BPS: Difference between revisions

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{{Infobox regtemp

| Title = BPS

| Subgroups = 3.5.7

| Comma basis = [[245/243]]

| Edo join 1 = b13 | Edo join 2 = b17

| Mapping = 1; 2 -1

| Generators = 9/7

| Generators tuning = 440.7

| Optimization method = CWE

| MOS scales = [[4L 1s (3/1-equivalent)|4L 1s <3/1>]], [[4L 5s (3/1-equivalent)|4L 5s <3/1>]]

| Color name =

| Odd limit 1 = 7 | Mistuning 1 = 4.73 | Complexity 1 = 4

| Odd limit 2 = 3.5.7 49 | Mistuning 2 = 9.46 | Complexity 2 = 9

}}

'''BPS''' (for ''Bohlen–Pierce–Stearns'') is a [[non-octave]] [[regular temperament|temperament]] in the 3.5.7 [[subgroup]] generated by a sharp [[~]][[9/7]] (or equivalently a flat ~[[7/3]]), [[tempering out]] the sensamagic comma ([[245/243]]) so that a stack of two generators represents [[5/3]] in addition to 81/49. This generates a [[mos scale]] of {{mos scalesig|4L 5s<3/1>|link=1}}, known as the Bohlen–Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13edt]], representing the equal-tempered [[Bohlen–Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17edt]], [[30edt|7\30edt]], and [[43edt|10\43edt]].

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<nowiki/>* In 3.5.7-targeted [[DKW theory|DKW]] tuning

== Tuning spectrum ==

== Tunings ==

=== Norm-based tunings ===

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 3.5.7-subgroup norm-based tunings

|-

! rowspan="2" |

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Tenney

| CTE: ~9/7 = 441.1431{{c}}

| CWE: ~9/7 = 440.6646{{c}}

| POTE: ~9/7 = 440.4881{{c}}

|}

=== Other tunings ===

* [[DKW theory|DKW]] (3.5.7): ~3 = 1901.955{{c}}, ~9/7 = 440.554{{c}}

=== Tuning spectrum ===

{| class="wikitable center-all left-4"

|-

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|}

<nowiki/>* Besides the [[3/1|tritave]]

~~=== Other tunings ===~~

* [[DKW theory|DKW]] (3.5.7): ~3 = 1901.955, ~9/7 = 440.554

[[Category:BPS| ]]

@@ Line 1: / Line 1: @@
+{{Infobox regtemp
+| Title = BPS
+| Subgroups = 3.5.7
+| Comma basis = [[245/243]]
+| Edo join 1 = b13 | Edo join 2 = b17
+| Mapping = 1; 2 -1
+| Generators = 9/7
+| Generators tuning = 440.7
+| Optimization method = CWE
+| MOS scales = [[4L 1s (3/1-equivalent)|4L 1s <3/1>]], [[4L 5s (3/1-equivalent)|4L 5s <3/1>]]
+| Color name =
+| Odd limit 1 = 7 | Mistuning 1 = 4.73 | Complexity 1 = 4
+| Odd limit 2 = 3.5.7 49 | Mistuning 2 = 9.46 | Complexity 2 = 9
+}}
 {{Wikipedia|Bohlen–Pierce scale}}
 '''BPS''' (for ''Bohlen–Pierce–Stearns'') is a [[non-octave]] [[regular temperament|temperament]] in the 3.5.7 [[subgroup]] generated by a sharp [[~]][[9/7]] (or equivalently a flat ~[[7/3]]), [[tempering out]] the sensamagic comma ([[245/243]]) so that a stack of two generators represents [[5/3]] in addition to 81/49. This generates a [[mos scale]] of {{mos scalesig|4L 5s<3/1>|link=1}}, known as the Bohlen–Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13edt]], representing the equal-tempered [[Bohlen–Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17edt]], [[30edt|7\30edt]], and [[43edt|10\43edt]].
@@ Line 67: / Line 81: @@
 <nowiki/>* In 3.5.7-targeted [[DKW theory|DKW]] tuning
-== Tuning spectrum ==
+== Tunings ==
+=== Norm-based tunings ===
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 3.5.7-subgroup norm-based tunings
+|-
+! rowspan="2" |
+! colspan="3" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+! Destretched
+|-
+! Tenney
+| CTE: ~9/7 = 441.1431{{c}}
+| CWE: ~9/7 = 440.6646{{c}}
+| POTE: ~9/7 = 440.4881{{c}}
+|}
+=== Other tunings ===
+* [[DKW theory|DKW]] (3.5.7): ~3 = 1901.955{{c}}, ~9/7 = 440.554{{c}}
+=== Tuning spectrum ===
 {| class="wikitable center-all left-4"
 |-
@@ Line 191: / Line 226: @@
 |}
 <nowiki/>* Besides the [[3/1|tritave]]
-=== Other tunings ===
-* [[DKW theory|DKW]] (3.5.7): ~3 = 1901.955, ~9/7 = 440.554
 [[Category:BPS| ]] <!-- main article -->