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'''Buzzard''' is a [[regular temperament|temperament]] that splits a tempered [[3/1|perfect twelfth (3/1)]] into four [[generator]]s of [[21/16]] subfourths, tempering out [[65536/64827]].  
{{Infobox regtemp
| Title = Buzzard
| Subgroups = 2.3.7, 2.3.5.7, 2.3.5.7.11.13
| Comma basis = [[65536/64827]] (2.3.7); <br>[[1728/1715]], [[5120/5103]] (7-limit);<br>[[176/175]], [[351/350]], [[540/539]], [[676/675]]<br>(13-limit)
| Edo join 1 = 53 | Edo join 2 = 58
| Mapping = 1; 4 21 -3 39 27
| Generators = 21/16
| Generators tuning = 475.7
| Optimization method = CWE
| MOS scales = 3L 2s
| Ploidacot =
| Pergen =
| Color name =
| Odd limit 1 = 2.3.7 9 | Mistuning 1 = 3.42 | Complexity 1 = 13
| Odd limit 2 = 15 | Mistuning 2 = 4.09 | Complexity 2 = 43
}}
'''Buzzard''' is a [[regular temperament|temperament]] that splits a tempered [[3/1|perfect twelfth (3/1)]] into four [[generator]]s 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 [[2/1|octave]] by a [[64/63|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 [[256/243|Pythagorean limma]] into four, one for 64/63, two for 49/48, and three for [[28/27]].  


If harmonic 5 is desired, it is found by twenty-one generators octave-reduced, [[tempering out]] [[1728/1715]] and [[5120/5103]]. It extends to the [[13-limit]] by tempering out [[176/175]], [[351/350]], [[540/539]], and [[676/675]].  
By finding [[harmonic]] [[5/1|5]] twenty-one generators away, buzzard is [[extension and restriction|extended]] to the full [[7-limit]], where it tempers out [[1728/1715]] and [[5120/5103]]. This equates the [[81/80|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 [[100/99|ptolemisma]] (100/99, {{S|10}}). 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]] ({{S|16}}), and merges [[26/15]] and [[19/11]], tempering out [[286/285]].
 
Buzzard can be tuned to [[53edo]], [[58edo]], or [[111edo]]. [[Mos scale]]s 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|2.3.7-]][[subgroup]] buzzard include [[subfourth]] (58 & 63) and [[lemongrass]] (63 & 68).  


Buzzard was named by [[Herman Miller]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html#10551 Yahoo! Tuning Group (Archive) | ''Names for important high-complexity temperaments'']</ref>.  
Buzzard was named by [[Herman Miller]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html#10551 Yahoo! Tuning Group (Archive) | ''Names for important high-complexity temperaments'']</ref>.  


See [[Buzzardsmic clan #Buzzard]] and [[Buzzardsmic clan #Septimal buzzard]] for technical data.
See [[Buzzardsmic clan #Buzzard]] for technical data.  


== Interval chain ==
== Interval chain ==
Line 46: Line 68:
| 5
| 5
| 1178.38
| 1178.38
| 63/32, 160/81
| 63/32, 99/50, 160/81
|
|
|-
|-
Line 91: Line 113:
| 14
| 14
| 659.46
| 659.46
| 35/24
| 22/15
| 19/13
| 19/13
|-
|-
| 15
| 15
| 1135.14
| 1135.14
| 27/14
| 25/13, 27/14, 52/27
|
|
|-
|-
Line 163: Line 185:
| 15/14
| 15/14
|  
|  
|-
| 29
| 594.60
| 45/32, 55/39
| 24/17
|}
|}
<nowiki/>* In 13-limit CWE tuning
<nowiki/>* In 13-limit CWE tuning


== Chords ==
=== As a detemperament of 5et ===
{{Main| Chords of buzzard }}
[[File: Buzzard 5et Detempering.png|thumb|Buzzard as a 58-tone 5et detempering]]
 
Buzzard is naturally a [[detemperament]] of the [[5edo|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 ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~21/16 = 475.7273{{c}}
| CWE: ~21/16 = 475.8328{{c}}
| POTE: ~21/16 = 475.8717{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~21/16 = 475.5546{{c}}
| CWE: ~21/16 = 475.6144{{c}}
| POTE: ~21/16 = 475.6361{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~21/16 = 475.6153{{c}}
| CWE: ~21/16 = 475.6760{{c}}
| POTE: ~21/16 = 475.6972{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
|-
|
| [[21/16]]
| 470.7809
|
|-
| [[48edo|19\48]]
|
| 475.0000
| 48eef val, lower bound of 7- and 9-odd-limit diamond monotone
|-
| [[53edo|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
|
|-
| [[164edo|65\164]]
|
| 475.6098
| 164d val
|-
|
| [[13/9]]
| 475.6115
|
|-
|
| [[11/8]]
| 475.6748
|
|-
| [[111edo|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
|
|-
| [[169edo|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
|
|-
| [[58edo|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
|
|-
| [[63edo|25\63]]
|
| 476.1905
| 63ceef val
|-
|
| [[7/4]]
| 477.0580
|
|-
| [[5edo|2\5]]
|
| 480.0000
| 5e val, upper bound of 7- and 9-odd-limit diamond monotone
|}
<nowiki>*</nowiki> Besides the octave


== Notes ==
== References ==


[[Category:Buzzard| ]] <!-- main article -->
[[Category:Buzzard| ]] <!-- main article -->
[[Category:Temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Vulture family]]
[[Category:Vulture family]]
[[Category:Buzzardsmic clan]]
[[Category:Orwellismic temperaments]]
[[Category:Orwellismic temperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Hemifamity temperaments]]

Latest revision as of 09:57, 9 February 2026

Buzzard
Subgroups 2.3.7, 2.3.5.7, 2.3.5.7.11.13
Comma basis 65536/64827 (2.3.7);
1728/1715, 5120/5103 (7-limit);
176/175, 351/350, 540/539, 676/675
(13-limit)
Reduced mapping ⟨1; 4 21 -3 39 27]
ET join 53 & 58
Generators (CWE) ~21/16 = 475.7 ¢
MOS scales 3L 2s
Ploidacot alpha-tetracot
Minimax error 2.3.7 9-odd-limit: 3.42 ¢;
15-odd-limit: 4.09 ¢
Target scale size 2.3.7 9-odd-limit: 13 notes;
15-odd-limit: 43 notes

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[1].

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

References

  1. ↑ Yahoo! Tuning Group (Archive) | Names for important high-complexity temperaments
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