A-team

A-Team is an oneirotonic-based temperament or harmonic framework, based on the oneirotonic MOS with period 1\1 and a generator chain with generator a subfourth between 13edo's 5\13 (461.5¢) and 18edo's 7\18 (466.7¢).

Tuning range

A-Team tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.

A short definition of A-Team is "meantone oneirotonic". This is because A-Team tunings share the following features with meantone diatonic tunings:

• The large step is a "meantone", somewhere between near-10/9 (as in 13edo) and near-9/8 (as in 18edo). Thus A-Team tempers out 81/80 like meantone does.
• The major mosthird (made of two large steps) is a meantone- to flattone-sized major third, thus is a stand-in for the classical diatonic major third.

EDOs that support A-Team include 13edo, 18edo, and 31edo.

• 13edo has characteristically small major mosseconds of about 185c. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best 11/8 out of all A-team tunings.
• 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. 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.
• 31edo is very close to the 2.9.5.21 POTE tuning, and can be used to make the major mos3rd a near-just 5/4.
• 44edo (generator 17\44 = 463.64¢), 57edo (generator 22\57 = 463.16¢), and 70edo (generator 27\70 = 462.857¢) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.

The sizes of the generator, large step and small step of oneirotonic are as follows in various A-Team tunings.

	13edo	18edo	31edo	Optimal (POTE) tuning	JI intervals represented (2.9.5.21 subgroup)
generator (g)	5\13, 461.54	7\18, 466.67	12\31, 464.52	464.39	21/16
L (3g - octave)	2\13, 184.62	3\18, 200.00	5\31, 193.55	193.16	9/8, 10/9
s (-5g + 2 octaves)	1\13, 92.31	1\18, 66.67	2\31, 77.42	78.07	21/20

Trivia: A-Team can be tuned by ear, by tuning a chain of pure harmonic sevenths and taking every other note. This corresponds to using a generator of 64/49 = 462.34819 cents. A chain of fourteen 7/4's are needed to tune the 8-note oneirotonic MOS. This produces a tuning close to 13edo.

Basic 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 an A-Team 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

We call modes (see oneirotonic 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.

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".

The Oneiro Falling Suspended Pentatonic, i.e. P1-M2-P4-M5-M7 (on J, J-K-M-N-P), 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 P1-M2-P4-P6-M7 (J-K-M-O-P) can be used for similar effect.

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.

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 for comparison)

(31edo for comparison)

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

Kata modes

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.)

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-M3-M5: Falling Major Triad; Squashed Major Triad
• R-m3-M5: Falling Minor Triad; Squashed Minor Triad
• R-m3-m5: Squashed Dim Triad
• R-M3-A5: Squashed Aug Triad
• R-M3-M5-A6: Falling Major Triad Add6
• R-m3-M5-A6: Falling Minor Triad Add6
• R-M3-M5-M7: Falling Major Tetrad
• R-m3-M5-M7: Falling Minor Tetrad
• R-m3-m5-M7: Half-Diminished Tetrad
• R-m3-m5-m7: Orwell Tetrad, Diminished Tetrad
• R-M3-A6: Squashed 1st Inversion Minor Triad; Sephiroth Triad (approximates 8:10:13 in 13edo and 31edo)
• R-M3-A6-M8: Sephiroth Triad Add7
• R-M3-A6-(M2)-(P4): Sephiroth Triad Add9 Sub11
• R-M3-A6-(m2)-(P4): Sephiroth Triad Addm9 Sub11
• R-M3-A6-(P4): Sephiroth Triad Sub11
• R-m3-P6: Rising Minor Triad; Squashed 1st Inversion Major Triad
• R-M3-P6: Rising Major Triad
• R-m3-M7: Minor add6 no5
• R-m3-m7: Minor addm6 no5
• R-m5-M7: Falling no3 add6
• R-m5-m7: Falling no3 add6
• R-M3-M8: Major 7th no5
• R-m3-M8: Minor Major 7th no5
• R-M3-M5-M8: Falling Major Seventh Tetrad
• R-m3-M5-M8: Falling Minor Major Seventh Tetrad
• R-M3-M7-M8: no5 Major Seventh Add6
• R-m3-M7-M8: no5 Minor Major Seventh Add6
• R-M3-P6-M8: Rising Major Seventh
• R-m3-P6-M8: Rising Oneiro Minor Major Seventh
• R-M3-(M2): Oneiro Major Add9
• R-m3-(M2): Oneiro Minor Add9
• R-M3-M5-(M2): Falling Major Triad Add9
• R-m3-M5-(M2): Falling Minor Triad Add9
• R-M3-(M2)-(P4): no5 Major Add9 Sub11
• R-m3-(M2)-(P4): no5 Minor Add9 Sub11
• R-m3-P6-M7-(M2)-(P4)-(A6)-(M8)
• R-M2-P4: Sus24 No5
• R-M2-M5: Falling Sus2 Triad
• R-P4-M5: Falling Sus4 Triad
• R-M2-P4-M5: Falling Sus24
• R-P4-M7: Oneiro Quartal Triad
• R-P4-M7-(M2): Oneiro Quartal Tetrad, Core Tetrad
• R-P4-M7-(M2)-(M5): Oneiro Quartal Pentad, Core Pentad
• R-P4-M7-(M2)-(M5)-(M8): Oneiro Quartal Hexad
• R-P4-M7-M8: Oneiro Quartal Seventh Tetrad
• R-P4-m8: Expanding Quartal Triad
• R-M2-P4-m8: Expanding Quartal Triad add2
• R-m3-P4-m8: Expanding Quartal Triad Addm3
• R-m5-m8: Contracting Quartal Triad
• R-m5-m7-m8: Contracting Quartal Triad Addm7
• R-M3-M5-m8: Falling Major Triad addm7

Primodal theory

18edo may be a better basis for a style of A-Team primodality using comma sharp and comma flat fifths than 13edo (in particular diesis sharp and diesis flat fifths; diesis is a category with a central region of 32 to 40¢). In 18edo both the major fifth (+31.4¢) and the minor fifth (-35.3¢) are about a diesis off from a just perfect fifth. In 13edo only the major fifth is a diesis sharp, and it is +36.5¢ off from just; so there's less wiggle room for a neji if you want every major fifth to be at most a diesis sharp).

31nejis (though 34edo is not an A-Team tuning) also provide opportunities to use dieses directly, since 1\31 (38.71¢) is a diesis.

Primodal chords

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

/11

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

/13

• 13:15:19 Tridecimal Falling Ultraminor Triad
• 13:16:19 Tridecimal Falling Submajor Triad
• 13:16:21 Tridecimal Squashed 1st Inversion Minor Triad
• 13:17:19 Tridecimal Naiadic Maj2; Tridecimal Falling Sus4
• 13:17:20 Tridecimal Rising Sus4
• 13:17:21 Tridecimal Squashed 2nd Inversion Major Triad
• 13:16:19:22 Tridecimal Falling Major Tetrad
• 26:29:38 Tridecimal Falling Sus2 Triad
• 26:31:38 Tridecimal Falling Bright Minor Triad
• 26:33:38 Tridecimal Falling Bright Major Triad
• 26:29:34:38 Tridecimal Falling Sus24

/17

• 17:20:25 Septendecimal Falling Minor Triad
• 17:21:25 Septen Falling Major Triad
• 17:20:26 Septen Rising Minor Triad
• 17:20:25:29 Septen Falling Minor Tetrad
• 17:21:25:29 Septen Falling Major Tetrad
• 17:20:26:29 Septen Rising Minor Triad addM6
• 34:41:50 Septen Falling Bright Minor Triad
• 34:43:50 Septen Falling Octodecous Major Triad (octodecous means '18edo-like')
• 34:40:47:55 Septen Orwell Tetrad
• 34:40:52:58:76:89:102:129 (Celephaïsian + P5; R-min3-r5-M6-M9-sub11-P12(fc)-M14)
• 34:40:52:58:76:89:102:110:129 (Celephaïsian + P5; R-min3-r5-M6-M9-sub11-P12(fc)-supmin13-M14)
• 34:40:50:58:89:102:129 (R-min3-f5-M6-M9-sub11-P12(rc)-M14)
• 34:40:50:58:89:102:110:129 (R-min3-f5-M6-M9-sub11-P12(rc)-supmin13-M14)
• 34:40:50:58:76:89:110:129 (R-m3-f5-M6-M9-sub11-supm13-M7)
• 34:40:50:58:76:89:102:110:129:208 (R-m3-f5-M6-M9-sub11-P12(rc)-supm13-M14-r19(rc^2))
• 34:38:40:44:49:52:58:64:68 Septen Celephaïsian

/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 A-Team (in particular, 18edo) in the prime family /19.

• 19:24:28 Novemdecimal Falling Bright Major Triad
• 19:23:28 Novem Falling Supraminor Triad
• 19:22:28 Novem Falling Ultraminor Triad
• 19:24:29 Novem Rising Major Triad
• 19:24:30 Novem Augmented Triad
• 19:24:43 Novem Major no5 add9
• 19:24:43:50 Novem Major no5 add9sub11
• 19:24:28:43:50 Novem Falling Major Triad add9 sub11
• 19:24:29:43:50 Novem Rising Major Triad add9 sub11
• 19:25:34 Novem Expanding Quartal
• 19:26:34 Novem Contracting Quartal
• 38:43:56 Novem Falling Minor Triad
• 38:45:56 Novemdecimal Falling Dark Major Triad
• 38:48:56:65 Novem Falling Major Tetrad
• 38:48:73 Novem Major Seventh no5
• 38:48:63 Novem Falling Major Triad
• 38:50:65 Novem Oneiro Quartal Triad
• 38:50:65:73 Novem Oneiro Quartal Seventh Tetrad
• 38:50:65:86 Novem Oneiro Core Tetrad
• 38:50:65:86:112 Novem Oneiro Core Pentad
• 38:50:65:86:112:146 Novem Oneiro Core Hexad
• 38:50:63 Novem Squashed First Inversion Neutral Triad
• 38:43:45:50:56:58:65:72:76 Novem Bright Celephaïsian
• 38:42:44:49:55:58:65:72:76 Novem Dark Celephaïsian

/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

• 23:27:30 Vicesimotertial Falling Min4 no5
• 23:27:30:35:44 Vice Falling Min4 addM5,M7
• 23:27:37 Vice Orwell Tetrad no4
• 23:29:34 Vice Octodecous Falling Major Triad
• 46:54:68 Vice Octodecous Falling Minor Triad
• 46:54:60:67 Vice Falling Min4
• 46:54:63 Vice Squashed Dim
• 46:54:63:68 Vice Oneiro Half-diminished Tetrad
• 46:54:63:74 Vice Orwell Tetrad
• 46:54:67 Vice Tridecous Falling Minor Triad (tridecous means '13edo-like')
• 46:57:67 Vice Tridecous Falling Major Triad
• 46:54:67:78 Vice Tridecous Falling Minor Tetrad
• 46:57:67:78 Vice Tridecous Falling Major Tetrad
• 46:54:60:67:78 Vice Falling Minor Tetrad Add Min4
• 46:60:67 Vice Falling Sus4
• 46:54:60:67 Vice Falling Min3 Sus4
• 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

/29

• 29:34:38 Vicesimononal Falling Sus4
• 29:34:42 Vicenon Falling Minor Triad
• 29:36:42 Vicenon Falling Major Triad
• 29:34:40:47 Vicenon Orwell Tetrad
• 29:38:52 Vicenon Expanding Quartal Triad
• 29:40:52 Vicenon Contracting Quartal Triad
• 29:38:65:84:99 Vicenon Oneiro Core Pentad
• 29:38:65:84:99:110 Vicenon Oneiro Core Hexad
• 58:65:76:84:99:116 Vicenon Oneiro Falling Suspended Pentatonic
• 58:65:76:89:99:116 Vicenon Oneiro Rising Suspended Pentatonic
• 58:72:76:89:99:116 Vicenon Oneiro Symmetrical Pentatonic
• 58:72:76:89:104:116 Vicenon Oneiro Expanding Quartal Pentatonic
• 58:72:80:89:104:116 Vicenon Oneiro Diminished Pentatonic
• 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 Sarnathian
• 58:61:68:72:80:84:94:104:116 Vicenon Sarlocrian

/47

• 47:52:55:61:68:72:80:89:94 Quadseptimal Celephaïsian

/61

• 61:68:72:80:89:93:104:116:122 Sessantunesimal Celephaïsian

Some A-Team 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

31nejis

1. 92:94:96:98:101:103:105:108:110:113:115:118:120:123:126:129:132:135:138:141:144:147:150:154:157:161:165:168:172:176:180:184

Samples

WT13C Prelude II (J Oneirominor) (score) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

(13edo, first 30 seconds is in J Celephaïsian)

(13edo, L Illarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

See also

• Well-Tempered 13-Tone Clavier (collab project to create 13edo oneirotonic keyboard pieces in a variety of keys and modes)