14edo: Difference between revisions

The '''14 equal divisions of the octave''' ('''14edo'''), or the '''14(-tone) equal temperament''' ('''14tet''', '''14et''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into fourteen equal steps of about 86 [[cent]]s. 14edo contains [[7edo]], doubling its number of tones.

The character of 14edo does not well serve those seeking low-limit JI approaches, with the exception of 5:7:9:11:17:19 (which is quite well approximated, relative to other JI approximations of the low-numbered EDOs). However, the ratios 7/5, 7/6, 9/7, 10/7, 10/9, 11/7, 11/9, and 11/10 are all recognizably approximated, and if you accept that 14edo offers approximations of these intervals, you end up with a low-complexity, high-damage 11-limit temperament where the commas listed at the bottom of this page are tempered out. This leads to some of the bizarre equivalences described in the second "Approximate Ratios" column in the table below.

14et has quite a bit of xenharmonic appeal, in a similar way to 17et, on account of having three types of 3rd and three types of 6th, rather than the usual two of 12et. Since 14et also has a recognizable 4th and 5th, this makes it good for those wishing to explore alternative triadic harmonies without adding significantly more notes. It possesses a triad-rich 9-note MOS scale of [[5L 4s]], wherein 7 of 9 notes are tonic to a subminor, supermajor, and/or neutral triad.

== Intervals ==

! Approximate<br>Ratios 1 <ref>based on treating 14edo as a 2.7/5.9/5.11/5.17/5.19/5 [[subgroup]]; other approaches are possible.</ref>

! Approximate<br>Ratios 2 <ref>based on treating 14edo as an 11-limit temperament of {{val| 14 22 32 39 48}} (14c)</ref>

! Approximate<br>Ratios 3 <ref>nearest 15-odd-limit intervals by direct mapping</ref>

! colspan="3" | [[Ups and Downs Notation]]

=== Chord names ===

For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]].

== JI approximation ==

! rowspan="2" | Subgroup

| [{{val| 14 22 39 }}]

| [{{val| 14 22 39 48 }}]

{{Uniform map|13|13.5|14.5}}

14edo [[tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 14 22 33 39 48 52 }}.  

! [[Harmonic limit|Prime<br>Limit]]

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

| Apotome

| Septimal quartertone

| Slendro diesis

| Hemimage

| Superleap

<references/>

* 5 4 5 - [[MOS]] of [[2L 1s]]

* 3 3 2 3 3 - MOS of [[4L 1s]]

* 3 1 3 3 1 3 - MOS of [[4L 2s]]

=== Titanium[9] ===

14edo is also the largest edo whose patent val [[support]]s [[titanium]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. Titanium is one of the simplest 7-limit temperaments, although rather inaccurate (the 7:5 is mapped onto 6\14, over 70 cents flat). Its otonal/major and utonal/minor tetrads are inversions of one another, which allows a greater variety of chord progressions (since different inversions of the same chord may have very different expressive qualities). Despite being so heavily tempered, the tetrads are still recognizable and aren't unpleasant-sounding as long as one uses the right timbres ("bell-like" or opaque-sounding ones probably work best). Titanium forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Titanium[9] has similarities to mavila, slendro, and pelog scales as well.

Using titanium[9], we could name the intervals of 14edo as follows. The 3, 5, 6, 8, 9, and 11-step intervals are all consonant, while 1, 2, 4, 7, 10, 12, and 13 steps are dissonant. There is no distinction between "perfect" (modulatory) and "imperfect" (major/minor) consonances here; there are enough chords here that root motion may occur by ''any'' consonant interval, and thus ''all'' six consonances are "perfect" intervals, rather than just two of them as in the diatonic system. As in the diatonic scale, the perfect intervals come in pairs separated by a major second, and with a characteristic dissonance between them; in titanium[9] there are three such pairs rather than just one.