14edo: Difference between revisions
Dave Keenan (talk | contribs) Change "Intervals" heading to "Notation". Add subheadings for "Ups and downs notation" and "Ivor Darreg's notation". Change "Chord names" to a main section heading (separate from the Notation section). |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | == Theory == | ||
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 | The character of 14edo does not well serve those seeking low-[[limit]] JI approaches, with the exception of [[Subgroup|5:7:9:11:17:19]] (which is quite well approximated, relative to other JI approximations of the low-numbered edos). However, the [[ratio]]s 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 [[comma]]s listed later in this page are [[tempered out]]. This leads to some of the bizarre equivalences described in the second "Approximate ratios" column in the table. | ||
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 | 14et has quite a bit of [[xenharmonic]] appeal, in a similar way to [[17edo|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. | ||
14edo contains an [[omnidiatonic]] scale that can replace the standard diatonic scale, allowing for recognizable triadic harmony using the chords [[6:7:9]] and [[14:18:21]], as well as a neutral chord which can be seen as [[2:sqrt(6):3]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|14}} | {{Harmonics in equal|14}} | ||
=== Octave stretch === | |||
14edo benefits from [[octave stretch]] as harmonics 3, 7, and 11 are all tuned flat. [[22edt]], [[36ed6]] and [[42zpi]] are among the possible choices. | |||
=== Subsets and supersets === | |||
Since 14 factors into primes as 2 × 7, 14edo contains [[2edo]] and [[7edo]] as subsets. | |||
== Notation == | == Notation == | ||
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! Cents | ! Cents | ||
! Approximate<br>[[Harmonic]]s | ! Approximate<br>[[Harmonic]]s | ||
! Approximate<br>Ratios 1 <ref> | ! Approximate<br>Ratios 1 <ref group="note">{{sg|limit=2.7/5.9/5.11/5.17/5.19/5 [[subgroup]]}}</ref> | ||
! Approximate<br>Ratios 2 <ref> | ! Approximate<br>Ratios 2 <ref group="note">Based on treating 14edo as an 11-limit temperament of {{val| 14 22 32 39 48}} (14c).</ref> | ||
! Approximate<br>Ratios 3 <ref> | ! Approximate<br>Ratios 3 <ref group="note">Nearest 15-odd-limit intervals by [[direct approximation]].</ref> | ||
! colspan="3" | [[Ups and | ! colspan="3" | [[Ups and downs notation]] | ||
! Interval Type | ! Interval Type | ||
! Audio | ! Audio | ||
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| [[File:piano_1_1edo.mp3]] | | [[File:piano_1_1edo.mp3]] | ||
|} | |} | ||
<references group="note" /> | |||
< | === Sagittal notation === | ||
This notation uses the same sagittal sequence as [[9edo#Sagittal notation|9-EDO]], is a subset of the notations for EDOs [[28edo#Sagittal notation|28]] and [[42edo#Second-best fifth notation|42b]], and is a superset of the notation for [[7edo#Sagittal notation|7-EDO]]. | |||
<imagemap> | |||
File:14-EDO_Sagittal.svg | |||
desc none | |||
rect 80 0 300 50 [[Sagittal_notation]] | |||
rect 423 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] | |||
rect 20 80 423 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]] | |||
default [[File:14-EDO_Sagittal.svg]] | |||
</imagemap> | |||
=== Ivor Darreg's notation === | === Ivor Darreg's notation === | ||
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0-3-8-11 = C vE G vB = Cv7 = C down-seven | 0-3-8-11 = C vE G vB = Cv7 = C down-seven | ||
For a more complete list, see [[Ups and | For a more complete list, see [[Ups and downs notation #Chords and Chord Progressions]]. | ||
== Approximation to JI == | == Approximation to JI == | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
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=== Uniform maps === | === Uniform maps === | ||
{{Uniform map| | {{Uniform map|edo=14}} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
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=== Commas === | === Commas === | ||
14et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 14 22 33 39 48 52 }}. | |||
{| class="commatable wikitable center-all left-3 right-4 left-6" | {| class="commatable wikitable center-all left-3 right-4 left-6" | ||
|- | |- | ||
! [[Harmonic limit|Prime<br>limit]] | ! [[Harmonic limit|Prime<br>limit]] | ||
! [[Ratio]]<ref> | ! [[Ratio]]<ref group="note">{{rd}}</ref> | ||
! [[Monzo]] | ! [[Monzo]] | ||
! [[Cent]]s | ! [[Cent]]s | ||
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| Sagugu | | Sagugu | ||
| Diaschisma | | Diaschisma | ||
|- | |||
|7 | |||
|[[21/20]] | |||
|[-2 1 -1 1⟩ | |||
|84.47 | |||
|Zogu | |||
|Chroma | |||
|- | |- | ||
| 7 | | 7 | ||
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| Island comma | | Island comma | ||
|} | |} | ||
<references/> | <references group="note" /> | ||
== Scales == | == Scales == | ||
=== MOS scales === | |||
{{Main|List of MOS scales in {{PAGENAME}}}} | |||
Here are the modes that create MOS scales in 14edo shown on horagrams from Scala, skipping multiples of 14: | Here are the modes that create MOS scales in 14edo shown on horagrams from Scala, skipping multiples of 14: | ||
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[[File:Screen Shot 2020-04-23 at 11.47.30 PM.png|none|thumb|870x870px|5\14 MOS using 1L 1s, 2L 1s, 3L 2s, 3L 5s, 3L 8s]] | [[File:Screen Shot 2020-04-23 at 11.47.30 PM.png|none|thumb|870x870px|5\14 MOS using 1L 1s, 2L 1s, 3L 2s, 3L 5s, 3L 8s]] | ||
=== Beep[9] === | ==== Beep[9] ==== | ||
14edo is also the largest edo whose patent val [[support]]s [[beep]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. beep 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). beep forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Beep[9] has similarities to mavila, slendro, and pelog scales as well. | 14edo is also the largest edo whose patent val [[support]]s [[beep]] temperament, tempering out the chromatic semitone (21:20), and falling toward the "brittle" (fifths wider than in 9edo) end of that spectrum. beep 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). beep forms enneatonic modes which are melodically strong and are very similar to diatonic modes, only with two mediants and submediants instead of one. Beep[9] has similarities to mavila, slendro, and pelog scales as well. | ||
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* 13\14: Major 9th<sub>9</sub>: A high, incisive leading tone. | * 13\14: Major 9th<sub>9</sub>: A high, incisive leading tone. | ||
* 14\14: The 10th<sub>9</sub> or "enneatonic decave" (i. e. the octave, 2:1). | * 14\14: The 10th<sub>9</sub> or "enneatonic decave" (i. e. the octave, 2:1). | ||
=== Others === | |||
* 2 2 2 2 2 2 2 - [[Equiheptatonic]] (exactly [[7edo]]) | |||
* 2 2 2 2 1 4 1 - Fennec{{idiosyncratic}} (original/default tuning) | |||
* 1 4 1 2 2 2 2 - Inverse fennec{{idiosyncratic}} (original/default tuning) | |||
* 3 1 4 1 4 1 - Pseudo-[[augmented]] | |||
* 1 4 1 2 1 4 1 - Pseudo-double harmonic minor | |||
== Diagrams == | == Diagrams == | ||
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== See also == | == See also == | ||
* [[Lumatone mapping for 14edo]] | * [[Lumatone mapping for 14edo]] | ||
* [[MisterShafXen’s take on 14edo harmony]] | |||
== Further reading == | == Further reading == | ||