Second

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 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.

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

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

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

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

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

Just ratio	Limit	FJS	Neutral FJS	Size	Notes	Intervals separated
17/16	17	m2^17		105c		4/3-17/12
18/17	17	A1_17		99c		17/16-9/8
19/18	19	m2^19		94c		19/16-9/8
20/19	19	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	m2_23		74c		3/2-23/16

Delta-2

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	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

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	d3^7_25		196c		8/7-32/25
35/32	7	M2^35		155c		8/7-5/4
52/49	13	P1^13_49	sA1^13_49	103c		13/8-49/32
55/52	13	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

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	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

Just ratio	Limit	FJS	Neutral FJS	Size	Notes	Intervals separated
44/39	13	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	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	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

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

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