User:Unque/Buzzard - Revision history

Fredg999: /* Subgroup Extensions */ Fix typo

2025-10-13T15:51:40Z

Subgroup Extensions: Fix typo

← Older revision		Revision as of 15:51, 13 October 2025
Line 9:		Line 9:

	=== Canonical Buzzard ===		=== 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 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 aberrisma 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).		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).

Unque at 18:13, 12 October 2025

2025-10-12T18:13:58Z

← Older revision		Revision as of 18:13, 12 October 2025
Line 28:		Line 28:
	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.		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 [[~~413343~~/~~409600~~]]). 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.		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 ==		== Interval Chain ==

Unque at 18:07, 12 October 2025

2025-10-12T18:07:06Z

← Older revision		Revision as of 18:07, 12 October 2025
Line 28:		Line 28:
	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.		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 413343/409600). 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.		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 [[413343/409600]]). 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 ==		== Interval Chain ==
Line 271:		Line 271:

	=== Buzzard[5] ===		=== Buzzard[5] ===
	<pre>! buzzard5.scl		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]].<pre>! buzzard5.scl
	!		!
	Buzzard[5] 2\|2 mode (LsLsL) in 19\48 tuning		Buzzard[5] 2\|2 mode (LsLsL) in 19\48 tuning
Line 281:		Line 281:

	=== Buzzard[8] ===		=== Buzzard[8] ===
	<pre>		The 8-note [[MOS scale\|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.<pre>
	! buzzard8.scl		! buzzard8.scl
	!		!
Line 296:		Line 296:

	=== Buzzard[13] ===		=== Buzzard[13] ===
	<pre>		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]].<pre>
	! buzzard13.scl		! buzzard13.scl
	!		!
Line 326:		Line 326:
	\|0-comma		\|0-comma
	\|470.781		\|470.781
	\|21/1		\|21
	\|Degenerate case; lower boundary of Buzzard targets.		\|Degenerate case; lower boundary of Buzzard targets.
	\|-		\|-
	\|1/4-comma		\|1/4-comma
	\|475.489		\|475.489
	\|3/1		\|3
	\|		\|
	\|-		\|-
Line 346:		Line 346:
	\|1/3-comma		\|1/3-comma
	\|477.058		\|477.058
	\|7/1		\|7
	\|		\|
	\|-		\|-
	\|1/2-comma		\|1/2-comma
	\|480.192		\|480.192
	\|147/1		\|147
	\|5-limit MOS is almost completely equalized.		\|5-limit MOS is almost completely equalized.
	\|-		\|-
	\|1-comma		\|1-comma
	\|489.612		\|489.612
	\|3087/1		\|3087
	\|Degenerate case; upper boundary of Buzzard targets.		\|Degenerate case; upper boundary of Buzzard targets.
	\|}		\|}
Line 369:		Line 369:
	\|[[28edo\|11\28]]		\|[[28edo\|11\28]]
	\|471.429		\|471.429
	\|		\|Semi-Aberrismic
	\|Near-just 5/4		\|Near-just 5/4
	\|-		\|-
	\|[[33edo\|13\33]]		\|[[33edo\|13\33]]
	\|472.727		\|472.727
	\|~~Unattested~~		\|Semi-Aberrismic
	\|		\|
	\|-		\|-

Unque: Created page with "'''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 giv..."

2025-10-12T17:55:23Z

Created page with "'''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 giv..."

New page

'''Buzzard''' is the 2.3.7-subgroup 53 & 58 temperament and its extensions. It splits the [[3/1|third harmonic]] into four equal parts, each part representing [[21/16]]; thus, it [[Temperament|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<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html#10551</ref>; this name was chosen due to 2.3.7 Buzzard's superficial resemblance to [[Vulture family|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 413343/409600). 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'''.
{| class="wikitable"
|+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] ===
<pre>! buzzard5.scl
!
Buzzard[5] 2|2 mode (LsLsL) in 19\48 tuning
250.0
475.0
725.0
950.0
2/1</pre>

=== Buzzard[8] ===
<pre>
! 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
</pre>

=== Buzzard[13] ===
<pre>
! 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
</pre>

== Tunings ==
Below is a table of useful target tunings for 7-limit Buzzard:
{| class="wikitable"
|+Buzzard Targets
!Tuning
!Generator (cents)
!Engenmonzo
!Notes
|-
|0-comma
|470.781
|21/1
|Degenerate case; lower boundary of Buzzard targets.
|-
|1/4-comma
|475.489
|3/1
|
|-
|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
|
|-
|1/2-comma
|480.192
|147/1
|5-limit MOS is almost completely equalized.
|-
|1-comma
|489.612
|3087/1
|Degenerate case; upper boundary of Buzzard targets.
|}
Below is a table of [[EDO]] tunings that support 2.3.7 Buzzard:
{| class="wikitable"
|+Buzzard EDOs
!Tuning
!Generator (cents)
!Extension
!Notes
|-
|[[28edo|11\28]]
|471.429
|
|Near-just 5/4
|-
|[[33edo|13\33]]
|472.727
|Unattested
|
|-
|[[38edo|15\38]]
|473.684
|Meantone
|
|-
|[[43edo|17\43]]
|474.419
|Meantone
|
|-
|[[48edo|19\48]]
|475.000
|Buteo
|
|-
|[[53edo|21\53]]
|475.472
|Buteo/Canonical
|Near-just prime 3
|-
|[[58edo|23\58]]
|475.862
|Canonical
|
|-
|[[63edo|25\63]]
|476.190
|Lemongrass
|Roughly equal error on primes 3 and 7, making for near-just 7/6
|-
|[[68edo|27\68]]
|476.471
|Lemongrass
|
|-
|[[73edo|29\73]]
|476.712
|Lemongrass
|Near-just prime 7
|-
|[[78edo|31\78c]]
|476.923
|Lemongrass
|Near-just prime 7; requires warts to support Lemongrass mapping of prime 5.
|-
|[[5edo|2\5]]
|480.000
|All/None
|Tempers out aberrisma; supports all extensions in theory but no extensions in practice.
|}

== Notes ==