12L 12s: Difference between revisions

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It is the 24-note mos scale of the [[compton]], [[catler]], and [[duodecim]] temperaments of the [[compton family]], which divide the octave into 12 parts. As such, this mos scale can be considered to be 2 rings of [[12edo]], and can be replicated with two 12edo instruments detuned from each other by a fixed amount. ~~Each~~ of these temperaments ~~maps~~ [[3/2]] to 7 steps of 12edo, tempering out the [[Pythagorean comma]]. Compton uses the [[3-limit]] of 12edo and adds an independent [[generator]] for [[5/4]]. Catler additionally maps 5/4 to 4\12, and ~~has~~ [[7/4]] as an independent generator. Duodecim is a low-accuracy temperament that further maps 7/4 to 10\12 and uses 11/8 as an independent generator.

It is the 24-note mos scale of the [[compton]], [[catler]], and [[duodecim]] temperaments of the [[compton family]], which divide the octave into 12 parts. As such, this mos scale can be considered to be 2 rings of [[12edo]], and can be replicated with two 12edo instruments detuned from each other by a fixed amount. All of these temperaments map [[3/2]] to 7 steps of 12edo, thus tempering out the [[Pythagorean comma]]. Compton uses the [[3-limit]] of 12edo, and adds an independent [[generator]] for [[5/4]] to improve the accuracy of [[5-limit]] harmony. Catler additionally maps 5/4 to 4\12, thus preserving the 5-limit of 12edo, and adds [[7/4]] as an independent generator. Duodecim is a low-accuracy temperament that further maps 7/4 to 10\12, thus keeping the full [[7-limit]] of 12edo, and uses [[11/8]] as an independent generator.

Using the [[TAMNAMS extension]], it can be named '''dodecawood''', since it has 12 periods per octave, each with one large step and one small step.

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=== Intervals ===

While 12L 12s can be treated as a 24-form system due to the scale containing 24 notes, it can also make sense as two rings of 12edo that differ by a small step. For example, in compton, the small step represents the [[81/80]] comma, and inflecting 12edo intervals by this step produces more accurate approximations of 5-limit intervals. In catler, the small step represents [[64/63]] and [[36/35]], and inflecting a 12edo minor seventh down by this step gives a more accurate ~7/4, with other ratios involving harmonic 7 also improved, while intervals within the 5-limit are represented as in 12edo.

~~=== Generator chain ===~~

~~{{MOS genchain}}~~

=== Modes ===

Since 12L 12s has only one large step and one small step per period, there are only two modes, which can be called the major and minor modes. In compton, catler, and duodecim, the major mode favors otonalities above the root, while the minor mode favors utonalities above the root. This is because [[5/4]], [[7/4]], and [[11/8]] are all closer to the 12edo step above it than the step below it.

== Scale tree ==

Softer tunings of 12L 12s are closer to an unequal derivative of [[24edo]], while harder tunings are closer to two rings of 12edo a comma step apart.

{{MOS tuning spectrum

| 3/2 = [[Duodecim]]

| 3/1 = [[Catler]]

| 5/2 = [[Catler]]

| 5/1 = [[Compton]]

| 6/1 = [[Compton]]

}}

@@ Line 2: / Line 2: @@
 {{MOS intro}}
-It is the 24-note mos scale of the [[compton]], [[catler]], and [[duodecim]] temperaments of the [[compton family]], which divide the octave into 12 parts. As such, this mos scale can be considered to be 2 rings of [[12edo]], and can be replicated with two 12edo instruments detuned from each other by a fixed amount. Each of these temperaments maps [[3/2]] to 7 steps of 12edo, tempering out the [[Pythagorean comma]]. Compton uses the [[3-limit]] of 12edo and adds an independent [[generator]] for [[5/4]]. Catler additionally maps 5/4 to 4\12, and has [[7/4]] as an independent generator. Duodecim is a low-accuracy temperament that further maps 7/4 to 10\12 and uses 11/8 as an independent generator.
+It is the 24-note mos scale of the [[compton]], [[catler]], and [[duodecim]] temperaments of the [[compton family]], which divide the octave into 12 parts. As such, this mos scale can be considered to be 2 rings of [[12edo]], and can be replicated with two 12edo instruments detuned from each other by a fixed amount. All of these temperaments map [[3/2]] to 7 steps of 12edo, thus tempering out the [[Pythagorean comma]]. Compton uses the [[3-limit]] of 12edo, and adds an independent [[generator]] for [[5/4]] to improve the accuracy of [[5-limit]] harmony. Catler additionally maps 5/4 to 4\12, thus preserving the 5-limit of 12edo, and adds [[7/4]] as an independent generator. Duodecim is a low-accuracy temperament that further maps 7/4 to 10\12, thus keeping the full [[7-limit]] of 12edo, and uses [[11/8]] as an independent generator.
 Using the [[TAMNAMS extension]], it can be named '''dodecawood''', since it has 12 periods per octave, each with one large step and one small step.
@@ Line 10: / Line 10: @@
 === Intervals ===
+While 12L 12s can be treated as a 24-form system due to the scale containing 24 notes, it can also make sense as two rings of 12edo that differ by a small step. For example, in compton, the small step represents the [[81/80]] comma, and inflecting 12edo intervals by this step produces more accurate approximations of 5-limit intervals. In catler, the small step represents [[64/63]] and [[36/35]], and inflecting a 12edo minor seventh down by this step gives a more accurate ~7/4, with other ratios involving harmonic 7 also improved, while intervals within the 5-limit are represented as in 12edo.
 {{MOS intervals}}
-=== Generator chain ===
-{{MOS genchain}}
 === Modes ===
+Since 12L 12s has only one large step and one small step per period, there are only two modes, which can be called the major and minor modes. In compton, catler, and duodecim, the major mode favors otonalities above the root, while the minor mode favors utonalities above the root. This is because [[5/4]], [[7/4]], and [[11/8]] are all closer to the 12edo step above it than the step below it.
 {{MOS mode degrees}}
 == Scale tree ==
+Softer tunings of 12L 12s are closer to an unequal derivative of [[24edo]], while harder tunings are closer to two rings of 12edo a comma step apart.
 {{MOS tuning spectrum
 | 3/2 = [[Duodecim]]
-| 3/1 = [[Catler]]
+| 5/2 = [[Catler]]
-| 5/1 = [[Compton]]
+| 6/1 = [[Compton]]
 }}