The 13th Harmonic
And its relationship to 31 tone equal temperament
The 13th harmonic is a poorly understood and poorly utilized harmonic relative to the other harmonics up through 16, which find uses in various traditional musics of the world. The closest I've found to this harmonic's traditional use occurs in Byzantine Chant, which uses a 2/3 tone...but the 2/3 tone of Byzantine Chant falls roughly halfway between the two 2/3rd tones involving this harmonic. As I show below, the two-thirds tone is a good way of intuitively understanding this harmonic.
Composers who wish to utilize this harmonic and its intervals in equal temperaments would do well to explore other temperaments, discussed at the end of this article.
31 tone equal temperament poorly matches the 13th harmonic: the closest match is off by 11.09 cents, which is worse even than the 10.36 cents by which the 9th harmonic is off. There are no accidental or serendipitous matches either, produced by differences between matches in harmonics cancelling out (as with 11:9 in 31-ET). All but one of the intervals introduced by the 13th harmonic fall close to halfway between two scale degrees.
Intuitively understanding the 13th harmonic
The 13th harmonic lies between the 12th and 14th, which are just octaves above the 6th and 7th harmonics. The 13th harmonic can thus be seen as dividing the interval of the septimal minor third into two close, but slightly asymmetrical intervals, much in the same way the 11th harmonic divides the minor third, or the 9th harmonic divides the major third.
The asymmetry in the space between the 12th and 14th harmonics, however, is small, only about 10.27 cents. A match close to halfway between the two intervals will thus only be off by a little over 5 cents. These differences will be imperceptible to some people, and even people who perceive them will tend to perceive them as a minor tweak in intonation rather than a distinct musical interval.
Measuring in 12-ET interval widths, the 7:6 ratio falls very close to a third of a step short of a minor third, making it about 8/3rds of a half step, or 4/3rds of a whole tone. Half of this interval is thus close to 4/3rds of a half step, or what is called a two-thirds tone.
To develop a more fine-tuned intuition, realize that the true harmonic lies a little bit above this "2/3rds" point. The amount by which it lies above this point is a little less than the amount by which the 11th harmonic lies above the halfway point between the two points of a minor third.
Stacked major thirds: the 25th vs 26th harmonics
Another way of understanding the 13th harmonic is its relationship to stacked major thirds. An octave above the 13th harmonic is the 26th, lying between 25 and 27, both of which conveniently factor.
Stacking two just (5:4) major thirds corresponds to the 25:16 ratio. On the other hand, stacking three just (3:2) perfect fifths corresponds to the 27:8 ratio, or 27:16 plus an octave. The thirteenth harmonic lies in between these two intervals, again, slightly above the halfway point.
The intervals of the 13th harmonic
The intervals involving the 13th harmonic are described as tridecimal. Because the 14:13 and 13:12 ratios fall in between the width of intervals typically called semitones, like the diatonic semitone, and those called neutral seconds, like 11:10 and 12:11, they can be called 2/3 tones.
- Greater tridecimal 2/3 tone (13:12) - This interval is about 138.57 cents wide, making it noticeably wider than a 12-ET half-step, and slightly narrower than a 12:11 neutral second, more noticeably narrower than the 11:10 neutral second which is matched in 31-ET. 31-ET's closest match to this interval is poor: 4 scale degrees more closely matches the neutral second.
- Lesser tridecimal 2/3 tone (14:13) - At about 128.3 cents wide, this interval is a bit closer to a typical semitone in width, and farther from the two neutral seconds. It is also very close to the 15:14 septimal semitone. 31-ET matches this interval most closely with 3 scale degrees, but the match is poor.
- Tridecimal neutral third (16:13) - This interval is close to the undecimal neutral third. At about 359.47 cents, it is about 12.06 cents wider. 31-ET's neutral third, however, is narrower than the undecimal interval, making the closest approximation a full 11.09 cents off from this interval.
- Tridecimal minor third (13:11) - At 289.21 cents, this interval is between a (6:5) minor third and a septimal (7:6) minor third in width, slightly closer to the septimal interval but very close to the halfway point between these two intervals. 31-ET matches both 7:6 and 6:5 well, so it makes sense that it poorly matches this interval, which falls closest to 7 scale degrees.
- Tridecimal major third (13:10) - At 454.21 cents, this interval is wider than the septimal (9:7) major third, but much closer to this interval than to a perfect fourth. In 31-ET, this is the best-matched interval involving the 13th harmonic, but it's still off by 10.3 cents: 12 scale degrees produces an interval that is a little wider than this interval.
- Tridecimal tritone (18:13) - Yet another tritone, at about 563.38 cents width, this interval is wider than the undecimal (11:8) tritone and narrower than the septimal (7:5) tritone; it is closer to the undecimal interval.
- (15:13) - At 247.74 cents, this interval is somewhere between the intervals usually considered minor thirds and those considered wide whole tones. It is close to halfway between the 8:7 and 7:6 ratios, a little closer to the smaller, 8:7 interval. 31-ET matches both 8:7 and 7:6 closely, so accordingly, this interval is poorly matched (6 scale degrees is closest).
The 13th Harmonic in other equal temperaments
- 17 tone equal temperament - 17-ET provides a very good match to the 13th harmonic. The 13:12 and 18:13 ratios are matched exceptionally well, and the matches to 16:13 and 13:11 are reasonable. 13:10 is not usably matched.
- 24 tone equal temperament - Or Quarter Tones - 24-ET comes closer to the 13th harmonic than some tunings, but is off by 9.47 cents. Only one interval, 13:10, is matched within less than 5 cents, although 16:13 is matched within less than 10 cents.
- 34 tone equal temperament - This tuning matches the 13th harmonic better than 31-ET, but not super closely. The matches to 13:12, 18:13, and 15:13 are exceptionally close, 16:13, 13:11, and 13:10 are good, and 14:13 is poor.
- 41 tone equal temperament - 41-ET matches this harmonic better than 31-ET, but just by a little bit. The 13:11 interval is matched well, 13:12, 16:13, and 18:13 less so, and 14:13 and 13:10 very poorly.
- 43 tone equal temperament - This esoteric tuning is a meantone tuning which also provides a good match to the 13th harmonic; it also matches all the other harmonics up through 16 decently. In many respects it is the most similar to 31-ET among tunings closely matching the 13th harmonic, and represents a natural progression for people who like 31-ET but wish to utilize this harmonic.
- 46 tone equal temperament - This little-known tuning provides a pretty good match to the 13th harmonic, and giving a good match to some intervals, including 14:13, 16:13, and 13:11 well.
- 50 tone equal temperament - Another esoteric meantone tuning, 50-ET matches the 13th harmonic exceptionally well, at the cost of a poorer match to the 3rd, 7th, and 9th harmonics relative to its cousin 43-ET. The 11th harmonic is also exceptionally closely matched.
- 53 tone equal temperament - 53-ET provides a very close match (~2.79 cents) to the 13th harmonic. It is the smallest tuning that matches most smaller harmonics well and also matches this harmonic. In this tuning, all intervals involving this harmonic are matched well, with 14:13 and 13:11 being the only one that are perceptibly off...the others are all matched with less than 3 cents difference.
- 72 tone equal temperament - This very fine tuning, which gives near-perfect matches to all intervals up through the 12th harmonic, falls apart at 13th, being 7.19 cents off from this harmonic, which is a huge amount for such a fine division of the octave. It is a poor choice of tuning for people wishing to utilize this harmonic. 58-ET, 66-ET, or perhaps 87-ET or 90-ET match the intervals of this harmonic better.