14edo: Difference between revisions
General cleanup |
|||
| Line 6: | Line 6: | ||
}} | }} | ||
== Intervals == | |||
= | |||
{| class="wikitable center-all right- | {| class="wikitable center-all right-2 left-4 left-5" | ||
|- | |- | ||
! Degree | ! Degree | ||
! Cents | ! Cents | ||
! Nearest <br> [[Harmonic]] | ! Nearest<br>[[Harmonic]] | ||
! Approximate <br> Ratios 1 <ref>based on treating | ! 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</ref> | ! Approximate<br>Ratios 2 <ref>based on treating 14edo as an 11-limit temperament</ref> | ||
! colspan="3" | [[Ups and Downs Notation | ! colspan="3" | [[Ups and Downs Notation]] | ||
! Interval Type | ! Interval Type | ||
|- | |- | ||
| Line 33: | Line 32: | ||
| 20/19, 19/18, 18/17 | | 20/19, 19/18, 18/17 | ||
| 22/21, 28/27, 21/20 | | 22/21, 28/27, 21/20 | ||
| up-unison,<br | | up-unison,<br>down-2nd | ||
| ^1, v2 | | ^1, v2 | ||
| ^D, vE | | ^D, vE | ||
| Line 41: | Line 40: | ||
| 171.429 | | 171.429 | ||
| 71 | | 71 | ||
| 11/10, 10/9, | | 11/10, 10/9, 19/17 | ||
| 9/8, 10/9, | | 9/8, 10/9, 11/10, 12/11 | ||
| 2nd | | 2nd | ||
| 2 | | 2 | ||
| E | | E | ||
| Neutral 2nd, or<br | | Neutral 2nd, or<br>Narrow Major 2nd | ||
|- | |- | ||
| | | 3 | ||
| 257.143 | | 257.143 | ||
| 37 | | 37 | ||
| 22/19, 20/17 | | 22/19, 20/17 | ||
| 7/6, 8/7 | | 7/6, 8/7 | ||
| up-2nd,<br | | up-2nd,<br>down-3rd | ||
| ^2, v3 | | ^2, v3 | ||
| ^E, vF | | ^E, vF | ||
| Line 71: | Line 70: | ||
| 428.571 | | 428.571 | ||
| 41 | | 41 | ||
| 9/7, 14/11, | | 9/7, 14/11, 22/17 | ||
| 9/7, 14/11 | | 9/7, 14/11 | ||
| up-3rd,<br | | up-3rd,<br>down-4th | ||
| ^3, v4 | | ^3, v4 | ||
| ^F, vG | | ^F, vG | ||
| Line 93: | Line 92: | ||
| 7/5, 10/7 | | 7/5, 10/7 | ||
| 7/5, 10/7 | | 7/5, 10/7 | ||
| up-4th,<br | | up-4th,<br>down-5th | ||
| ^4, v5 | | ^4, v5 | ||
| ^G, vA | | ^G, vA | ||
| Line 108: | Line 107: | ||
| Narrow 5th | | Narrow 5th | ||
|- | |- | ||
| | | 9 | ||
| 771.429 | | 771.429 | ||
| 25 | | 25 | ||
| 14/9, 11/7, | | 14/9, 11/7, 17/11 | ||
| | | 14/9, 11/7 | ||
| up-5th,<br | | up-5th,<br>down-6th | ||
| ^5, v6 | | ^5, v6 | ||
| ^A, vB | | ^A, vB | ||
| Line 128: | Line 127: | ||
| Neutral 6th | | Neutral 6th | ||
|- | |- | ||
| | | 11 | ||
| 942.857 | | 942.857 | ||
| 55 | | 55 | ||
| 19/11, 17/10 | | 19/11, 17/10 | ||
| 12/7, 7/4 | | 12/7, 7/4 | ||
| up-6th,<br | | up-6th,<br>down-7th | ||
| ^6, v7 | | ^6, v7 | ||
| ^B, vC | | ^B, vC | ||
| Line 141: | Line 140: | ||
| 1028.571 | | 1028.571 | ||
| 29 | | 29 | ||
| 20/11, 9/5, | | 20/11, 9/5, 34/19 | ||
| 16/9, 9/5, | | 16/9, 9/5, 20/11, 11/6 | ||
| 7th | | 7th | ||
| 7 | | 7 | ||
| C | | C | ||
| Neutral 7th, or<br | | Neutral 7th, or<br>Wide Minor 7th | ||
|- | |- | ||
| 13 | | 13 | ||
| 1114.286 | | 1114.286 | ||
| 61 | | 61 | ||
| 19/10, 36/19, | | 19/10, 36/19, 17/9 | ||
| 21/11, 27/14, 40/21 | | 21/11, 27/14, 40/21 | ||
| up-7th,<br />down-8ve | | up-7th,<br />down-8ve | ||
| Line 158: | Line 157: | ||
| Wide Major 7th | | Wide Major 7th | ||
|- | |- | ||
| | | 14 | ||
| 1200 | | 1200.000 | ||
| 2 | | 2 | ||
| colspan="2" | 2/1 | | colspan="2" | 2/1 | ||
| Line 170: | Line 169: | ||
<references /> | <references /> | ||
[[ | [[Ivor Darreg]] wrote in [http://www.tonalsoft.com/sonic-arts/darreg/dar15.htm this article]: | ||
''The 14-tone scale presents a new situation: while one might use ordinary sharps and flats in addition to conventional naturals for the notes of the 7-tone-equal temperament, it would be misleading and confusing to do so, because there is a 7-tone circle of fifths (admittedly quite distorted) already notatable and nameable as F C G D A E B in the usual manner. But there is no 14-tone circle of fifths. There is simply a second set of 7 fifths in a circle which does not intersect the with the first set. Thus is we think of B-flat and B, or B-natural and F-sharp, the 14-tone-system interval would NOT be a fifth of that system and would not sound like one, since B F would be the very same kind of distorted fifth that C G or A E happens to be in 7 or 14. Our suggestion is to call the new notes of 14, the second set of 7, F* C* G* D* A* E* B*, and use asterisks or arrows or whatever you please on the staff. Or just number the tones as for 13.'' | ''The 14-tone scale presents a new situation: while one might use ordinary sharps and flats in addition to conventional naturals for the notes of the 7-tone-equal temperament, it would be misleading and confusing to do so, because there is a 7-tone circle of fifths (admittedly quite distorted) already notatable and nameable as F C G D A E B in the usual manner. But there is no 14-tone circle of fifths. There is simply a second set of 7 fifths in a circle which does not intersect the with the first set. Thus is we think of B-flat and B, or B-natural and F-sharp, the 14-tone-system interval would NOT be a fifth of that system and would not sound like one, since B F would be the very same kind of distorted fifth that C G or A E happens to be in 7 or 14. Our suggestion is to call the new notes of 14, the second set of 7, F* C* G* D* A* E* B*, and use asterisks or arrows or whatever you please on the staff. Or just number the tones as for 13.'' | ||
| Line 183: | Line 182: | ||
|} | |} | ||
=Chord Names= | == Chord Names == | ||
Ups and downs can be used to name 14edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). | Ups and downs can be used to name 14edo chords. Because every interval is perfect, the quality can be omitted, and the words major, minor, augmented and diminished are never used. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). | ||
| Line 199: | Line 198: | ||
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. | For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. | ||
= | == Just approximation == | ||
=== Selected 13-limit intervals === | |||
[[File:14ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:14ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
== Rank two temperaments == | |||
* [[List of 14edo rank two temperaments by badness]] | |||
=Rank two temperaments= | |||
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: | ||
[[File:Screen Shot 2020-04-23 at 11.47.09 PM.png|none|thumb|877x877px|3\14 MOS using 1L 1s, 1L 2s, 1L 3s, 4L 1s, 5L 4s]] | [[File:Screen Shot 2020-04-23 at 11.47.09 PM.png|none|thumb|877x877px|3\14 MOS using 1L 1s, 1L 2s, 1L 3s, 4L 1s, 5L 4s]] | ||
[[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]] | ||
=Harmony= | == Scales == | ||
5 5 4 - [[MOSScales|MOS]] of [[2L_1s|2L1s]]<br /> | |||
5 4 5 - [[MOSScales|MOS]] of [[2L_1s|2L1s]]<br /> | |||
4 1 4 4 1 - [[MOSScales|MOS]] of [[3L_2s|3L2s]]<br /> | |||
4 1 4 1 4 - [[MOSScales|MOS]] of [[3L_2s|3L2s]]<br /> | |||
3 3 3 3 2 - [[MOSScales|MOS]] of [[4L_1s|4L1s]]<br /> | |||
3 2 3 3 3 - [[MOSScales|MOS]] of [[4L_1s|4L1s]]<br /> | |||
3 2 2 2 2 3 - [[MOSScales|MOS]] of [[2L_4s|2L4s]]<br /> | |||
2 2 3 2 2 3 - [[MOSScales|MOS]] of [[2L_4s|2L4s]]<br /> | |||
'''3 3 1 3 3 1 -''' [[MOSScales|MOS]] of [[4L_2s|4L2s]]<br /> | |||
3 1 3 3 1 3 - [[MOSScales|MOS]] of [[4L_2s|4L2s]]<br /> | |||
3 1 3 1 3 3 - [[MOSScales|MOS]] of [[4L_2s|4L2s]]<br /> | |||
2 2 1 2 2 2 2 1 - [[MOSScales|MOS]] of [[6L_2s|6L2s]]<br /> | |||
2 2 2 1 2 2 2 1 - [[MOSScales|MOS]] of [[6L_2s|6L2s]]<br /> | |||
'''2 2 2 2 1 2 2 1 -''' [[MOSScales|MOS]] of [[6L_2s|6L2s]]<br /> | |||
2 1 2 2 1 2 2 2 - [[MOSScales|MOS]] of [[6L_2s|6L2s]]<br /> | |||
2 1 2 1 2 1 2 1 2 - [[MOSScales|MOS]] of [[5L_4s|5L4s]]<br /> | |||
2 1 2 1 2 1 2 2 1 - [[MOSScales|MOS]] of [[5L_4s|5L4s]]<br /> | |||
2 1 2 1 2 2 1 2 1 - [[MOSScales|MOS]] of [[5L_4s|5L4s]]<br /> | |||
2 1 1 2 1 2 1 1 2 1 - [[MOSScales|MOS]] of [[4L_6s|4L6s]]<br /> | |||
2 1 1 1 2 1 1 2 1 1 1 - [[MOSScales|MOS]] of [[3L_8s|3L8s]]<br /> | |||
'''1 1 2 1 1 1 2 1 1 1 2''' - [[MOSScales|MOS]] of [[3L_8s|3L8s]]<br /> | |||
== Harmony == | |||
The character of 14-EDO 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 above. | The character of 14-EDO 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 above. | ||
14-EDO has quite a bit of xenharmonic appeal, in a similar way to 17-EDO, on account of having three types of 3rd and three types of 6th, rather than the usual two of 12-TET. Since 14-EDO 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 5L4s, wherein 7 of 9 notes are tonic to a subminor, supermajor, and/or neutral triad. | 14-EDO has quite a bit of xenharmonic appeal, in a similar way to 17-EDO, on account of having three types of 3rd and three types of 6th, rather than the usual two of 12-TET. Since 14-EDO 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 5L4s, wherein 7 of 9 notes are tonic to a subminor, supermajor, and/or neutral triad. | ||
===Titanium[9]=== | === Titanium[9] === | ||
14edo is also the largest edo whose patent val supports [[ | 14edo is also the largest edo whose patent val supports [[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. | 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. | ||
1\14: Minor 2nd< | * 1\14: Minor 2nd<sub>9</sub>: functions similarly to the diatonic minor second, but is more incisive. | ||
2\14: Major 2nd< | * 2\14: Major 2nd<sub>9</sub>: functions similarly to the diatonic major second, but is narrower and has a rather different quality. | ||
3\14: Perfect 3rd< | * 3\14: Perfect 3rd<sub>9</sub>: the generator, standing in for 8:7, 7:6, ''and'' 6:5, but closest to 7:6. | ||
4\14: Augmented 3rd< | * 4\14: Augmented 3rd<sub>9</sub>, diminished 4th<sub>9</sub>: A dissonance, falling in between two perfect consonances and hence analogous to the tritone. | ||
5\14: Perfect 4th< | * 5\14: Perfect 4th<sub>9</sub>: technically represents 5:4 but is quite a bit wider. | ||
6\14: Perfect 5th< | * 6\14: Perfect 5th<sub>9</sub>: represents 4:3 and 7:5, much closer to the former. | ||
7\14: Augmented 5th< | * 7\14: Augmented 5th<sub>9</sub>, diminished 6th<sub>9</sub>: The so-called "tritone" (but no longer made up of three whole tones). Like 4\14 and 10\14, this is a characteristic dissonance separating a pair of perfect consonances. | ||
8\14: Perfect 6th< | * 8\14: Perfect 6th<sub>9</sub>: represents 10:7 and 3:2, much closer to the latter. | ||
9\14: Perfect 7th< | * 9\14: Perfect 7th<sub>9</sub>: technically represents 5:8 but noticeably narrower. | ||
10\14: Augmented 7th< | * 10\14: Augmented 7th<sub>9</sub>, diminished 8th<sub>9</sub>: The third and final characteristic dissonance, analogous to the tritone. | ||
11\14: Perfect 8th< | * 11\14: Perfect 8th<sub>9</sub>: Represents 5:3, 12:7 and 7:4. | ||
12\14: Minor 9th< | * 12\14: Minor 9th<sub>9</sub>: Analogous to the diatonic minor seventh, but sharper than usual. | ||
13\14: Major 9th< | * 13\14: Major 9th<sub>9</sub>: A high, incisive leading tone. | ||
14\14: The 10th< | * 14\14: The 10th<sub>9</sub> or "enneatonic decave" (i. e. the octave, 2:1). | ||
=Commas= | == Commas == | ||
14 EDO [[tempering_out|tempers out]] the following [[Comma| | 14 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes the [[val]] {{val|14 22 33 39 48 52}}.) | ||
{| class="wikitable" | {| class="wikitable center-all left-2 right-3" | ||
|- | |- | ||
! | ! Comma | ||
! | ! Monzo | ||
! | ! Cents | ||
![[Color notation | ! [[Color notation|Color Names]] | ||
! | ! Name 1 | ||
! | ! Name 2 | ||
|- | |- | ||
| 2187/2048 | |||
| |<nowiki> | -11 7 </nowiki>> | | |<nowiki> | -11 7 </nowiki>> | ||
| 113.69 | |||
| Lawa | |||
| Apotome | |||
| | |||
|- | |- | ||
| 2048/2025 | |||
| |<nowiki> | 11 -4 -2 </nowiki>> | | |<nowiki> | 11 -4 -2 </nowiki>> | ||
| 19.55 | |||
| Sagugu | |||
| Diaschisma | |||
| | |||
|- | |- | ||
| 36/35 | |||
| |<nowiki> | 2 2 -1 -1 </nowiki>> | | |<nowiki> | 2 2 -1 -1 </nowiki>> | ||
| 48.77 | |||
| Rugu | |||
| Septimal Quarter Tone | |||
| | |||
|- | |- | ||
| 49/48 | |||
| |<nowiki> | -4 -1 0 2 </nowiki>> | | |<nowiki> | -4 -1 0 2 </nowiki>> | ||
| 35.70 | |||
| Zozo | |||
| Slendro Diesis | |||
| | |||
|- | |- | ||
| 1728/1715 | |||
| |<nowiki> | 6 3 -1 -3 </nowiki>> | | |<nowiki> | 6 3 -1 -3 </nowiki>> | ||
| 13.07 | |||
| Triru-agu | |||
| Orwellisma | |||
| Orwell Comma | |||
|- | |- | ||
| 10976/10935 | |||
| |<nowiki> | 5 -7 -1 3 </nowiki>> | | |<nowiki> | 5 -7 -1 3 </nowiki>> | ||
| 6.48 | |||
| Satrizo-agu | |||
| Hemimage | |||
| | |||
|- | |- | ||
| | |||
| |<nowiki> | 47 -7 -7 -7 </nowiki>> | | |<nowiki> | 47 -7 -7 -7 </nowiki>> | ||
| 0.34 | |||
| Trisa-seprugu | |||
| Akjaysma | |||
| 5\7 Octave Comma | |||
|- | |- | ||
| 99/98 | |||
| |<nowiki> | -1 2 0 -2 1 </nowiki>> | | |<nowiki> | -1 2 0 -2 1 </nowiki>> | ||
| 17.58 | |||
| Loruru | |||
| Mothwellsma | |||
| | |||
|- | |- | ||
| 243/242 | |||
| |<nowiki> | -1 5 0 0 -2 </nowiki>> | | |<nowiki> | -1 5 0 0 -2 </nowiki>> | ||
| 7.14 | |||
| Lulu | |||
| Rastma | |||
| | |||
|- | |- | ||
| 385/384 | |||
| |<nowiki> | -7 -1 1 1 1 </nowiki>> | | |<nowiki> | -7 -1 1 1 1 </nowiki>> | ||
| 4.50 | |||
| Lozoyo | |||
| Keenanisma | |||
| | |||
|- | |- | ||
| 91/90 | |||
| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>> | | |<nowiki> | -1 -2 -1 1 0 1 </nowiki>> | ||
| 19.13 | |||
| Thozogu | |||
| Superleap | |||
| | |||
|- | |- | ||
| 676/675 | |||
| |<nowiki> | 2 -3 -2 0 0 2 </nowiki>> | | |<nowiki> | 2 -3 -2 0 0 2 </nowiki>> | ||
| 2.56 | |||
| Bithogu | |||
| Parizeksma | |||
| | |||
|} | |} | ||
=Images= | == Images == | ||
[[File:14edo_wheel.png|alt=14edo wheel.png|343x343px|14edo wheel.png]] | [[File:14edo_wheel.png|alt=14edo wheel.png|343x343px|14edo wheel.png]] | ||
= | == Books == | ||
= | |||
[[File:Libro_Tetradecafónico.PNG|alt=Libro_Tetradecafónico.PNG|Libro_Tetradecafónico.PNG]] | [[File:Libro_Tetradecafónico.PNG|alt=Libro_Tetradecafónico.PNG|Libro_Tetradecafónico.PNG]] | ||
''Sword, Ron. "Tetradecaphonic Scales for Guitar" IAAA Press. First Ed: June 2009.'' | ''Sword, Ron. "Tetradecaphonic Scales for Guitar" IAAA Press. First Ed: June 2009.'' | ||
=Compositions= | == Compositions == | ||
[http://split-notes.com/004/ NANA WODORI] by knowsur | [http://split-notes.com/004/ NANA WODORI] by knowsur | ||
| Line 396: | Line 394: | ||
[http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/Fourteen_EDO_CBobro_r8b.mp3 Fourteen EDO] by [[Cameron_Bobro|Cameron Bobro]] | [http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/Fourteen_EDO_CBobro_r8b.mp3 Fourteen EDO] by [[Cameron_Bobro|Cameron Bobro]] | ||
= Software Support = | == Software Support == | ||
[[File:SA14 for Mus2.zip]] | [[File:SA14 for Mus2.zip]] | ||