Buzzard: Difference between revisions

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{{Infobox ~~Regtemp~~

{{Infobox regtemp

| Title = Buzzard

| Subgroups = 2.3.7, 2.3.5.7~~, 2.3.5.7.11~~, 2.3.5.7.11.13

| 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]], [[540/539]], [[5120/5103]] (11~~-limit); [[176/175]], [[351/350]], [[540/539]], [[676/675]] (13-limit)

| Comma basis = [[65536/64827]] (2.3.7); [[1728/1715]], [[5120/5103]] (7-limit); [[176/175]], [[351/350]], [[540/539]], [[676/675]] (13-limit)

| Edo join 1 = 53 | Edo join 2 = 58

| Mapping = 1; 4 21 -3 39 27

| ~~Generator~~ = 21/16

| Generators = 21/16

| ~~Generator~~ tuning = 475.7

| Generators tuning = 475.7

| Optimization method = CWE

| MOS scales = 3L 2s

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| Pergen =

| Color name =

| Odd limit 1 = (2.3.7) 9 | Mistuning 1 = 3.42 | Complexity 1 = 13

| 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 [[65536/64827]]. If [[harmonic]] [[5/1|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]].

'''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]].

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>.

See [[Buzzardsmic clan #Buzzard]] for technical data.

~~__TOC__~~

~~{{Todo|improve synopsis}}~~

~~{{Clear}}~~

== Interval chain ==

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

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|}

<nowiki/>* In 13-limit CWE tuning

=== As a detemperament of 5et ===

[[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 ==

@@ Line 1: / Line 1: @@
-{{Infobox Regtemp
+{{Infobox regtemp
 | Title = Buzzard
-| Subgroups = 2.3.7, 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
+| 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]], [[540/539]], [[5120/5103]] (11-limit);<br>[[176/175]], [[351/350]], [[540/539]], [[676/675]] (13-limit)
+| 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
-| Generator = 21/16
+| Generators = 21/16
-| Generator tuning = 475.7
+| Generators tuning = 475.7
 | Optimization method = CWE
 | MOS scales = 3L 2s
@@ Line 12: / Line 12: @@
 | Pergen =
 | Color name =
-| Odd limit 1 = (2.3.7) 9 | Mistuning 1 = 3.42 | Complexity 1 = 13
+| 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 [[65536/64827]]. If [[harmonic]] [[5/1|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]].
+'''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]].
+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>.
 See [[Buzzardsmic clan #Buzzard]] for technical data.
-__TOC__
-{{Todo|improve synopsis}}
-{{Clear}}
 == Interval chain ==
 In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.
@@ Line 186: / Line 192: @@
 |}
 <nowiki/>* In 13-limit CWE tuning
+=== As a detemperament of 5et ===
+[[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 ==