4L 3s: Difference between revisions

{{Infobox MOS

| Name = smitonic

| Periods = 1

| nLargeSteps = 4

| nSmallSteps = 3

'''4L 3s''' refers to the structure of [[MOS]] scales with generators ranging from 1\4edo (one degree of [[4edo]], 300¢) to 2\7edo (two degrees of [[7edo]], or approx. 342.857¢). The name '''smitonic''' ''smy-TON-ik'' /smaɪˈtɒnɪk/ has been proposed (derived from the obsolete temperament name ''smite'' for [[sixix]], from 'sharp minor third', taking sharp to mean sharp of the 12edo minor third).

There are also other temperaments in the 4L 3s range, particularly [[amity]] and [[myna]], but 7 notes in the generator chain are not enough to contain the most concordant chords in these temperaments; you would need to use a [[MODMOS]] or use a larger MOS gamut.

== Notation==

'''J/Q&''' J&/K@ '''K/L@''' '''L/K&''' L&/M@ '''M/N@''' '''N/M&''' N&/O@ '''O/P@''' '''P/O@''' P&/J@ '''J'''

== Scale tree ==

The spectrum looks like this:

{| class="wikitable"

! colspan="8" | Generator

| style="text-align:center;" |

| style="text-align:center;" | [[Myna]] is around here

| style="text-align:center;" |

| style="text-align:center;" |

| | 947.368

| style="text-align:center;" | L/s = 4

| style="text-align:center;" | Kleismic is around here

| style="text-align:center;" | L/s = 3. Orgone starts here

| | 321.951

| | 643.902

| style="text-align:center;" | Orgone minmax tuning

| | 323.809

| | 4 3 4

| | 156.522

| style="text-align:center;" | [[Amity]] is around here

| | 342.857

There are two notable harmonic entropy minima: [[Kleismic_family|kleismic]], in which the generator is 6/5 and 6 of them make a 3/1, and [[Starling_temperaments|myna]], in which the generator is also 6/5 but now '''10''' of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).

=== Sixix ===

Sixix tunings (with generator a supraminor third sharper than 5\18 and flatter than 9\32) have step ratios between 5/4 and 3/2.

Sixix can be considered "meantone smitonic". This is because sixix tunings share the following features with [[meantone]] diatonic tunings:

* The large step is a "meantone", somewhere between near-10/9 (as in [[32edo]]) and near-9/8 (as in [[18edo]]). Thus sixix tempers out [[81/80]] like meantone does.

* The major mosthird (made of two large steps) is a roughly [[meantone]]-sized major third, thus is a stand-in for the classical diatonic major third.

* 18edo can be used to make large and small steps more distinct (the step ratio is 3/2, thus 18edo smitonic is distorted [[19edo]] diatonic), or for its nearly pure 9/8. It also makes rising fifths (733.3c, a perfect mossixth) and falling fifths (666.7c, a major mosfifth) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.

* [[25edo]] can be used to make the major mosthird a good [[5/4]] (384¢).

=== Hyposoft smitonic ===

These tunings (with generator a supraminor third sharper than 3\11 and flatter than 5\18) have [[step ratio]]s between 3/2 and 2/1.

The large step is a sharper major second in these tunings than in sixix tunings. These tunings could be considered "[[parapyth]] smitonic" or "[[archy]] smitonic", in analogy to sixix being meantone smitonic.

{| class="wikitable right-2 right-3 right-4 right-5"

| 2\11, 218.18

| 5\29, 206.90

[[Orgone]] tunings (with generator a minor third sharper than 4\15 and flatter than 3\11) have step ratios between 2/1 and 3/1. It nominally approximates the 2.7.11 subgroup, on which the [[26edo]] tuning is very accurate and pretty much optimal. The large step approximates [[8/7]], and the major smifourth (2 large steps + 1 small step) approximates [[11/8]].

EDOs that support orgone include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].

The sizes of the generator, large step and small step of smitonic are as follows in various orgone tunings.

{| class="wikitable right-2 right-3 right-4 right-5"

! [[11edo]]

! [[15edo]]

! [[26edo]]

| 5\26, 230.77

[[Kleismic]] (aka hanson or keemun) tunings (with generator a minor third sharper than 5\19 and flatter than 4\15) have step ratios between 3/1 and 4/1. Kleismic is a [[5-limit]] microtemperament that tempers out the [[kleisma]] 15625/15552. The generator is close to a pure [[6/5]] minor third, and 6 minor thirds are used to reach [[3/2]]. The 7-note MOS only has one perfect fifth, so bigger MOSes, such as the [[4L 7s]] 11-note MOS, are suggested for getting 5-limit harmony.

The sizes of the generator, large step and small step of smitonic are as follows in various kleismic tunings.

{| class="wikitable right-2 right-3 right-4 right-5"

| generator (g)

| 5\19, 315.79

| 9\34, 317.65

| 6/5

| L (octave - 3g)

| 3\15, 240.00

| 15/13, 23/20

| s (4g - octave)

| 1\19, 63.16

| 2\34, 70.59

| 68.03

{| class="wikitable center-all"

|-

! Heptatonic interval category name

! Notation of 2/1 inverse

| 2

| colspan="6" style="text-align:left" | The chromatic 11-note MOS (either [[7L 4s]] or [[4L 7s]]) also has the following intervals (from some root):

| -10

== Pseudo-diatonic theory ==

=== Orgone ===

=== Sixix ===

== Primodal theory ==

=== Primodal chords ===

== Rank-2 temperaments ==

=== Sixix (18&25) ===

[[Sixix]] can be viewed as a [[dual-fifth]] temperament, i.e. a temperament on the 2.3+.3-.5 "subgroup" (3+ = sharp 3, 3- = flat 3):

* It has both a sharp fifth and a flat fifth but no near-just 3/2.

* Combining the sharp fifth and the flat fifth yields a good approximation of 9/8; two 9/8's make a 5/4, so it tempers out 81/80 in the underlying 2.9.5 subgroup.

* The chroma of sixix[7] is the difference between the sharp fifth and the flat fifth, and functions much like a(n untempered) comma in sixix harmony, giving two slightly different flavors of fifths, minor thirds, major thirds, etc, much like in [[porcupine]] harmony. Tempering out this comma leads to [[7edo]].

== Samples ==

[[Category:Scales]]

[[Category:MOS scales]]

[[Category:Abstract MOS patterns]]