

The two sizes of second in a 7-form rank-2 temperament provide a set of generators for the entire system in question.

== Name ==
The term ''second'' comes from conventional music theory. In the diatonic interval naming convention, intervals are 1-indexed, so the heptatonic 1-step is referred to as a ''second'' due to involving two adjacent notes.

''Whole tone'' and ''semitone'' come from diatonic scale theory, which comprises the diatonic scale of five whole tones and two semitones; the step ratio of diatonic is either exactly or roughly 2:1 in most historical tunings, thus a semitone is roughly half of a whole tone. This is not to be confused with the sense of ''tone'' to refer to a note.

== Qualities ==
The two main qualities of second in diatonic interval arithmetic, major and minor, are discussed at [[Diatonic major second]] and [[Diatonic semitone]] respectively. Also of note is the diminished second, which serves as the distance between the two types of semitone, and is the "Pythagorean comma" tempered out in 12edo. Intervals separated by a diminished second are called ''enharmonically equivalent''. The diminished second and chromatic semitone form a basis for diatonic temperaments, corresponding to the edo join 7 & 12.

=== Neutral seconds ===
Neutral seconds are not found in diatonic, but they do exist in a diatonic functional context as "neutral" diatonic intervals.

== As an interval region ==
Seconds as an interval region generally range from about 60 to 240 cents, with major seconds in the larger portion of the range and minor seconds in the smaller portion, and [[interordinal]] intervals from 240-260 and 40-60 cents can also function as thirds. Note that as a size range this includes most chromatic semitones in diatonic interval arithmetic, which are therefore considered seconds when discussing interval regions despite their functional distinctness.

=== Major seconds ===
Major seconds are approximately 200 cents in size. A flat major second of ~170 cents generates a heptatonic scale that is very close to equal-tempered.

=== Minor seconds ===
Minor seconds are approximately 100 cents in size. As an interval region this may be called "semitone" so that chromatic semitones are not mislabeled. A major second and minor second generally stack to a [[minor third]].

=== Neutral seconds ===
Neutral seconds are approximately 150 cents in size, distinct from minor seconds and major seconds. They characteristically are half of some kind of [[minor third]]. They are one of the most distinctive-sounding yet versatile xenharmonic intervals, which makes them highly valuable.

Supraminor/submajor scales, such as 2.3.17/7 blackdye, have a large number of neutral seconds. In large edos, it's possible to have a neutral second of about 165 cents 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.

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

== As the generators of temperaments ==
As mentioned previously, a major and minor second together can serve as generators for any diatonic temperament. Against the octave, however, there are a wide range of temperaments generated by second-sized intervals.
{| class="wikitable"
|+
!Temperament
!Form
!Tuning range
!JI
!Notes
|-
|Valentine
|15
|75-80c
|25/24
|Also [[Carlos Alpha]]
|-
|Passion
|12
|95-100c
|17/16~18/17
|
|-
|Miracle
|10
|115-120c
|15/14~16/15
|An interval of this size, representing 15/14~16/15, is called a "secor".
|-
|Negri
|10
|~125c
|16/15
|
|-
|Bohpier
|8
|~146c
|27/25
|Also [[13edt]] without octaves
|-
|Porcupine
|7
|160-165c
|11/10
|Porcupine neutral seconds 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.
|-
|Tetracot
|7
|~175c
|10/9
|
|-
|Didacus
|6
|190-195c
|28/25~9/8
|
|-
|Slendric
|5
|231-236c
|8/7
|
|}

== In just intonation ==
Unlike with other interval categories, it is often somewhat prudent to consider each second-sized interval by its delta rather than its prime limit, due to the general asymmetry of interval complexity when it comes to seconds (the m3-complement of 11/10 is ''320/297''!). Therefore, ''superparticular'' seconds (8/7 through 28/27) will be considered first (although themselves sorted by prime-limit), and then other miscellaneous second-sized intervals after that. Emphasis will be placed on the intervals separated by each second; this and a corresponding section on the [[Diesis]] page will serve to help readers understand the "Edostep interpretations" sections of edo pages. The FJS notation for each interval will also be provided, to illuminate whether the interval is actually considered a second in JI notation systems (and thus interval arithmetic) or something else like an imperfect or augmented unison. Also, intervals provided are at most up to the 23-limit, and up to the 13-limit for delta-2 and above.

=== Superparticular ===

==== 5-limit ====
{| class="wikitable"
|+
!Just ratio
!FJS
!Size
!Notes
!Intervals separated
|-
|9/8
|M2
|204c
|
|3/2-4/3
|-
|10/9
|M2^5
|182c
|
|9/8-5/4, 3/2-5/3
|-
|16/15
|m2_5
|112c
|See [[Father]]
|5/4-4/3, 9/8-6/5
|-
|25/24
|A1^25
|70c
|See [[Dicot]]
|6/5-5/4, 16/15-10/9
|}

==== 7-limit ====
{| class="wikitable"
!Just ratio
!FJS
!Size
!Notes
!Intervals separated
|-
|8/7
|M2_7
|231c
|
|3/2-7/4
|-
|15/14
|A1^5_7
|119c
|
|7/6-5/4, 7/5-3/2
|-
|21/20
|m2^7_5
|84c
|
|4/3-7/5, 5/3-7/4
|-
|28/27
|m2^7
|63c
|See [[Trienstonian]]
|9/8-7/6, 9/7-4/3
|}

==== 11-limit ====
{| class="wikitable"
!Just ratio
!FJS
!Neutral FJS
!Size
!Notes
!Intervals separated
|-
|11/10
|m2^11/5
|n2^11/5
|165c
|
|10/9-11/9
|-
|12/11
|M2_11
|n2_11
|151c
|
|11/9-4/3
|-
|22/21
|P1^11_7
|sA1^11_7
|81c
|This is 33/32 * 64/63, hence 14/11 being an imperfect fourth.
|14/11-4/3
|}

==== 13-limit ====
{| class="wikitable"
!Just ratio
!FJS
!Neutral FJS
!Size
!Notes
!Intervals separated
|-
|13/12
|m2^13
|n2^13
|139c
|
|3/2-13/8
|-
|14/13
|M2^7_13
|n2^7_13
|128c
|
|13/8-7/4
|-
|26/25
|d2^13_25
|sd2^13_25
|68c
|
|5/4-13/10, 13/8-25/16
|-
|27/26
|A1_13
|sA1_13
|65c
|
|27/16-13/8
|}

==== Higher limit ====
{| class="wikitable"
!Just ratio
!Limit
!FJS
!Neutral FJS
!Size
!Notes
!Intervals separated
|-
|17/16
|17
| colspan="2" |m2^17
|105c
|
|4/3-17/12
|-
|18/17
|17
| colspan="2" |A1_17
|99c
|
|17/16-9/8
|-
|19/18
|19
| colspan="2" |m2^19
|94c
|
|19/16-9/8
|-
|20/19
|19
| colspan="2" |A1^5_19
|89c
|
|5/4-19/16
|-
|23/22
|23
|A1^23_11
|sA1^23_11
|77c
|
|23/16-11/8
|-
|24/23
|23
| colspan="2" |m2_23
|74c
|
|3/2-23/16
|}

=== Delta-2 ===
{| class="wikitable"
|+
!Just ratio
!Limit
!FJS
!Neutral FJS
!Size
!Notes
!Intervals separated
|-
|15/13
|13
|A2^5_13
|sA2^5_13
|247c
|[[Interordinal]].
|13/10-3/2
|-
|27/25
|5
| colspan="2" |m2_25
|133c
|Well-approximated by 9edo.
|25/24-9/8
|-
|35/33
|11
|M2^35_11
|n2^35_11
|102c
|
|3/2-99/70
|}

=== Delta-3 ===
{| class="wikitable"
!Just ratio
!Limit
!FJS
!Neutral FJS
!Size
!Notes
!Intervals separated
|-
|25/22
|11
|A2^25_11
|sA2^25_11
|221c
|
|11/10-5/4
|-
|28/25
|7
| colspan="2" |d3^7_25
|196c
|
|8/7-32/25
|-
|35/32
|7
| colspan="2" |M2^35
|155c
|
|8/7-5/4
|-
|52/49
|13
|P1^13_49
|sA1^13_49
|103c
|
|13/8-49/32
|-
|55/52
|13
| colspan="2" |A1^55_13
|97c
|
|13/11-5/4, 13/10-11/8
|-
|80/77
|11
|A1^5_77
|sA1^5_77
|66c
|
|11/10-8/7
|}

=== Delta-4 ===
{| class="wikitable"
!Just ratio
!Limit
!FJS
!Neutral FJS
!Size
!Notes
!Intervals separated
|-
|39/35
|13
|m2^13_35
|n2^13_35
|187c
|
|7/6-13/10
|-
|49/45
|7
| colspan="2" |d3^49_5
|147c
|
|9/7-7/5, 15/14-7/6
|-
|81/77
|11
|A1_77
|sA1_77
|88c
|
|9/7-11/9
|}

=== Delta-5 and -6 ===
{| class="wikitable"
!Just ratio
!Limit
!FJS
!Neutral FJS
!Size
!Notes
!Intervals separated
|-
|44/39
|13
| colspan="2" |M2^11_13
|209c
|
|13/11-4/3
|-
|49/44
|11
|m3^49/11
|sd3^49/11
|186c
|
|11/7-7/4
|-
|54/49
|7
| colspan="2" |A1_49
|168c
|Septimal chromatic semitone.
|7/6-9/7
|-
|77/72
|11
|m2^77
|n2^77
|116c
|
|12/11-7/6, 8/7-11/9, 9/7-11/8
|-
|96/91
|13
|A1_7,13
|sA1_7,13
|93c
|
|
|-
|104/99
|13
| colspan="2" |m2^13_11
|85c
|
|
|-
|117/112
|13
|P1^13_7
|sA1^13_7
|76c
|
|
|-
|126/121
|11
|M2^7_121
|m2^7_121
|70c
|
|
|-
|55/49
|11
|A1^55_49
|3/2-A1^55_49
|200c
|
|7/5-11/7
|}

=== More complex ratios ===
{| class="wikitable"
!Just ratio
!FJS
!Size
!Notes
!Intervals separated
|-
|135/128
|A1^5
|92c
|See [[Mavila]]
|16/15-9/8
|-
|800/729
|M2^25
|161c
|
|27/20-40/27
|}

== Ground's scheme for neutral second 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
|-
|121.243
|7/6 / 4√(7/5)
|Quadranseptimal Supraminor Second
|1\10
|120.000
| -1.243
|-
|124.511
|4√(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
|5√(3/2)
|Quintanpyth Neutral Second
|2\17
|141.176
|0.785
|-
|145.628
|4√(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
|3√(4/3)
|Trienpyth Submajor Second
|4\29
|165.517
| -0.498
|-
|170.013
|6/5 / 4√(7/5)
|Quadranseptimal Submajor Second
|1\7
|171.429
|1.415
|-
|175.489
|4√(3/2)
|Quadranpyth Major Second
|6\41
|175.610
|0.121
|}

Second - Revision history

Vector at 10:01, 6 March 2026