Buzzard

Buzzard is a temperament that splits a tempered perfect twelfth (3/1) into four generators of 21/16 subfourths, tempering out the buzzardsma (ratio: 65536/64827). Two generators therefore give us a semitwelfth, and five give us a sub-octave just short of the octave by a septimal comma. Bending the semitwelfth up by a septimal comma results in ~7/4, and down results in ~12/7, with the implication that the septimal diesis of 49/48 is equated to two septimal commas. In fact, buzzard slices the Pythagorean limma into four, one for 64/63, two for 49/48, and three for 28/27.

By finding harmonic 5 twenty-one generators away, buzzard is extended to the full 7-limit, where it tempers out 1728/1715 and 5120/5103. This equates the syntonic comma with the septimal comma and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals. Buzzard then extends naturally to the 13-limit by identifying the semitwelfth as 26/15, and identifying the comma step as the ptolemisma (100/99, S10). This means 176/175, 351/350, 540/539, and 676/675 all vanish.

Finally, it is possible to extend buzzard to the 19-limit, where it merges 17/16 and 16/15, tempering out 256/255 (S16), and merges 26/15 and 19/11, tempering out 286/285.

Buzzard can be tuned to 53edo, 58edo, or 111edo. Mos scales of buzzard cluster strongly around 5edo, similar to those of rodan (see #As a detemperament of 5et). Rather than directly using mos scales, which are either extremely imbalanced or overly large, an approach to buzzard may involve picking and choosing which intervals from each pentatonic category to keep in the scale.

Alternative extensions of 2.3.7-subgroup buzzard include subfourth (58 & 63) and lemongrass (63 & 68).

Buzzard was named by Herman Miller in 2004.

See Buzzardsmic clan #Buzzard for technical data.

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are in bold.

#	Cents*	Approximate ratios
		13-limit	19-limit extension
0	0.00	1/1
1	475.68	21/16
2	951.35	26/15	19/11
3	227.03	8/7
4	702.70	3/2
5	1178.38	63/32, 99/50, 160/81
6	454.06	13/10
7	929.73	12/7
8	205.41	9/8
9	681.08	40/27
10	1156.76	35/18, 39/20, 96/49
11	432.44	9/7
12	908.11	22/13, 27/16
13	183.79	10/9
14	659.46	22/15	19/13
15	1135.14	25/13, 27/14, 52/27
16	410.82	33/26	19/15
17	886.49	5/3
18	162.17	11/10
19	637.84	13/9
20	1113.52	40/21	19/10
21	389.20	5/4
22	864.87	33/20	28/17
23	140.55	13/12
24	616.22	10/7
25	1091.90	15/8	32/17
26	367.58	26/21	21/17
27	843.25	13/8
28	118.93	15/14
29	594.60	45/32, 55/39	24/17

* In 13-limit CWE tuning

As a detemperament of 5et

Buzzard as a 58-tone 5et detempering

Buzzard is naturally a detemperament of the 5 equal temperament. The diagram on the right shows a 58-tone detempered scale, with a generator range of -28 to +29. 58 is the largest number of tones for a mos where intervals in the 5 categories do not overlap. Each category is divided into eleven or twelve qualities separated by 5 generator steps, which represent the generic comma step.

Notice also the little interval between the largest of a category and the smallest of the next, which represents the differences between 16/15 and 15/14, between 16/13 and 26/21, between 7/5 and 45/32, between 21/13 and 13/8, and between 28/15 and 15/8. It spans 53 generator steps, so it vanishes in 53edo, but is tuned to the same size as the comma step in 58edo. 111edo tunes it to one half the size of the comma step, which may be seen as a good compromise.

Since the intervals cluster around 5edo, a notation system based on 5 tones per octave may be preferred to the standard diatonic one; see Pentatonic Functional Just System for how such a system could work.

Chords and harmony

See also: Chords of buzzard

Tunings

Norm-based tunings

2.3.7-subgroup norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~21/16 = 475.7273 ¢	CWE: ~21/16 = 475.8328 ¢	POTE: ~21/16 = 475.8717 ¢

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~21/16 = 475.5546 ¢	CWE: ~21/16 = 475.6144 ¢	POTE: ~21/16 = 475.6361 ¢

13-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~21/16 = 475.6153 ¢	CWE: ~21/16 = 475.6760 ¢	POTE: ~21/16 = 475.6972 ¢

Tuning spectrum

Edo generator	Unchanged interval (eigenmonzo)	Generator (¢)	Comments
	21/16	470.7809
19\48		475.0000	48eef val, lower bound of 7- and 9-odd-limit diamond monotone
21\53		475.4717	Lower bound of 11- through 15-odd-limit diamond monotone
	3/2	475.4888
	15/8	475.5307
	5/4	475.5387
	5/3	475.5505
	9/5	475.5695
	13/8	475.5751
	13/12	475.5901
65\164		475.6098	164d val
	13/9	475.6115
	11/8	475.6748
44\111		475.6757
	15/14	475.6944
	11/6	475.6961
	15/13	475.7023
	11/9	475.7228
	13/7	475.7234
	7/5	475.7287
67\169		475.7396	169cdf val
	21/13	475.7595
	11/7	475.7736
	21/20	475.7766
	21/11	475.8036
	11/10	475.8336
23\58		475.8621	Upper bound of 11- through 15-odd-limit diamond monotone
	13/11	475.8992
	9/7	475.9167
	15/11	475.9321
	15/13	476.1295
	7/6	476.1613
25\63		476.1905	63ceef val
	7/4	477.0580
2\5		480.0000	5e val, upper bound of 7- and 9-odd-limit diamond monotone

* Besides the octave