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 cents. A rough tuning range for the neutral third is 330 to 370 cents 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 cents 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 cents 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 cents 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 cents 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 cents respectively.

By delta

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

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	343c
17	353c
24	350c
25	-	336c
26	-	*	369c
27	356c
29	-	331c	*
31	348c
34	353c
41	351c
53	-	340c	362c

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