Neutral third

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Neutral third

A neutral third (n3) is an interval that spans two steps of the diatonic scale with a quality between major and minor. It exists in neutralized diatonic scales as exactly one half of a perfect fifth.

In just intonation, an interval may be classified as a neutral third if it is reasonably mapped to 2\7 and 7\24 (precisely two steps of the diatonic scale and three and a half steps of the chromatic scale).

As a concrete interval region, it is typically near 350 cents in size, distinct from the minor third of roughly 300 cents and the major third of roughly 400 ¢. A rough tuning range for the neutral third is 330 to 370 ¢ according to Margo Schulter's theory of interval regions.

Two neutral thirds stack to a perfect fifth, and as such the neutral third range is generally divided at roughly 350 ¢ into artoneutral (flatter) and tendoneutral (sharper) thirds. As such, neutral thirds tend to exist in pairs.

In just intonation

By prime limit

The 3-limit and 5-limit do not have simple neutral thirds, so we start with the 7-limit:

• The 7-limit artoneutral and tendoneutral thirds are the ratios of 60/49 and 49/40 respectively, and they are slightly flat of and slightly sharp of 351 ¢ respectively.
• The 11-limit alpharabian artoneutral and tendoneutral thirds are the ratios of 11/9 and 27/22 respectively, and they are about 347 and 355 ¢ respectively.
• The 13-limit artoneutral and tendoneutral thirds are the ratios of 39/32 and 16/13 respectively, and they are about 342 and 359 ¢ respectively.
• The 17-limit supraminor and submajor thirds are the ratios of 17/14 and 21/17 respectively, and they are about 336 and 366 ¢ respectively.

By delta

See Delta-N ratio.

Delta 2		Delta 3		Delta 4		Delta 5
11/9	347 ¢	16/13	359 ¢	21/17	365 ¢	26/21	370 ¢
		17/14	336 ¢	23/19	330 ¢	27/22	355 ¢
						28/23	341 ¢

In edos

The following table lists the best tuning of 39/32 and 16/13 in various significant edos. For applicable edos, it also lists one half of the edo's perfect fifth, approximating 1\2edf, which, while not a just interval, is the "canonical" neutral third tuning, as stacking two of them gives 3/2.

Edo	1\2edf	39/32	16/13
7	343 ¢
17	353 ¢
24	350 ¢
25	—	336 ¢
26	—	*	369 ¢
27	356 ¢
29	—	331 ¢	*
31	348 ¢
34	353 ¢
41	351 ¢
53	—	340 ¢	362 ¢

In regular temperaments

Temperaments generated by neutral thirds often involve tempering a pair of neutral thirds together. As such, each pair of neutral thirds has a corresponding temperament, which equates both neutral thirds to half of a perfect fifth:

Pair of neutral thirds	Temperament
60/49, 49/40	Breedsmic
11/9, 27/22	Rastmic
39/32, 16/13	Temperament of 512/507
17/14, 21/17	Temperament of 294/289

In moment-of-symmetry scales

Intervals between 327 and 400 cents generate the following MOS scales:

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

MOSes with more than 12 notes are not included.

Range	MOS
327-343c	1L 2s	3L 1s	4L 3s	7L 4s
343-360c			3L 4s	7L 3s
360-400c				3L 7s