5L 2s: Difference between revisions
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{{Infobox MOS | {{Infobox MOS}} | ||
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{{MOS intro}} | {{MOS intro}} | ||
Among the most well-known forms of this scale are the diatonic scale of [[12edo]], the Pythagorean diatonic scale, and scales produced by meantone systems. | Among the most well-known forms of this scale are the diatonic scale of [[12edo]], the Pythagorean diatonic scale, and scales produced by meantone systems. | ||
==Name == | ==Name== | ||
[[TAMNAMS]] suggests the temperament-agnostic name '''diatonic''' for this scale, which commonly refers to a scale with 5 whole steps and 2 small steps. Under TAMNAMS and for all scale pattern pages on the wiki, '''the term ''diatonic'' exclusively refers to 5L 2s'''. | [[TAMNAMS]] suggests the temperament-agnostic name '''diatonic''' for this scale, which commonly refers to a scale with 5 whole steps and 2 small steps. Under TAMNAMS and for all scale pattern pages on the wiki, '''the term ''diatonic'' exclusively refers to 5L 2s'''. | ||
The term ''diatonic'' may also refer to scales produced using [[Tetrachord|tetrachords]], [[just intonation]], or in general have more than one size of whole tone. Such scales, such as [[Zarlino]], [[blackdye]] and [[diasem]], are specifically called ''[[Detempering|detempered]] diatonic scales'' (for an RTT-based philosophy) or ''deregularized diatonic scales'' (for an RTT-agnostic philosophy). The terms ''diatonic-like'' or ''diatonic-based'' may also be used to refer such scales, depending on what's contextually the most appropriate. | The term ''diatonic'' may also refer to scales produced using [[Tetrachord|tetrachords]], [[just intonation]], or in general have more than one size of whole tone. Such scales, such as [[Zarlino]], [[blackdye]] and [[diasem]], are specifically called ''[[Detempering|detempered]] diatonic scales'' (for an RTT-based philosophy) or ''deregularized diatonic scales'' (for an RTT-agnostic philosophy). The terms ''diatonic-like'' or ''diatonic-based'' may also be used to refer such scales, depending on what's contextually the most appropriate. | ||
==Notation== | ==Notation== | ||
===Intervals === | ===Intervals=== | ||
Intervals are identical to that of standard notation. As such, the usual [[Interval quality|interval qualities]] of major/minor and augmented/perfect/diminished apply here. | Intervals are identical to that of standard notation. As such, the usual [[Interval quality|interval qualities]] of major/minor and augmented/perfect/diminished apply here. | ||
{| class="wikitable" | {| class="wikitable" | ||
! rowspan="2" |Interval class | ! rowspan="2" |Interval class | ||
! colspan="2" |Large variety | ! colspan="2" |Large variety | ||
! colspan="2" | Small variety | ! colspan="2" |Small variety | ||
|- | |- | ||
! Size | |||
! Quality | |||
!Size | !Size | ||
!Quality | !Quality | ||
|- | |- | ||
|'''1st (unison)''' | |'''1st (unison)''' | ||
| 0 | |0 | ||
|Perfect | |Perfect | ||
|0 | |0 | ||
| Line 35: | Line 33: | ||
|- | |- | ||
|2nd | |2nd | ||
| L | |L | ||
|Major | |Major | ||
|s | |s | ||
| Line 42: | Line 40: | ||
|3rd | |3rd | ||
|2L | |2L | ||
| Major | |Major | ||
|L + s | |L + s | ||
|Minor | |Minor | ||
|- | |- | ||
|4th | |4th | ||
|3L | | 3L | ||
|Augmented | |Augmented | ||
|2L + 1s | |2L + 1s | ||
|Perfect | |Perfect | ||
|- | |- | ||
|5th | | 5th | ||
| 3L + 1s | |3L + 1s | ||
|Perfect | |Perfect | ||
|2L + 2s | | 2L + 2s | ||
| Diminished | |Diminished | ||
|- | |- | ||
|6th | |6th | ||
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|Minor | |Minor | ||
|- | |- | ||
| 7th | |7th | ||
| 5L + 1s | |5L + 1s | ||
| Major | |Major | ||
|4L + 2s | |4L + 2s | ||
|Minor | |Minor | ||
|- | |- | ||
|'''8th (octave)''' | |'''8th (octave)''' | ||
|5L + 2s | |||
|Perfect | |||
|5L + 2s | |5L + 2s | ||
| Perfect | | Perfect | ||
|} | |} | ||
===Note names === | === Note names=== | ||
Note names are identical to that of standard notation. Thus, the basic (12edo) gamut for 5L 2s is the following: | Note names are identical to that of standard notation. Thus, the basic (12edo) gamut for 5L 2s is the following: | ||
{{MOS gamut | {{MOS gamut}} | ||
==Theory == | ==Theory== | ||
===Introduction to step sizes===<!-- The 5L 2s page already had an introduction to step sizes, but this may be worth moving to its own page. --> | ===Introduction to step sizes===<!-- The 5L 2s page already had an introduction to step sizes, but this may be worth moving to its own page. --> | ||
:''Main article: [[Scale tree]] and [[TAMNAMS#Step ratio spectrum]]'' | :''Main article: [[Scale tree]] and [[TAMNAMS#Step ratio spectrum]]'' | ||
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|+ | |+ | ||
!Step ratio (L:s) | !Step ratio (L:s) | ||
!Step pattern | ! Step pattern | ||
!EDO | !EDO | ||
!Selected multiples | !Selected multiples | ||
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|- | |- | ||
|4:3 | |4:3 | ||
| 4 4 3 4 4 4 3 | |4 4 3 4 4 4 3 | ||
|[[26edo]] | |[[26edo]] | ||
| | | | ||
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|[[24edo]], [[36edo]], etc. | |[[24edo]], [[36edo]], etc. | ||
|- | |- | ||
|5:2 | | 5:2 | ||
|5 5 2 5 5 5 2 | |5 5 2 5 5 5 2 | ||
|[[29edo]] | |[[29edo]] | ||
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*[[Parapyth]], with generators around 704.7¢. | *[[Parapyth]], with generators around 704.7¢. | ||
*[[Archy]], with generators around 709.3¢. This includes: | *[[Archy]], with generators around 709.3¢. This includes: | ||
** Supra, with generators around 707.2¢ | **Supra, with generators around 707.2¢ | ||
**Superpyth, with generators around 710.3¢ | ** Superpyth, with generators around 710.3¢ | ||
** Ultrapyth, with generators around 713.7¢. | **Ultrapyth, with generators around 713.7¢. | ||
==Tuning ranges== | ==Tuning ranges== | ||
===Simple tunings=== | ===Simple tunings=== | ||
[[17edo]] and [[19edo]] are the smallest edos that offer a greater variety of pitches than 12edo. Note that any enharmonic equivalences that 12edo has no longer hold for either 17edo or 19edo, as shown in the table below.{{MOS degrees | [[17edo]] and [[19edo]] are the smallest edos that offer a greater variety of pitches than 12edo. Note that any enharmonic equivalences that 12edo has no longer hold for either 17edo or 19edo, as shown in the table below.{{MOS degrees|Step Ratio=2/1; 3/1; 3/2|Genchain Extend=7}} | ||
===Parasoft tunings=== | ===Parasoft tunings=== | ||
:''Main article: [[Flattone]]'' | :''Main article: [[Flattone]]'' | ||
Parasoft tunings (4:3 to 3:2) correspond to flattone temperaments, characterized by flattened perfect 5ths ([[3/2]], flat of 702¢) to produce major 3rds that are flatter than [[5/4]] (386¢). | Parasoft diatonic tunings (4:3 to 3:2) correspond to flattone temperaments, characterized by flattened perfect 5ths ([[3/2]], flat of 702¢) to produce major 3rds that are flatter than [[5/4]] (386¢). | ||
Edos include [[19edo]], [[26edo]], [[45edo]], and [[64edo]].{{MOS degrees | Edos include [[19edo]], [[26edo]], [[45edo]], and [[64edo]].{{MOS degrees|Step Ratio=3/2; 4/3; 7/5; 10/7|Genchain Extend=0, 5}} | ||
===Hyposoft tunings === | ===Hyposoft tunings=== | ||
:''Main article: [[Meantone]]'' | :''Main article: [[Meantone]]'' | ||
Hyposoft tunings (3:2 to 2:1) correspond to meantone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce diatonic major 3rds that approximate 5/4 (386¢). | Hyposoft diatonic tunings (3:2 to 2:1) correspond to meantone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce diatonic major 3rds that approximate 5/4 (386¢). | ||
Edos include [[19edo]], [[31edo]], [[43edo]], and [[50edo]].{{MOS degrees | Edos include [[19edo]], [[31edo]], [[43edo]], and [[50edo]].{{MOS degrees|Step Ratio=3/2; 5/3; 7/4; 8/5|Genchain Extend=0, 5}} | ||
===Hypohard tunings=== | === Hypohard tunings=== | ||
:''Main article: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]'' | :''Main article: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]'' | ||
The range of hypohard tunings can be divided into a minihard range (2:1 to 5:2) and quasihard range (5:2 to 3:1). | The range of hypohard tunings can be divided into a minihard range (2:1 to 5:2) and quasihard range (5:2 to 3:1). | ||
==== Minihard tunings==== | ====Minihard tunings==== | ||
Minihard tunings correspond to Pythagorean tuning and schismatic temperament, characterized by having a perfect 5th that is as close to just (701.96¢) as possible, resulting in a major 3rd of [[81/64]] (407¢). | Minihard diatonic tunings correspond to Pythagorean tuning and schismatic temperament, characterized by having a perfect 5th that is as close to just (701.96¢) as possible, resulting in a major 3rd of [[81/64]] (407¢). | ||
Edos include [[41edo]] and [[53edo]].{{MOS degrees | Edos include [[41edo]] and [[53edo]].{{MOS degrees|Step Ratio=7/3; 9/4|Genchain Extend=0, 5}} | ||
==== Quasihard tunings ==== | ====Quasihard tunings==== | ||
Quasihard tunings correspond to "neogothic" or "parapyth" systems whose perfect 5th is slightly sharper than just, resulting in major 3rds that are sharper than 81/64 and minor 3rds that are slightly flat of [[32/27]] (294¢). | Quasihard diatonic tunings correspond to "neogothic" or "parapyth" systems whose perfect 5th is slightly sharper than just, resulting in major 3rds that are sharper than 81/64 and minor 3rds that are slightly flat of [[32/27]] (294¢). | ||
Edos include [[17edo]], [[29edo]], and [[46edo]]. 17edo is considered to be on the sharper end of the neogothic spectrum, with a major 3rd that is more discordant than flatter neogothic tunings.{{MOS degrees | Edos include [[17edo]], [[29edo]], and [[46edo]]. 17edo is considered to be on the sharper end of the neogothic spectrum, with a major 3rd that is more discordant than flatter neogothic tunings.{{MOS degrees|Step Ratio=3/1; 5/2; 8/3|Genchain Extend=0, 5}} | ||
===Parahard and ultrahard tunings=== | ===Parahard and ultrahard tunings=== | ||
:''Main article: [[Archy]]'' | :''Main article: [[Archy]]'' | ||
Parahard (3:1 to 4:1) and ultrahard | Parahard (3:1 to 4:1) and ultrahard (4:1 to 1:0) diatonic tunings correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢. | ||
Edos include [[17edo]], [[22edo]], [[27edo]], and [[32edo]], among others.{{MOS degrees | Edos include [[17edo]], [[22edo]], [[27edo]], and [[32edo]], among others.{{MOS degrees|Step Ratio=3/1; 4/1; 5/1; 6/1|Genchain Extend=0, 5}} | ||
==Modes== | ==Modes== | ||
Diatonic modes have standard names from classical music theory | Diatonic modes have standard names from classical music theory. | ||
{{MOS modes}} | {{MOS modes}} | ||
Each mode has the following scale degrees, reached by raising or lowering certain naturals by a chroma. | Each mode has the following scale degrees, reached by raising or lowering certain naturals by a chroma. | ||
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! colspan="8" |Scale degree (on C) | ! colspan="8" |Scale degree (on C) | ||
|- | |- | ||
!UDP | ! UDP | ||
!Step pattern | !Step pattern | ||
!1st | !1st | ||
!2nd | ! 2nd | ||
!3rd | !3rd | ||
!4th | !4th | ||
!5th | !5th | ||
!6th | ! 6th | ||
!7th | !7th | ||
!8th | !8th | ||
|- | |- | ||
|<nowiki>6|0</nowiki> | |<nowiki>6|0</nowiki> | ||
|LLLsLLs | | LLLsLLs | ||
|Perfect (C) | |Perfect (C) | ||
|Major (D) | |Major (D) | ||
|Major (E) | | Major (E) | ||
|Augmented (F#) | |Augmented (F#) | ||
|Perfect (G) | |Perfect (G) | ||
| Line 237: | Line 234: | ||
|- | |- | ||
|<nowiki>3|3</nowiki> | |<nowiki>3|3</nowiki> | ||
|LsLLLsL | | LsLLLsL | ||
|Perfect (C) | |Perfect (C) | ||
|Major (D) | |Major (D) | ||
| Line 243: | Line 240: | ||
|Perfect (F) | |Perfect (F) | ||
|Perfect (G) | |Perfect (G) | ||
|Major (A) | | Major (A) | ||
|Minor (Bb) | | Minor (Bb) | ||
|Perfect (C) | |Perfect (C) | ||
|- | |- | ||
| Line 256: | Line 253: | ||
|Minor (Ab) | |Minor (Ab) | ||
|Minor (Bb) | |Minor (Bb) | ||
|Perfect (C) | | Perfect (C) | ||
|- | |- | ||
|<nowiki>1|5</nowiki> | |<nowiki>1|5</nowiki> | ||
| Line 282: | Line 279: | ||
==Scales== | ==Scales== | ||
===Subset and superset scales=== | ===Subset and superset scales=== | ||
5L 2s has a parent scale of [[2L 3s]], a pentatonic scale, meaning 2L 3s is a subset. 5L 2s also has | 5L 2s has a parent scale of [[2L 3s]], a pentatonic scale, meaning 2L 3s is a subset. 5L 2s also has two child scales, which are supersets of 5L 2s: | ||
*[[7L 5s]], a chromatic scale produced using soft-of-basic step ratios. | *[[7L 5s]], a chromatic scale produced using soft-of-basic step ratios. | ||
*[[5L 7s]], a chromatic scale produced using hard-of-basic step ratios. | *[[5L 7s]], a chromatic scale produced using hard-of-basic step ratios. | ||
12edo | 12edo, the equalized form of both 7L 5s and 5L 7s, is also a superset of 5L 2s. | ||
===MODMOS scales and muddles=== | ===MODMOS scales and muddles=== | ||
{{main| 5L 2s MODMOSes }} ''and [[5L 2s Muddles]]'' | {{main| 5L 2s MODMOSes }} ''and [[5L 2s Muddles]]'' | ||
===Scala files=== | === Scala files=== | ||
*[[Meantone7]] – 19edo and 31edo tunings | *[[Meantone7]] – 19edo and 31edo tunings | ||
*[[Nestoria7]] – 171edo tuning | *[[Nestoria7]] – 171edo tuning | ||
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==Scale tree== | ==Scale tree== | ||
{{Scale tree|depth=6|Comments=7/5:[[Flattone]] is in this region;21/13:[[Golden meantone]] (696.2145¢);5/3:[[Meantone]] is in this region;2/1:(Generators smaller than this are proper);9/4:The generator closest to a just [[3/2]] for EDOs less than 200;16/7:[[Garibaldi]] / [[Cassandra]];21/8:Golden neogothic (704.0956¢);8/3:[[Neogothic]] is in this region;4/1:[[Archy]] is in this region}} | {{Scale tree|depth=6|Comments=7/5:[[Flattone]] is in this region;21/13:[[Golden meantone]] (696.2145¢);5/3:[[Meantone]] is in this region;2/1:(Generators smaller than this are proper);9/4:The generator closest to a just [[3/2]] for EDOs less than 200;16/7:[[Garibaldi]] / [[Cassandra]];21/8:Golden neogothic (704.0956¢);8/3:[[Neogothic]] is in this region;4/1:[[Archy]] is in this region}} | ||
=== Step ratio diagram=== | |||
=== Step ratio diagram === | |||
[[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]] | [[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]] | ||
==See also== | ==See also== | ||
* [[Diatonic functional harmony]] | *[[Diatonic functional harmony]] | ||