Neutral third

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Neutral third

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Name	Neutral third
Lower bound	330¢ – 340¢
Upper bound	360¢ – 370¢
Generated MOSes	4L 3s, 3L 4s, 7L 3s, 3L 7s
Example JI intervals
Intervals	11/9 (347.4¢) 16/13 (359.5¢)
Related regions
Complement	Neutral sixth

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Neutral third

A neutral third (n3), as a concrete interval region, 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.

Diatonically, neutral thirds appear as part of the variant of diatonic with generators halved, where the neutral third is the generator and the 600-cent tritone is the period.

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 (though hemipythagorean has an irrational sqrt(3/2) interval that might be considered the "canonical" neutral third), 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 √(3/2), 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	Breed*
11/9, 27/22	Neutral
39/32, 16/13	Temperament of 512/507
17/14, 21/17	Temperament of 294/289

* Breed is a rank-3 temperament, the other generator being ~7/5

In moment-of-symmetry scales

Intervals between 327 and 400 ¢ generate the following mos scales:

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

Scales with more than 12 notes are not included.

Range	Mos
327–343 ¢	1L 2s	3L 1s	4L 3s	7L 4s
343–360 ¢			3L 4s	7L 3s
360–400 ¢				3L 7s