5L 2s: Difference between revisions

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}}

One way of distinguishing the '''diatonic''' scale is by considering it a [[MOS scale|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. Among the most well-known variants of this MOS proper are [[12edo|12EDO]]'s diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.

== On the term ''diatonic'' ==

In [[TAMNAMS]] (which is the convention on all pages on scale patterns on the wiki), [[diatonic]] exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized'' (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.

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* L L s L L L s

Insert 2 for L and 1 for s and you'll get the ~~12edo~~ diatonic of standard practice.

Insert 2 for L and 1 for s and you'll get the 12EDO diatonic of standard practice.

* 2 2 1 2 2 2 1

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=== Hypohard ===

The near-just part of the region is of interest mainly for those interested in [[Pythagorean tuning]] and large, accurate ~~edo~~ systems based on close-to-Pythagorean fifths, such as [[41edo|41EDO]] and [[53edo|53EDO]]. This class of tunings is called [[schisma|schismic]] temperament; these tunings can approximate 5-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12EDO tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)~~.~~

The near-just part of the region is of interest mainly for those interested in [[Pythagorean tuning]] and large, accurate EDO systems based on close-to-Pythagorean fifths, such as [[41edo|41EDO]] and [[53edo|53EDO]]. This class of tunings is called [[schisma|schismic]] temperament; these tunings can approximate 5-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12EDO tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)

The sharp-of-just part of this range includes so-called "[[neogothic]]" or "parapyth" systems, which tune the diatonic major third slightly sharply of [[81/64]] (around [[14/11]]) and the diatonic minor third slightly flatly of [[32/27]] (around [[13/11]]). Good neogothic EDOs include [[29edo|29EDO]] and [[46edo|46EDO]]. [[17edo|17EDO]] is often considered the sharper end of the neogothic spectrum; its major third at 423 cents is considerably more discordant than in flatter neogothic tunings.

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== Scale tree ==

If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO:

If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

~~{| class="wikitable"~~

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

|-

| 4\7 ||

|-

Chroma-positive generator: 685.7143 cents (4\7) to 720 cents (3\5)

~~| || 7\12~~

|-

| 3\5 ||

|}

~~If we carry this freshman~~-~~summing out a little further, new, larger [[EDO]]s pop up in our continuum~~.

Chroma-negative generator: 480 cents (2\5) to 514.2857 cents (3\7)

{| class="wikitable center-all"

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| || || || || || || 67\116 || 693.103 || 18 || 13 || 1.385 ||

|-

| || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 ||

| || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 || [[Flattone]] is in this region

|-

| || || || || || || 63\109 || 693.578 || 17 || 12 || 1.417 ||

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| || || || || || || 48\83 || 693.976 || 13 || 9 || 1.444 ||

|-

| || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 || ~~L/s = 3/2~~

| || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 ||

|-

| || || || || || || 51\88 || 695.455 || 14 || 9 || 1.556 ||

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| || || || || 29\50 || || || 696.000 || 8 || 5 || 1.600 ||

|-

| || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || Golden meantone

| || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || [[Golden meantone]] (696.2145¢)

|-

| || || || || || 47\81 || || 696.296 || 13 || 8 || 1.625 ||

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| || || || || || || 65\112 || 696.429 || 18 || 11 || 1.636 ||

|-

| || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region

| || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || [[Meantone]] is in this region

|-

| || || || || || || 61\105 || 697.143 || 17 || 10 || 1.700 ||

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| || || || || || || 57\98 || 697.959 || 16 || 9 || 1.778 ||

|-

| || || || || ||32\55 || || 698.182 || 9 || 5 || 1.800 ||

| || || || || || 32\55 || || 698.182 || 9 || 5 || 1.800 ||

|-

| || || || || || || 39\67 || 698.507 || 11 || 6 || 1.833 ||

|-

| || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)

| || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic <br>(Generators smaller than this are proper)

|-

| || || || || || || 38\65 || 701.539 || 11 || 5 || 2.200 ||

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| || || || || || 31\53 || || 701.887 || 9 || 4 || 2.250 || The generator closest to a just [[3/2]] for EDOs less than 200

|-

| || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 ||

| || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 || [[Garibaldi]] / [[Cassandra]]

|-

| || || || || 24\41 || || || 702.409 || 7 || 3 || 2.333 ||

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| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||

|-

| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden neogothic

| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden neogothic (704.0956¢)

|-

| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 || Neogothic is in this region

| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 || [[Neogothic]] is in this region

|-

| || || || || || || 64\109 || 704.587 || 19 || 7 || 2.714 ||

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| || || || || || || 47\80 || 705.000 || 14 || 5 || 2.800 ||

|-

| || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 || ~~L/s = 3/1~~

| || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 ||

|-

| || || || || || || 43\73 || 706.849 || 13 || 4 || 3.250 ||

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| || || || || || || 49\83 || 708.434 || 15 || 4 || 3.750 ||

|-

| || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region

| || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || [[Archy]] is in this region

|-

| || || || || || || 42\71 || 709.859 || 13 || 3 || 4.333 ||

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[[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]]

5L 2s contains the pentatonic MOS [[~~2L_3s|~~2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[~~7L_5s|~~7L 5s]] or [[~~5L_7s|~~5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (~~700c~~).

5L 2s contains the pentatonic MOS [[2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L 5s]] or [[5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (700¢).

== Related Scales ==

@@ Line 16: / Line 16: @@
 }}
-One way of distinguishing the '''diatonic''' scale is by considering it a [[MOS scale|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. Among the most well-known variants of this MOS proper are [[12edo|12EDO]]'s diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems.  Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.
+One way of distinguishing the '''diatonic''' scale is by considering it a [[MOS scale|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. Among the most well-known variants of this MOS proper are [[12edo|12EDO]]'s diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.
 == On the term ''diatonic'' ==
 In [[TAMNAMS]] (which is the convention on all pages on scale patterns on the wiki), [[diatonic]] exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized''  (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.
@@ Line 25: / Line 26: @@
 * L L s L L L s
-Insert 2 for L and 1 for s and you'll get the 12edo diatonic of standard practice.
+Insert 2 for L and 1 for s and you'll get the 12EDO diatonic of standard practice.
 * 2 2 1 2 2 2 1
@@ Line 61: / Line 62: @@
 === Hypohard ===
-The near-just part of the region is of interest mainly for those interested in [[Pythagorean tuning]] and large, accurate edo systems based on close-to-Pythagorean fifths, such as [[41edo|41EDO]] and [[53edo|53EDO]]. This class of tunings is called [[schisma|schismic]] temperament; these tunings can approximate 5-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12EDO tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)<!-- (see [[5L 2s/Temperaments#Schismic]])-->.
+The near-just part of the region is of interest mainly for those interested in [[Pythagorean tuning]] and large, accurate EDO systems based on close-to-Pythagorean fifths, such as [[41edo|41EDO]] and [[53edo|53EDO]]. This class of tunings is called [[schisma|schismic]] temperament; these tunings can approximate 5-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12EDO tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)
 The sharp-of-just part of this range includes so-called "[[neogothic]]" or "parapyth" systems, which tune the diatonic major third slightly sharply of [[81/64]] (around [[14/11]]) and the diatonic minor third slightly flatly of [[32/27]] (around [[13/11]]). Good neogothic EDOs include [[29edo|29EDO]] and [[46edo|46EDO]]. [[17edo|17EDO]] is often considered the sharper end of the neogothic spectrum; its major third at 423 cents is considerably more discordant than in flatter neogothic tunings.
@@ Line 115: / Line 116: @@
 == Scale tree ==
-If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO:
+If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.
-{| class="wikitable"
+If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
-|-
-| 4\7 ||
-|-
+Chroma-positive generator: 685.7143 cents (4\7) to 720 cents (3\5)
-| || 7\12
-|-
-| 3\5 ||
-|}
-If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
+Chroma-negative generator: 480 cents (2\5) to 514.2857 cents (3\7)
 {| class="wikitable center-all"
@@ Line 160: / Line 157: @@
 | || || || || || || 67\116 || 693.103 || 18 || 13 || 1.385 ||
 |-
-| || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 ||
+| || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 || [[Flattone]] is in this region
 |-
 | || || || || || || 63\109 || 693.578 || 17 || 12 || 1.417 ||
@@ Line 168: / Line 165: @@
 | || || || || || || 48\83 || 693.976 || 13 || 9 || 1.444 ||
 |-
-| || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
+| || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 ||
 |-
 | || || || || || || 51\88 || 695.455 || 14 || 9 || 1.556 ||
@@ Line 178: / Line 175: @@
 | || || || || 29\50 || || || 696.000 || 8 || 5 || 1.600 ||
 |-
-| || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || Golden meantone
+| || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || [[Golden meantone]] (696.2145¢)
 |-
 | || || || || || 47\81 || || 696.296 || 13 || 8 || 1.625 ||
@@ Line 184: / Line 181: @@
 | || || || || || || 65\112 || 696.429 || 18 || 11 || 1.636 ||
 |-
-| || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region
+| || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || [[Meantone]] is in this region
 |-
 | || || || || || || 61\105 || 697.143 || 17 || 10 || 1.700 ||
@@ Line 196: / Line 193: @@
 | || || || || || || 57\98 || 697.959 || 16 || 9 || 1.778 ||
 |-
-| || || || || ||32\55 || || 698.182 || 9 || 5 || 1.800 ||
+| || || || || || 32\55 || || 698.182 || 9 || 5 || 1.800 ||
 |-
 | || || || || || || 39\67 || 698.507 || 11 || 6 || 1.833 ||
 |-
-| || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)
+| || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic <br>(Generators smaller than this are proper)
 |-
 | || || || || || || 38\65 || 701.539 || 11 || 5 || 2.200 ||
@@ Line 206: / Line 203: @@
 | || || || || || 31\53 || || 701.887 || 9 || 4 || 2.250 || The generator closest to a just [[3/2]] for EDOs less than 200
 |-
-| || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 ||
+| || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 || [[Garibaldi]] / [[Cassandra]]
 |-
 | || || || || 24\41 || || || 702.409 || 7 || 3 || 2.333 ||
@@ Line 222: / Line 219: @@
 | || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
 |-
-| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden neogothic
+| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden neogothic (704.0956¢)
 |-
-| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 || Neogothic is in this region
+| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 || [[Neogothic]] is in this region
 |-
 | || || || || || || 64\109 || 704.587 || 19 || 7 || 2.714 ||
@@ Line 232: / Line 229: @@
 | || || || || || || 47\80 || 705.000 || 14 || 5 || 2.800 ||
 |-
-| || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
+| || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 ||
 |-
 | || || || || || || 43\73 || 706.849 || 13 || 4 || 3.250 ||
@@ Line 248: / Line 245: @@
 | || || || || || || 49\83 || 708.434 || 15 || 4 || 3.750 ||
 |-
-| || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region
+| || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || [[Archy]] is in this region
 |-
 | || || || || || || 42\71 || 709.859 || 13 || 3 || 4.333 ||
@@ Line 273: / Line 270: @@
 [[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]]
-L 2s contains the pentatonic MOS [[2L_3s|2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L_5s|7L 5s]] or [[5L_7s|5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (700c).
+L 2s contains the pentatonic MOS [[2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L 5s]] or [[5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (700¢).
 == Related Scales ==