User:Unque/Buzzard

Buzzard is the 2.3.7-subgroup 53 & 58 temperament and its extensions. It splits the third harmonic into four equal parts, each part representing 21/16; thus, it tempers out the comma 65536/64827. This temperament provides a useful interpretation of radically near-equal pentatonic scales using the 2.3.7 subgroup, and it can be used to generate scales that further extend this structure via small inflections.

The name Buzzard was given to this temperament by Herman Miller in 2004; this name was chosen due to 2.3.7 Buzzard's superficial resemblance to Vulture temperament, similar to how Buzzards in real life superficially resemble Vultures.

For technical data on temperaments that temper out 65536/64827, see Buzzardsmic clan.

Subgroup Extensions

Prime 5 is notably lacking from Buzzard's subgroup; several extensions exist to introduce it, as well as extensions to reach higher primes 11 and 13.

Canonical Buzzard

The interval at five generators up falls short of the octave by a small interval (namely 64/63, the septimal comma), which in Buzzard functions as an aberrisma. In 2.3.7 Buzzard, this aberrisma can be used to inflect between 3-limit and 7-limit consonances, such as between 21/16 and 4/3, or between 9/8 and 8/7. The canonical extension to include prime 5 is created by using this amerrisma as an all-purpose inflection, allowing 5-limit consonances to be accessed by inflecting in the opposite direction; for instance, 9/8 can be inflected upwards to reach 8/7, or downwards to reach 10/9. To do this, it tempers out 5120/5103, equating 64/63 with 81/80.

The canonical extension to 11-limit further equates this aberrisma with 100/99 (thus tempering out 176/175), allowing 11-limit consonances to be reached via similar inflections; for instance, 11/9 can be found as an inflection downwards from 5/4 (or equivalently, two inflections below 81/64, or three below 9/7), and 15/11 can be found as an inflection upwards from 27/20 (or two inflections above 4/3, or three above 21/16).

Finally, we can attain 13-limit consonances in this structure by equating the aberrisma with 105/104 (thus tempering out 351/350). This allows us to find 13/11 as an inflection downwards from 7/6 (or two inflections below 32/27, or three below 6/5), and 15/13 as an inflection upwards from 8/7 (or two inflections above 9/8, or three above 10/9). Note that, if one does the math, these two intervals fall on the same place in the genchain (thus tempering out 169/165 by proxy).

Buteo

Buteo is a different way to extend this inflection system. It tempers out 10976/10935 just like canonical buzzard, allowing the 5-limit consonances to be reached in the same manner; however, it introduces the 11-limit consonances instead by equating the aberrisma with 55/54, thus tempering out 2200/2187. 11/8, for instance, can be reached as an inflection upwards from 27/20 (or two inflections above 4/3, or three above 21/16), and 11/9 can be reached as an inflection upwards from 6/5 (or two inflections above 32/27, or three above 7/6).

Just like in canonical Buzzard, the 13-limit consonances in Buteo are achieved by equating the aberrisma with 105/104 (thus tempering out 351/350).

Lemongrass

Another useful extension to Buzzard is known as Lemongrass. This extension considers two types of aberrisma intervals rather than just the one type that was considered by canonical Buzzard: the basic aberrisma at five generators, which represents 64/63, and the double aberrisma found at ten generators, which represents 49/48. Lemongrass finds the 5-limit consonances by using the double aberrisma as an inflection (equating 49/48 with 81/80, thus tempering out 245/243), rather than using the basic aberrisma as the previous extensions did.

In order to create this property, the basic aberrisma in Lemongrass is narrower than it is in canonical Buzzard.

Other Extensions

While it makes little sense from the perspective of a target tuning, Meantone occurs quite often in low EDO tunings of 2.3.7 Buzzard, such as 38edo and 43edo, making it a reasonable extension to the Buzzard structure. This extension finds 5-limit consonances by equating them with 3-limit consonances, or equivalently by equating the aberrisma with 36/35 (thus tempering out 81/80). 5/4, for instance, can be reached at one inflection below 9/7, and 5/3 at one inflection below 12/7.

Another possibility is to split the aberrisma into two "semi-aberrisma" intervals; this allows the 5-limit consonances to be reached as a midpoint between existing 3-limit and 7-limit consonances (thus tempering out 729/700). 5/4, for instance, can be reached as the midpoint of 81/64 and 9/7; and 5/3 as the midpoint of 27/16 and 12/7. While this extension is of lower accuracy than Buteo or Canonical, it is supported by many low-EDO tunings of 2.3.7 Buzzard, such as 28edo and 33edo.

Interval Chain

The following is the list of intervals generated by a continuous chain of 21/16 generators. Octave-reduced odd harmonics or subharmonics are noted in bold.

Interval Chain

Gens Up	7-limit (Canonical)	7-limit (Lemongrass)	7-limit (Meantone)	11-limit (Canonical)	11-limit (Buteo)	13-limit (Canonical)
1	21/16	21/16	21/16
2	441/256	441/256	441/256	33/25		22/13, 26/15
3	8/7	8/7	8/7	25/22
4	3/2	3/2	3/2
5	160/81	63/32	35/18	99/50	108/55	208/105
6				176/135		13/10
7	12/7	12/7	12/7
8	9/8	9/8	9/8, 10/9
9	40/27	189/128	35/24
10	96/49	160/81		88/45		39/20
11	9/7	9/7	9/7
12	27/16	27/16	5/3
13	10/9
14	72/49	40/27		22/15	16/11
15	27/14	27/14	27/14
16	81/64	81/64	5/4
17	5/3
18	54/49	10/9	54/49	11/10
19	81/56	81/56
20	243/128	243/128	15/8
21	5/4
22	81/49	5/3		33/20	18/11
23	243/224		15/14
24	729/512
25	15/8
26	243/196	5/4		11/9
27						13/8
28	2187/2048	2187/2048	25/24

Chords and Scales

Main article: Buzzard/Chords

Buzzard[5]

Buzzard gives us a roughly-equal pentatonic scale of 3L 2s; the large step and small step differ by a tiny step that represents 64/63.

! buzzard5.scl
!
Buzzard[5] 2|2 mode (LsLsL) in 19\48 tuning
250.0
475.0
725.0
950.0
2/1

Buzzard[8]

The 8-note MOS introduces three small steps of 64/63, and thus it has the pattern 5L 3s. These small steps act as inflections and allow more complex consonances to be reached, while the disparity between the step sizes allows the structure to largely adhere to the pentatonic form despite the greater number of notes.

! buzzard8.scl
!
Buzzard[8] 4|3 mode (LsLLsLsL) in 19\48 tuning
225.0
250.0
475.0
700.0
725.0
950.0
975.0
2/1

Buzzard[13]

The 13-note MOS functions much the same as the 8-note one, using small inflections to reach nearby consonances in the largely pentatonic structure; however, unlike the 8-note MOS, it includes certain chains of two consecutive inflections in a row, which allows for more complex inflections. This scale has the pattern 5L 8s.

! buzzard13.scl
!
Buzzard[13] 6|6 mode (sLsLssLssLsLs) in 19\48 tuning
25.0
225.0
250.0
450.0
475.0
500.0
700.0
725.0
750.0
950.0
975.0
1175.0
2/1

Tunings

Below is a table of useful target tunings for 7-limit Buzzard:

Buzzard Targets

Tuning	Generator (cents)	Engenmonzo	Notes
0-comma	470.781	21	Degenerate case; lower boundary of Buzzard targets.
1/4-comma	475.489	3
3/11-comma	475.917	9/7
2/7-comma	476.161	7/3	Efficient way to divide error among primes 3 and 7.
1/3-comma	477.058	7
1/2-comma	480.192	147	5-limit MOS is almost completely equalized.
1-comma	489.612	3087	Degenerate case; upper boundary of Buzzard targets.

Below is a table of EDO tunings that support 2.3.7 Buzzard:

Buzzard EDOs

Tuning	Generator (cents)	Extension	Notes
11\28	471.429	Semi-Aberrismic	Near-just 5/4
13\33	472.727	Semi-Aberrismic
15\38	473.684	Meantone
17\43	474.419	Meantone
19\48	475.000	Buteo
21\53	475.472	Buteo/Canonical	Near-just prime 3
23\58	475.862	Canonical
25\63	476.190	Lemongrass	Roughly equal error on primes 3 and 7, making for near-just 7/6
27\68	476.471	Lemongrass
29\73	476.712	Lemongrass	Near-just prime 7
31\78c	476.923	Lemongrass	Near-just prime 7; requires warts to support Lemongrass mapping of prime 5.
2\5	480.000	All/None	Tempers out aberrisma; supports all extensions in theory but no extensions in practice.