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'''~~Bohlen–Pierce–Stearns~~''' (~~BPS~~) is a [[non-octave]] [[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 {{sl|4L 5s|3/1}} ~~against the tritave~~, known as the Bohlen–Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13]]~~edt~~, representing the equal-tempered [[Bohlen–Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17]]~~edt~~, [[30edt|7\30]]~~edt~~, and [[43edt|10\43]]~~edt~~.

{{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]].

As the generator of the Bohlen–Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the ~~tetrad~~ 3:5:7:9 to serve as the ~~theory~~'s primary ~~consonant tetrad~~.

As the generator of the Bohlen–Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the 3:5:7:9 tetrad to serve as BPS' primary consonance, similar to how the 4:5:6 triad serves as meantone's primary consonance.

For technical data, see ''[[Sensamagic clan#BPS]]'' or ''[[No-twos subgroup temperaments#BPS]]'' (currently, extensions with 2 are stored on the former page and no-twos extensions are stored on the latter).

For technical data, see [[Sensamagic clan #BPS]] or [[No-twos subgroup temperaments #BPS]] (currently, extensions with 2 are stored on the former page and no-twos extensions are stored on the latter).

== Extensions ==

Several extensions of this temperament are possible to incorporate additional harmonics.

In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces ~~''Mintra''~~ temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[~~Mintaka~~]], which produces {{sl|5L 2s|3/1}} and {{sl|5L 7s|3/1}} ~~MOS~~ scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].

In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces [[mintra]] temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[mintaka]], which produces {{mos scalesig|5L 2s<3/1>|link=1}} and {{mos scalesig|5L 7s<3/1>|link=1}} mos scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].

Another weak extension to add prime 17, known as ''[[~~Dubhe~~]]'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces {{sl|8L 1s|3/1}} enneatonic and {{sl|9L 8s|3/1}} chromatic ~~MOS~~ scales. Simple tunings include [[17edt]] and [[26edt]].

Another weak extension to add prime 17, known as ''[[dubhe]]'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces {{mos scalesig|8L 1s<3/1>|link=1}} enneatonic and {{mos scalesig|9L 8s<3/1>|link=1}} chromatic mos scales. Simple tunings include [[17edt]] and [[26edt]].

=== Strong extensions ===

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=== Prime 2 ===

''Main article: [[Relationship between Bohlen–Pierce and octave-ful temperaments#Relationship of rank-2 Bohlen.E2.80.93Pierce.E2.80.93Stearns temperament to octave-ful temperaments|Relationship between Bohlen–Pierce and octave-ful temperaments]].

: ''Main article: [[Relationship between Bohlen–Pierce and octave-ful temperaments#Relationship of rank-2 Bohlen.E2.80.93Pierce.E2.80.93Stearns temperament to octave-ful temperaments|Relationship between Bohlen–Pierce and octave-ful temperaments]].

It is also possible to incorporate octaves into BPS. The logical choices for a mapping of 2 are 7 generators up (equating [[2/1]] to [[125/63]]), which produces [[sensi]], and 6 generators down (equating 2/1 to [[49/25]]), which produces [[hedgehog]].

== Interval chains ==

These interval chains cover strong extensions of BPS. For ~~Mintra~~, see [[Mintaka#Mintra]].

These interval chains cover strong extensions of BPS. For mintra, see [[Mintaka #Mintra]].

In the below, tritave-reduced harmonics below 243 are indicated in '''bold'''.

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|+ style="font-size: 105%;" | Basic BPS (extension-agnostic)

|-

! rowspan="2" | &#~~35;~~ !! rowspan="2" | Cents* !! colspan="2" | Approximate ~~Ratios~~

! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios

|-

! 3.5.7 subgroup !! Add-19 extension

|-

| ~~−4~~ || 139.7 || 27/25, 49/45 || 21/19, 133/125

| −4 || 139.7 || 27/25, 49/45 || 21/19, 133/125

|-

| ~~−3~~ || 580.3 || 7/5, 243/175 || 27/19, 171/125

| −3 || 580.3 || 7/5, 243/175 || 27/19, 171/125

|-

| ~~−2~~ || 1020.8 || 9/5, '''49/27''' || 35/19, 133/75, 243/133

| −2 || 1020.8 || 9/5, '''49/27''' || 35/19, 133/75, 243/133

|-

| ~~−1~~ || 1461.4 || '''7/3''', 81/35 || 45/19, 57/25

| −1 || 1461.4 || '''7/3''', 81/35 || 45/19, 57/25

|-

| 0 || 0.0 || '''1/1''', 245/243 || 135/133, 175/171, 375/361

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| 9 || 161.1 || 375/343, 625/567 || 361/343

|}

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

<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"

|-

! Edt ~~Generator~~

! Edt generator

! [[Eigenmonzo|~~Eigenmonzo~~ (~~Unchanged-interval~~)]]*

! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*

! Generator~~ ~~(¢)

! Generator (¢)

! Comments

|-

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|

| 432.263

|

|-

| [[35edt|8\35]]

|

| 434.733

|

|-

|

| [[7/3]]

| [[9/7]]

| 435.084

| 0-~~comma~~

| Untempered

|-

| [[48edt|11\48]]

|

| 435.865

|

|-

| [[61edt|14\61]]

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|-

|

~~| 3645/2401~~

~~| 437.449~~

~~| 1/6-comma~~

|-

~~| [[100edt|23\100]]~~

|

~~| 437.450~~

|

| 438.038

| [[Delta-rational chord|DR]] 5:7:9, close to 5/24-comma

|-

|

| [[49~~/45~~]]

| [[135/49]]

| 438.632

| 1/4-comma

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|-

|

| [[7/5]]

| [[15/7]]

| 439.814

| 1/3-comma

|-

~~| [[108edt|25\108]]~~

|

~~| 440.267~~

|

|-

|

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| 442.921

|

|-

|

| [[2/1]]

| 443.136

| [[Sensi]] mapping of 2/1 to ~125/63

|-

| [[30edt|7\30]]

|

| 443.790

|

|-

|

~~| 175/81~~

~~| 444.544~~

~~| 2/3-comma~~

|-

~~| [[77edt|18\77]]~~

|

~~| 444.613~~

|

|-

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| Full comma

|}

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

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

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

* [[DKW theory|DKW]] (3.5.7): ~3 = 1\1, ~9/7 = 440.554

[[Category:~~Bohlen-Pierce~~]]

[[Category:BPS| ]]

[[Category:~~Temperaments~~]]

[[Category:Rank-2 temperaments]]

[[Category:~~Tritave~~-~~equivalent~~ temperaments]]

[[Category:Non-octave temperaments]]

[[Category:Sensamagic clan]]

[[Category:Bohlen–Pierce]]

@@ Line 1: / Line 1: @@
-'''Bohlen–Pierce–Stearns''' (BPS) is a [[non-octave]] [[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 {{sl|4L 5s|3/1}} against the tritave, known as the Bohlen–Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13]]edt, representing the equal-tempered [[Bohlen–Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17]]edt, [[30edt|7\30]]edt, and [[43edt|10\43]]edt.
+{{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]].
-As the generator of the Bohlen–Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad.
+As the generator of the Bohlen–Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the 3:5:7:9 tetrad to serve as BPS' primary consonance, similar to how the 4:5:6 triad serves as meantone's primary consonance.
-For technical data, see ''[[Sensamagic clan#BPS]]'' or ''[[No-twos subgroup temperaments#BPS]]'' (currently, extensions with 2 are stored on the former page and no-twos extensions are stored on the latter).
+For technical data, see [[Sensamagic clan #BPS]] or [[No-twos subgroup temperaments #BPS]] (currently, extensions with 2 are stored on the former page and no-twos extensions are stored on the latter).
 == Extensions ==
 Several extensions of this temperament are possible to incorporate additional harmonics.
-In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces ''Mintra'' temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[Mintaka]], which produces {{sl|5L 2s|3/1}} and {{sl|5L 7s|3/1}} MOS scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].
+In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces [[mintra]] temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[mintaka]], which produces {{mos scalesig|5L 2s<3/1>|link=1}} and {{mos scalesig|5L 7s<3/1>|link=1}} mos scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].
-Another weak extension to add prime 17, known as ''[[Dubhe]]'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces {{sl|8L 1s|3/1}} enneatonic and {{sl|9L 8s|3/1}} chromatic MOS scales. Simple tunings include [[17edt]] and [[26edt]].
+Another weak extension to add prime 17, known as ''[[dubhe]]'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces {{mos scalesig|8L 1s<3/1>|link=1}} enneatonic and {{mos scalesig|9L 8s<3/1>|link=1}} chromatic mos scales. Simple tunings include [[17edt]] and [[26edt]].
 === Strong extensions ===
@@ Line 20: / Line 35: @@
 === Prime 2 ===
-''Main article: [[Relationship between Bohlen–Pierce and octave-ful temperaments#Relationship of rank-2 Bohlen.E2.80.93Pierce.E2.80.93Stearns temperament to octave-ful temperaments|Relationship between Bohlen–Pierce and octave-ful temperaments]].
+: ''Main article: [[Relationship between Bohlen–Pierce and octave-ful temperaments#Relationship of rank-2 Bohlen.E2.80.93Pierce.E2.80.93Stearns temperament to octave-ful temperaments|Relationship between Bohlen–Pierce and octave-ful temperaments]].
 It is also possible to incorporate octaves into BPS. The logical choices for a mapping of 2 are 7 generators up (equating [[2/1]] to [[125/63]]), which produces [[sensi]], and 6 generators down (equating 2/1 to [[49/25]]), which produces [[hedgehog]].
 == Interval chains ==
-These interval chains cover strong extensions of BPS. For Mintra, see [[Mintaka#Mintra]].
+These interval chains cover strong extensions of BPS. For mintra, see [[Mintaka #Mintra]].
 In the below, tritave-reduced harmonics below 243 are indicated in '''bold'''.
@@ Line 32: / Line 47: @@
 |+ style="font-size: 105%;" | Basic BPS (extension-agnostic)
 |-
-! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios
+! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
 |-
 ! 3.5.7 subgroup !! Add-19 extension
 |-
-| &minus;4 || 139.7 || 27/25, 49/45 || 21/19, 133/125
+| −4 || 139.7 || 27/25, 49/45 || 21/19, 133/125
 |-
-| &minus;3 || 580.3 || 7/5, 243/175 || 27/19, 171/125
+| −3 || 580.3 || 7/5, 243/175 || 27/19, 171/125
 |-
-| &minus;2 || 1020.8 || 9/5, '''49/27''' || 35/19, 133/75, 243/133
+| −2 || 1020.8 || 9/5, '''49/27''' || 35/19, 133/75, 243/133
 |-
-| &minus;1 || 1461.4 || '''7/3''', 81/35 || 45/19, 57/25
+| −1 || 1461.4 || '''7/3''', 81/35 || 45/19, 57/25
 |-
 | 0 || 0.0 || '''1/1''', 245/243 || 135/133, 175/171, 375/361
@@ Line 64: / Line 79: @@
 | 9 || 161.1 || 375/343, 625/567 || 361/343
 |}
-<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning
+<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"
 |-
-! Edt<br />Generator
+! Edt<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]*
+! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
-! Generator<br />(¢)
+! Generator (¢)
 ! Comments
 |-
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 |
 | 432.263
+|
+|-
+| [[35edt|8\35]]
+|
+| 434.733
 |
 |-
 |
-| [[7/3]]
+| [[9/7]]
 | 435.084
-| 0-comma
+| Untempered
+|-
+| [[48edt|11\48]]
+|
+| 435.865
+|
 |-
 | [[61edt|14\61]]
@@ Line 90: / Line 136: @@
 |-
 |
-| 3645/2401
-| 437.449
-| 1/6-comma
-|-
-| [[100edt|23\100]]
-|
-| 437.450
 |
+| 438.038
+| [[Delta-rational chord|DR]] 5:7:9, close to 5/24-comma
 |-
 |
-| [[49/45]]
+| [[135/49]]
 | 438.632
 | 1/4-comma
@@ Line 110: / Line 151: @@
 |-
 |
-| [[7/5]]
+| [[15/7]]
 | 439.814
 | 1/3-comma
-|-
-| [[108edt|25\108]]
-|
-| 440.267
-|
 |-
 |
@@ Line 163: / Line 199: @@
 | 442.921
 |
+|-
+|
+| [[2/1]]
+| 443.136
+| [[Sensi]] mapping of 2/1 to ~125/63
 |-
 | [[30edt|7\30]]
 |
 | 443.790
-|
-|-
-|
-| 175/81
-| 444.544
-| 2/3-comma
-|-
-| [[77edt|18\77]]
-|
-| 444.613
 |
 |-
@@ Line 194: / Line 225: @@
 | Full comma
 |}
-<nowiki />* Besides the [[3/1|tritave]]
+<nowiki/>* Besides the [[3/1|tritave]]
-=== Other tunings ===
-* [[DKW theory|DKW]] (3.5.7): ~3 = 1\1, ~9/7 = 440.554
-[[Category:Bohlen-Pierce]]
+[[Category:BPS| ]] <!-- main article -->
-[[Category:Temperaments]]
+[[Category:Rank-2 temperaments]]
-[[Category:Tritave-equivalent temperaments]]
+[[Category:Non-octave temperaments]]
+[[Category:Sensamagic clan]]
+[[Category:Bohlen–Pierce]]