4L 5s (3/1-equivalent): Difference between revisions
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: ''For the octave equivalent 4L 5s pattern, see [[4L 5s]].'' | : ''For the octave equivalent 4L 5s pattern, see [[4L 5s]].'' | ||
{{Infobox MOS | {{Infobox MOS|Other names=Lambda}} | ||
| Other names = Lambda | {{MOS intro|Other Names=Lambda}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s|diatonic scale]]. | ||
}} | |||
{{MOS intro | |||
| Other Names = Lambda | |||
}} It is often considered to be [[Bohlen–Pierce]]'s equivalent of the ubiquitous [[5L 2s|diatonic scale]]. | |||
{{MOS scalesig|4L 5s <3/1>}} can be thought of as a | {{MOS scalesig|4L 5s <3/1>}} can be thought of as a mos generated by a sharpened 9/7 (or equivalently, a flat 7/3) such that two such intervals stack to an interval approximating [[5/3]]. This leads to [[BPS]] (''Bohlen–Pierce–Stearns''), a [[3.5.7 subgroup|3.5.7-subgroup]] [[rank-2 temperament]] that [[tempering out|tempers out]] [[245/243]]. BPS is considered to be a very good temperament on the 3.5.7 subgroup, and is [[support]]ed by many [[edt]]s (and even [[edo]]s) besides [[13edt]]. | ||
Some low-numbered | Some low-numbered edos that support BPS are {{EDOs| 19, 22, 27, 41, and 46 }}, and some low-numbered edts that support it are {{EDTs| 9, 13, 17, and 30 }}, all of which make it possible to play Bohlen–Pierce music to some reasonable extent. These equal temperaments contain not only this scale, but with the exception of 9edt they also contain the 13-note "BP chromatic" mos scale, or BPS[13], which can be thought of as a [[detempering|detempered]] version of the 13edt Bohlen–Pierce scale. This scale may be a suitable melodic substitute for the "BP chromatic" scale, and is basically the same as how [[meantone]] temperaments such as {{EDOs| 19, 31, and 43 }} and edos approximating [[Pythagorean tuning]] ({{EDOs| 41 and 53 }}) contain a 12-note chromatic scale as a subset despite not containing 12edo as a subset. | ||
When playing this scale in some | When playing this scale in some edo, it may be desired to [[stretched and compressed tuning|stretch or compress the octaves]] to make 3/1 just (or closer to just), rather than the octave being pure—or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1. | ||
One can add the octave to BPS by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This leads to [[sensi]], in essence treating it as a "3.5.7.2-subgroup" ("add-octave") extension of BPS. | One can add the octave to BPS by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This leads to [[sensi]], in essence treating it as a "3.5.7.2-subgroup" ("add-octave") extension of BPS. | ||
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=== Proposed mode names === | === Proposed mode names === | ||
[[User:Lériendil|Lériendil]] proposes mode names derived from the constellations of the northern sky. | [[User:Lériendil|Lériendil]] proposes mode names derived from the constellations of the northern sky. | ||
{{MOS modes | {{MOS modes| Mode Names= | ||
| Mode Names= | |||
Lyncian $ | Lyncian $ | ||
Aurigan $ | Aurigan $ | ||
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== Notation == | == Notation == | ||
Bohlen–Pierce theory possesses a well-established [[non-octave]] notation system for [[ | Bohlen–Pierce theory possesses a well-established [[non-octave]] notation system for [[edt]]s and no-2's music, which is based on this mos scale as generated by approximately [[7/3]], relating it to BPS. The preferred generator for any edt is its patent val approximation of 7/3. | ||
This notation uses 9 nominals: for compatibility with [[diamond- | This notation uses 9 nominals: for compatibility with [[diamond-mos notation]], the current recommendation is to use the notes {{nowrap| J K L M N O P Q R }} as presented in the J Cassiopeian (symmetric, sLsLsLsLs) mode, and represented by a circle of generators going as follows: {{dash|…, Q♯, O♯, M♯, K♯, R, P, N, L, J, Q, O, M, K, R♭, P♭, N♭, L♭, …|hair|med}} However, an alternative convention ({{w|Bohlen–Pierce scale #Intervals and scale diagrams|as seen on Wikipedia}} and some other articles of this wiki) labels them {{nowrap| C D E F G H J A B }} in the C Andromedan (LssLsLsLs) mode, which rotates to the E symmetric mode. | ||
An extension of [[ups and downs notation]], in the obvious way, can be found at [[Lambda ups and downs notation]]. | An extension of [[ups and downs notation]], in the obvious way, can be found at [[Lambda ups and downs notation]]. | ||
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| P8 | | P8 | ||
| P9 | | P9 | ||
|}<br | |}<br> | ||
{| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;" | {| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;" | ||
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| M8 | | M8 | ||
| P9 | | P9 | ||
|}<br | |}<br> | ||
{| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;" | {| class="wikitable article-table" style="text-align: center; margin: auto auto auto auto;" | ||
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|} | |} | ||
== | == Scale tree == | ||
Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5{{c}}. | Below is a list of equal temperaments which contain a 4L 5s scale using generators between 422.7 and 475.5{{c}}. | ||
{{MOS tuning spectrum | {{MOS tuning spectrum | ||
| Depth = 7 | | Depth = 7 | ||
| 2/1 = Equally-tempered [[Bohlen–Pierce scale]] | | 2/1 = Equally-tempered [[Bohlen–Pierce scale]] | ||
| 13/6 = [[BPS]] (Bohlen–Pierce–Stearns) | | 13/6 = [[BPS]] (Bohlen–Pierce–Stearns) region | ||
| 22/13 = Essentially just 7/3 | | 22/13 = Essentially just 7/3 | ||
}} | }} | ||
Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively. However, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by 37\ | Analogously to how the diatonic scale equalizes approaching [[7edo]] and its small steps collapse to 0 in [[5edo]], this scale equalizes approaching [[9edt]] and its small steps collapse in [[4edt]]; therefore, temperaments setting the 7/3 generator to precisely 7\9edt and to precisely 3\4edt are analogs of [[whitewood]] and [[blackwood]] respectively. However, unlike for the diatonic scale, the just point is not close to the center of the tuning range, but approximately 1/4 of the way between 9edt and 4edt, being closely approximated by [[48edt|37\48edt]] and extremely closely approximated by [[153edt|118\153edt]]. | ||
[[Category:Bohlen–Pierce]] | [[Category:Bohlen–Pierce]] | ||