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User:Inthar/5L 3s

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Revision as of 20:46, 9 May 2021 by Inthar (talk | contribs) (→Archeodim)

This page describes User:Inthar's approach to 5L 3s (aka oneirotonic).

Standing assumptions

The TAMNAMS system is used in this article to name 5L 3s intervals and step size ratios and step ratio ranges.

The notation used in this article is J Ultharian (LsLLsLsL) = JKLMNOPQJ, with reference pitch N = 261.6255653 Hz, unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".) Ultharian has been chosen as the default mode because we want to carry over the diatonic idea of sharpening the second-to-last degree to get the leading tone for minor keys and the sharpened "Vmaj", and we also have the "sharp V" for the oneiromajor tonality by default.

The chain of perfect mosfourths becomes: ... P@ K@ N@ Q L O J M P K N Q& L& O& ...

Thus the 13edo gamut is as follows:

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

The 18edo gamut is notated as follows:

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

The 21edo gamut:

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

Modes

Oneirotonic modes are named after cities in the Dreamlands.

Mode UDP Name
Ana LLsLLsLs 7|0 Dylathian (də-LA(H)TH-iən)
LLsLsLLs 6|1 Illarnekian (ill-ar-NEK-iən)
LsLLsLLs 5|2 Celephaïsian (kel-ə-FAY-zhən)
LsLLsLsL 4|3 Ultharian (ul-THA(I)R-iən)
Kata LsLsLLsL 3|4 Mnarian (mə-NA(I)R-iən)
sLLsLLsL 2|5 Kadathian (kə-DA(H)TH-iən)
sLLsLsLL 1|6 Hlanithian (lə-NITH-iən)
sLsLLsLL 0|7 Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"

We call modes with a major mos5th ana modes (from Greek for 'up'), because the sharper 5th degree functions as a flattened melodic fifth when moving from the tonic up. The ana modes of the MOS are the 4 brightest modes, namely Dylathian, Illarnekian, Celephaïsian and Ultharian. We call modes with a minor mos5th kata modes (from Greek for 'down'). The kata modes of the MOS are the 4 darkest modes, namely Mnarian, Kadathian, Hlanithian and Sarnathian. In kata modes, the melodically squashed fifth from the tonic downwards is the flatter 5th degree. Kata modes could be used to distort diatonic tropes that start from the tonic and work downwards or work upwards towards the tonic from below it. For example:

  • Mnarian (LSLSLLSL) and Kadathian (SLLSLLSL) are kata-Mixolydians
  • Hlanithian (SLLSLSLL) is a kata-melodic major (the 4th degree sounds like a major third; it's actually a perfect mosfourth.)
  • Sarnathian (SLSLLSLL) is a kata-melodic minor (When starting from the octave above, the 4th degree sounds like a minor third; it's actually a diminished mosfourth.)

Functional tonalities

For classical-inspired functional harmony, we use the terms (Functional) Oneiromajor and (Functional) Oneirominor: Oneiromajor for Illarnekian where the 6th degree (the rising fifth) can be sharpened, and Oneirominor for Ultharian where the 8th degree (the leading tone) can be sharpened. The respective purposes of these alterations are:

  1. in Oneiromajor, to have both major (requiring a sharpened 6th degree) on the flat fourth "subdominant" and the sharp fifth as "dominant"
  2. in Oneirominor, to have both the flat 8th degree as the dominant of the "mediant" (relative major) and the sharp 8th degree as leading tone

In key signatures, Oneirominor should be treated as Ultharian and Oneiromajor should be treated as Illarnekian. Note that Oneiromajor and Oneirominor still have the relative major-minor relationship; they are related by a major mosthird, just like diatonic major/minor.

Alterations

Archeodim

We call the LSLLLSLS pattern (independently of modal rotation) archeodim, because the "LLL" resembles the archeotonic scale in 13edo and the "LSLSLS" resembles the diminished scale. Archeodim is the most important oneirotonic MODMOS pattern (a MODMOS is a MOS with one or more alterations), because it allows one to evoke certain ana or kata diatonic modes where three whole steps in a row are important (Dorian, Phrygian, Lydian or Mixo) in an octatonic context. The MOS would not always be able to do this because it has at most two consecutive large steps. Archeodim modes exist in all oneirotonic tunings, since they use the same large and small steps as the oneirotonic scale itself.

As with the MOS, archeodim has four ana and four kata rotations:

Mode UDP Name
Ana LLLsLsLs 7|0 &4 Dylydian (Dylathian &4)
LLsLsLsL 6|1 @8 Illarmixian (Illarnekian @8)
LsLLLsLs 5|2 &6 Celdorian (Celephaïsian &6)
sLLLsLsL 4|3 @2 Ulphrygian (Ultharian @2)
Kata LsLsLLLs 3|4 &8 Mnionian (Mnarian &8)
sLsLLLsL 0|7 &7 Sardorian (Sarnathian &7)
LsLsLsLL 3|4 @7 Mnaeolian (Mnarian @7)
sLsLsLLL 0|7 @6 Sarlocrian (Sarnathian @6)

The cyclic order of archeodim modes are:

  1. LSLLLSLS is Celdorian
  2. SLLLSLSL is Ulphrygian
  3. LLLSLSLS is Dylydian
  4. LLSLSLSL is Illarmixian
  5. LSLSLSLL is Mnaeolian
  6. SLSLSLLL is Sarlocrian
  7. LSLSLLLS is Mnionian
  8. SLSLLLSL is Sardorian

Notably the cyclic order is almost exactly the "circle" of diatonic modes except the "circle" breaks at Sardorian-Celdorian.

In 13edo, archeodim is a subset of the 9-note 4L 5s MOS generated by the subminor third 3\13.

Other MODMOSes

Other potentially interesting oneirotonic MODMOSes (that do not use half-sharps or half-flats) are:

  • the distorted harmonic minor LSLLSALS (A = aug oneiron2nd = L + chroma)
  • the distorted Freygish SASLSLLS
  • Celephaïsian &4 &6 LsAsLsLs

Hypohard oneiro theory

Oneirotonic is often used as distorted diatonic. Because distorted diatonic modal harmony and functional harmony both benefit from a recognizable major third, the following theory essentially assumes a hypohard tuning, i.e. an oneirotonic tuning with generator between 5\13 and 7\18 (or possibly an approximation of such a tuning, such as a neji). The reader is encouraged to experiment and see what ideas work for other oneirotonic tunings.

Ana modes

The ana modes have squashed versions of the classical major and minor pentachords R-M2-M3-P4-P5 and R-M2-m3-P4-P5 and can be viewed as providing a distorted version of classical diatonic functional harmony. For example, in the Dylathian mode, the 4:5:9 triad on the sixth degree can sound like both "V" and "III of iv" depending on context.

In pseudo-classical functional harmony, the 6th scale degree (either an augmented mossixth or a perfect mossixth) could be treated as mutable. The perfect mossixth would be used when invoking the diatonic V-to-I trope by modulating by a perfect mosfourth from the sixth degree "dominant". The augmented mossixth would be used when a major key needs to be used on the fourth degree "subdominant".

Pentatonic subsets

The Oneiro Falling Suspended Pentatonic, i.e. R-Lon2-Pon4-Lon5-Lon7, is also an important subset in ana modes: it roughly implies the "least" tonality (In particular, it only implies ana-ness, not major or minor tonality), and it sounds floaty, and suspended, much like suspended and quartal chords do in diatonic contexts. The Oneiro Rising Suspended Pentatonic R-Lon2-Pon4-Pon6-Lon7 (J-K-M-O-P) can be used for similar effect.

Modes of the oneiro-pentatonic MOS:

  1. R-Lon2-Pon4-Lon5-Lon7 Oneiro Falling Suspended Pentatonic
  2. R-Lon2-Pon4-Pon6-Lon7 Oneiro Rising Suspended Pentatonic
  3. R-son3-Pon4-Pon6-Lon7 Oneiro Symmetrical Pentatonic
  4. R-son3-Pon4-Pon6-so8 Oneiro Expanding Quartal Pentatonic
  5. R-son3-son5-Pon6-so8 Oneiro Diminished Pentatonic

Functional harmony

Oneiro has at least two different types of "V-to-I" resolution because of the two fifth sizes:

  1. One uses the sharp fifth as the "V" and uses a true major third. The sharp "V" voiceleads naturally to the flat fifth in the resolved falling tonic triad on the I: e.g. P6-M8-P2 > M5-P1-(M/m)3.
  2. One uses the flat fifth as the "V" and the chord on the "V" is a "false major triad" R-P4-P6 (root-falling 4th-rising 5th).

Some suggested basic ana functional harmony progressions are listed below, outlined very roughly. Note that VI, VII and VIII are sharp 5th, 6th-like and 7th-like degrees respectively. A Roman numeral without maj or min means either major or minor. The "Natural" Roman numerals follow the Illarnekian mode.

  • I-IVmin-VImaj-I
  • Imaj-VIImin-IVmin-Imaj
  • Imin-@IIImaj-VImaj-Imaj
  • Imin-@IIImaj-Vdim-VImaj-Imin
  • Imin-@VIIImin-IIImaj-VImaj-Imin
  • Imin-IVmin-@VIIImin-@IIImaj-VImaj-Imin
  • Imin-IVmin-IIdim-VImaj-Imin
  • Imin-IVmin-IIdim-@IIImaj-Imin
  • I-VIImin-IImin-VImaj-I
  • Imaj-VIImin-IVmin-VImaj-Imaj
  • Modulations by major mos2nd:
    • I-IV-VII-II
    • I-IVmaj-II
    • I-VIImin-II
  • Modulations by major mos3rd:
    • Modulate up major mos2nd twice
    • Imin-VImin-III (only in 13edo)
    • Imaj-&VImin-III (only in 13edo)
  • Modulations by minor mos3rd:
    • I-VI-@III
    • I-IVmin-VImin-@VIIImaj-@III

Another Western-classical-influenced approach to oneirotonic chord progressions is to let the harmony emerge from counterpoint. This would allow, for example, using the perfect oneirofourth and minor oneirofifth (instead of the major oneirothird and the perfect oneirofourth) as stand-ins for major thirds and fourths in neobaroque contexts (this adds some dissonance which might be what you want sometimes, e.g. in a chord that is supposed to resolve to a more consonant chord).

Samples

(A short contrapuntal 13edo keyboard exercise, meant to be played in all 13 keys. The first part is in Oneiromajor, i.e. Illarnekian with mutable 6th degree, and the second part is in Oneirominor, i.e. Celephaïsian with mutable 7th degree.)

(18edo)

(31edo)

(21edo for comparison)

‎(A rather classical-sounding 3-part harmonization of the ascending J Illarnekian scale; tuning is 13edo)

Kata modes

Using triadic harmony is possible in Kata modes, but it requires the use of Rising triads (Rising Minor, Rising Major and Rising Naiadic) more, instead of Falling Triads (flat fifths). This is to make use of the sharp fifth and sharp fourth (flat fifth down from the tonic).

When used in an "ana" way, the kata modes are radically different in character than the ana modes. Particularly in 13edo and tunings close to it, the fifth and seventh scale degrees become the more concordant 11/8 and quasi-13/8 respectively, so they may sound more like stable scale functions. Hlanithian, in particular, may be like a more stable version of the Locrian mode in diatonic.

Chords and extended harmony

Chords are given in oneirotonic MOS interval notation. For example, M5 means major mosfifth (squashed fifth).

"Rising" means that a triad uses the perfect mos6th (major 5th); "falling" means that a triad uses a major mos5th (minor 5th)

  • R-Lon3-Lon5: Falling Major Triad; Squashed Major Triad
  • R-son3-Lon5: Falling Minor Triad; Squashed Minor Triad
  • R-son3-son5: Squashed Dim Triad
  • R-Lon3-Aon5: Squashed Aug Triad
  • R-Lon3-Lon5-Aon6: Falling Major Triad Add6
  • R-son3-Lon5-Aon6: Falling Minor Triad Add6
  • R-Lon3-Lon5-Lon7: Falling Major Tetrad
  • R-son3-Lon5-Lon7: Falling Minor Tetrad
  • R-son3-son5-Lon7: Oneiro Half-Diminished Tetrad
  • R-son3-son5-son7: Orwell Tetrad, Oneiro Diminished Tetrad
  • R-Lon3-Aon6: Squashed 1st Inversion Minor Triad; Sephiroth Triad (approximates 8:10:13 in 13edo and 31edo)
  • R-Lon3-Aon6-Lo8: Sephiroth Triad Add7
  • R-Lon3-Aon6-(Mon2)-(Pon4): Sephiroth Tetrad Add9
  • R-Lon3-Aon6-(mon2)-(Pon4): Sephiroth Tetrad Addm9
  • R-Lon3-Aon6-(Pon4): Sephiroth Tetrad
  • R-son3-Pon6: Rising Minor Triad; Squashed 1st Inversion Major Triad
  • R-Lon3-Pon6: Rising Major Triad
  • R-son3-Lon7: Minor add6 non5
  • R-son3-son7: Minor addm6 non5
  • R-son5-Lon7: Falling non3 add6
  • R-son5-son7: Falling non3 add6
  • R-Lon3-Lo8: Major 7th non5
  • R-son3-Lo8: Minor Major 7th non5
  • R-Lon3-Lon5-Lo8: Falling Major Seventh Tetrad
  • R-son3-Lon5-Lo8: Falling Minor Major Seventh Tetrad
  • R-Lon3-Lon7-Lo8: non5 Major Seventh Add6
  • R-son3-Lon7-Lo8: non5 Minor Major Seventh Add6
  • R-Lon3-Pon6-Lo8: Rising Major Seventh
  • R-son3-Pon6-Lo8: Rising Oneiro Minor Major Seventh
  • R-Lon3-(Mon2): Oneiro Major Add9
  • R-son3-(Mon2): Oneiro Minor Add9
  • R-Lon3-Lon5-(Mon2): Falling Major Triad Add9
  • R-son3-Lon5-(Mon2): Falling Minor Triad Add9
  • R-Lon3-(Mon2)-(Pon4): non5 Major Add9 Sub11
  • R-son3-(Mon2)-(Pon4): non5 Minor Add9 sub11
  • R-Lon2-Pon4: Sus24 Non5
  • R-Lon2-Lon5: Falling Sus2 Triad
  • R-Pon4-Lon5: Falling Sus4 Triad
  • R-Lon2-Pon4-Lon5: Falling Sus24
  • R-Pon4-Lon7: Oneiro Quartal Triad
  • R-Pon4-Lon7-(Mon2): Oneiro Quartal Tetrad, Core Tetrad
  • R-Pon4-Lon7-(Mon2)-(Mon5): Oneiro Quartal Pentad, Core Pentad
  • R-Pon4-Lon7-(Mon2)-(Mon5)-(Mo8): Oneiro Quartal Hexad
  • R-Pon4-Lon7-Lo8: Oneiro Quartal Seventh Tetrad
  • R-Pon4-so8: Expanding Quartal Triad
  • R-Lon2-Pon4-so8: Expanding Quartal Triad add2
  • R-son3-Pon4-so8: Expanding Quartal Triad Addm3
  • R-son5-so8: Contracting Quartal Triad
  • R-son5-son7-so8: Contracting Quartal Triad Addm7
  • R-Lon3-Lon5-so8: Falling Major Triad addm7

Hyposoft oneiro theory

21edo has the soft oneirotonic (5L 3s) MOS with generator 8\21; in addition to the naiadics (457.14¢) and extremely sharp fifths (742.85¢) that generate it, it has neutral thirds (instead of major thirds as in 13edo oneirotonic), neogothic minor thirds, and meantone-like diatonic semitones. The oneirofifths (4-step intervals) are more tritone-like than fifth-like, unlike in 13edo, although they do have a consonant, even JI-like quality to them. In terms of JI, it mainly approximates 5:9:11:13 (R-so8-Lon2-Pon4) and 16:23:30 (R-Lon5-Lo8). Importantly, the sharp fifth is now harmonically much more fifth-like than the flat fifth, unlike in 13edo and harder tunings. Rather than squashed tertian triads, it may be preferable to use triads with sharp fifths, quartal harmony, stacks of seconds and thirds, third+sixth and third+seventh chords, and using the JI approximations (subsets of 5:9:11:13 (R-so8-Lon2-Pon4), 9:10:11:13 (R-Lon2-Lon3-Lon5), and 8:15:23 (R-Lon7-Lon5)).

34edo (semisoft) oneirotonic is broadly similar, except the small steps are more 12edo-like and less meantone-like.

  • sus24 non5: J-K-M, L-M-O, M-N-P, O-P-J, Q-J-L
  • sus24 non5 addmin7: J-Q-K-M, M-L-N-P
  • falling fifth addmaj7 ≈ 16:23:30: O-K-N, L-P-K, Q-M-P
  • Falling Major Triad: Q-K-M, L-N-P
  • 16:23:29 (in 21edo): J-N-Q

Other chords:

  • Quartal Tetrad: Q-J-L-O, L-M-O-J, O-P-J-M, J-K-M-P, M-N-P-K
  • Quartal Pentad: L-M-O-P-J, O-P-J-K-M, J-K-M-N-P, M-N-P-Q-K
  • Falling Minor Triad: J-L-N, O-Q-K
  • squashed diminished: M-O-Q, P-J-L, K-M-O, N-P-J

Seconds, tritones and sevenths are consonances in addition to thirds and sixths. The naiadic is a semiconsonance. The sharp fifth is fifth-like but is still a dissonance.

It is suggested to use R-mos3-mos5-mos7 carefully in 21edo oneiro - it sounds more like a dissonant, unstable 12edo diminished 7th, unlike in 13edo and harder oneiro tunings.

A primodality-inspired perspective

The 21edo Dylathian mode is close to the /23 JI scale 46:51:56:60:66:73:78:86:92. The other 21edo oneirotonic modes could be tuned to modal rotations of this scale. 

Dylathian    1/1  :  178.636063  340.551559  459.994367  624.998596  799.515123  914.208315 1083.243358 1200.000000
Ultharian    51/46:  161.915496  281.358304  446.362533  620.879060  735.572252  904.607295 1021.363937 1200.000000
Hlanithian   28/23:  119.442808  284.447037  458.963564  573.656756  742.691799  859.448441 1038.084504 1200.000000
Illarnekian  30/23:  165.004228  339.520756  454.213948  623.248991  740.005633  918.641696 1080.557192 1200.000000
Mnarian      33/23:  174.516527  289.209719  458.244762  575.001404  753.637467  915.552963 1034.995772 1200.000000
Sarn         73/46:  114.693192  283.728235  400.484877  579.120940  741.036436  860.479244 1025.483473 1200.000000
Celephaisian 39/23:  169.035043  285.791685  464.427748  626.343244  745.786052  910.790281 1085.306808 1200.000000
Kadathian    43/23:  116.756642  295.392705  457.308201  576.751009  741.755238  916.271765 1030.964957 1200.000000

Chords

  • 16:23:30
  • 30:39:56:86

Oneirotonic nejis

Some relatively low-complexity oneirotonic-inspired primodal chords. They are grouped by prime family.

Note: These are quick and dirty and should be field-tested.

/11

  • 22:25:26:29:32:34:38:42:44 Undecimal Celephaïsian
  • 22:25:26:29:32:34:38:40:44 Undecimal Ultharian

/19

The notes 38:41:43:46:48:50:52:54:56:58:60:63:65:68:70:73:76 provide the best low complexity fit to oneirotonic (in particular, 18edo) in the prime family /19.

  • 38:43:45:50:56:58:65:72:76 Novem Bright Celephaïsian
  • 38:42:44:49:55:58:65:72:76 Novem Dark Celephaïsian
  • 38:43:45:50:56:58:65:68:76 Novem Bright Ultharian
  • 38:42:44:49:55:58:65:68:76 Novem Dark Ultharian

/23

23(2:4) has many oneiro pitches, some close to 13edo, and some close to 18edo: 46:48:50:51:52:54:56:57:58:60:63:65:67:68:70:73:74:76:79:82:83:85:87:88:92

  • 46:51:56:60:66:73:78:86:92 Vice Pseudosoft Dylathian
  • 46:52:58:60:68:76:79:89:92 Vice Bright Dylathian
  • 46:51:57:60:67:75:78:88:92 Vice Dark Dylathian
  • 46:52:58:60:68:71:79:89:92 Vice Bright Illarnekian
  • 46:51:57:60:67:70:78:88:92 Vice Dark Illarnekian
  • 46:52:54:60:68:71:79:89:92 Vice Bright Celephaïsian
  • 46:51:54:60:67:70:78:88:92 Vice Dark Celephaïsian
  • 46:52:54:60:68:71:79:83:92 Vice Bright Ultharian
  • 46:51:54:60:67:70:78:82:92 Vice Dark Ultharian
  • 46:52:54:60:64:71:79:83:92 Vice Bright Mnarian
  • 46:51:54:60:63:70:78:82:92 Vice Dark Mnarian
  • 46:49:54:60:64:71:79:83:92 Vice Bright Kadathian
  • 46:48:54:60:63:70:78:82:92 Vice Dark Kadathian
  • 46:49:54:60:64:71:75:83:92 Vice Bright Hlanithian
  • 46:48:54:60:63:70:74:82:92 Vice Dark Hlanithian
  • 46:49:54:58:64:71:75:83:92 Vice Bright Sarn
  • 46:48:54:57:63:70:74:82:92 Vice Dark Sarn
  • 46:49:54:58:64:71:75:83:92 Vice Pseudosoft Sarn
  • 46:48:54:56:63:70:73:82:92 Vice Pseudohard Sarn

/29

  • 58:65:72:80:84:94:99:110:116 Vicenon Dylydian
  • 58:65:72:76:84:94:99:110:116 Vicenon Dylathian
  • 58:65:72:76:84:89:99:110:116 Vicenon Illarnekian
  • 58:65:72:76:84:89:99:104:116 Vicenon Illarmixian
  • 58:65:68:76:84:94:99:110:116 Vicenon Celdorian
  • 58:65:68:76:84:89:99:110:116 Vicenon Celephaïsian
  • 58:65:68:76:84:89:99:104:116 Vicenon Ultharian
  • 58:65:68:76:80:89:99:104:116 Vicenon Mnarian
  • 58:65:68:76:80:89:99:110:116 Vicenon Mnionian
  • 58:65:68:76:80:89:94:104:116 Vicenon Mnaeolian
  • 58:61:68:76:80:89:99:104:116 Vicenon Kadathian
  • 58:61:68:76:84:89:99:104:116 Vicenon Ulphrygian
  • 58:61:68:76:80:89:94:104:116 Vicenon Hlanithian
  • 58:61:68:72:80:89:99:104:116 Vicenon Sardorian
  • 58:61:68:72:80:89:94:104:116 Vicenon Sarn
  • 58:61:68:72:80:84:94:104:116 Vicenon Sarlocrian

Complete nejis

The reader is encouraged to tweak these nejis and add more nejis that they like.

13nejis

  1. 58:61:65:68:72:76:80:84:89:94:99:104:110:116 - A low-complexity 13neji; has /13, /17, /19, and /29 prime modes
    • For lower complexity, can use 64 instead of 65 or 100 instead of 99
  2. 92:97:102:108:114:120:127:134:141:149:157:165:174:184 - Vice 13neji

18nejis

  1. 92:96:100:104:108:112:116:120:125:130:136:141:146:152:158:164:170:177:184 - 18neji with /13, /17, /19, /23, and /29 prime modes

21nejis

  1. 128:132:137:141:146:151:156:161:167:172:178:184:190:197:203:210:217:224:232:240:248:256
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