Valinorsmic clan
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The valinorsmic clan of rank-3 temperaments tempers out the valinorsma, 176/175, which equates 48/35 with 15/11 (rather than 11/8, which is what the keenanisma 385/384 does).
For the rank-4 valinorsmic temperament, see Rank-4 temperament #Valinorsmic (176/175).
Valinor
Valinor is the 2.5.7.11-subgroup temperament defined by tempering out 176/175.
Subgroup: 2.5.7.11
Comma list: 176/175
Subgroup-val mapping: [⟨1 0 0 -4], ⟨0 1 0 2], ⟨0 0 1 1]]
- mapping generators: ~2, ~5, ~7
- WE: ~2 = 1199.3113 ¢, ~5/4 = 389.5404 ¢, ~7/4 = 971.5534 ¢
- error map: ⟨-0.689 +1.849 +1.350 -2.061]
- CWE: ~2 = 1200.0000 ¢, ~5/4 = 389.3935 ¢, ~7/4 = 971.5196 ¢
- error map: ⟨0.000 +3.080 +2.694 -1.011]
Optimal ET sequence: 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd
Badness (Sintel): 0.0793
Overview to extensions
Subgroup extensions
When extending to prime 13, two strong extensions are considerable: valimar and valarin, both considered below; these intersect in tridecimal didacus.
Full 11-limit extensions
The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds 121/120 as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. Varda, adds 896/891, slicing the first generator in two with a semi-octave period. Nickel adds 36/35 as well as 45/44. Ares adds 64/63. Minerva adds 99/98. Thrush adds 126/125. These slice the last generator in two. Guanyin adds 540/539, slicing the first generator in three. Manwe adds 1331/1323, slicing the last generator in three. Clio adds 81/80, slicing the first generator in four. Lono adds 5120/5103, slicing the last generator in six. Mandos adds 243/242. Shrusus adds 245/243. These slice the last generator in five. Ulmo adds 2200/2187, slicing the last generator in seven. Finally, draco adds 19683/19600, slicing the last generator in nine. Most of these have natural extensions to the 13-limit via tempering out both 351/350 and 352/351.
Discussed elsewhere are:
- Zeus (+121/120) → Biyatismic clan
- Varda (+896/891) → Diaschismic rank-3 family
- Nickel (+36/35 or 45/44) → Mint family
- Ares (+64/63 or 100/99) → Archytas family
- Minerva (+99/98) → Marvel family
- Thrush (+126/125) → Starling family
- Guanyin (+540/539) → Orwellismic family
- Clio (+81/80) → Didymus rank-3 family
- Lono (+5120/5103) → Hemifamity family
- Mandos (+243/242) → Rastmic rank-3 clan
- Shrusus (+245/243) → Sensamagic family
- Ulmo (+2200/2187) → Ragismic family
- Draco (+19683/19600) → Cataharry family
Considered below are manwe and augenic.
Valimar
This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4's, where two form 11/7, so that four form 16/13. This logic is supported by rank-2 temperaments such as tridecimal didacus and the 13-limit magus/amigo extensions, as well as winston and valentino. This is also natural considering S-expressions, noting that 176/175 (S8/S10) is nontrivially equivalent to (S11⋅S12)/(S14⋅S15), and this extension tempers out 1573/1568 (S11/S14), and 3584/3575 (S12/S15). It remains well-tuned in equal temperaments with optimal valinorsmic 5/4's (such as 37, 40, and 43).
Subgroup: 2.5.7.11.13
Comma list: 176/175, 1573/1568
Subgroup-val mapping: [⟨1 0 0 -4 13], ⟨0 1 0 2 -4], ⟨0 0 1 1 0]]
- mapping generators: ~2, ~5, ~7
- WE: ~2 = 1199.433 ¢, ~5/4 = 389.226 ¢, ~7/4 = 971.559 ¢
- error map: ⟨-0.567 +1.779 +1.600 -2.440 -0.267]
- CWE: ~2 = 1200.000 ¢, ~5/4 = 389.466 ¢, ~7/4 = 971.517 ¢
- error map: ⟨0.000 +3.152 +2.691 -0.869 +1.607]
Optimal ET sequence: 6, 9, 15f, 22f, 28, 31, 37, 231cd, 268cd
Badness (Sintel): 0.336
Valarin
This extension equates two, ideally slightly flattened, 8/7's to 13/10, which is also natural given valinorsmic's tuning tendency towards sharpening 7/4. This extension is supported by rank-2 temperaments including tridecimal didacus, tridecimal orwell, lupercalia (an extension of valentine), and llywelyn.
Subgroup: 2.5.7.11.13
Comma list: 176/175, 640/637
Subgroup-val mapping: [⟨1 0 0 -4 7], ⟨0 1 0 2 1], ⟨0 0 1 1 -2]]
- mapping generators: ~2, ~5, ~7
- WE: ~2 = 1199.167 ¢, ~5/4 = 389.474 ¢, ~7/4 = 972.185 ¢
- error map: ⟨-0.833 +1.495 +1.694 -1.851 +0.413]
- CWE: ~2 = 1200.000 ¢, ~5/4 = 389.189 ¢, ~7/4 = 972.829 ¢
- error map: ⟨0.000 +2.875 +4.003 -0.110 +3.003]
Optimal ET sequence: 6, 10e, 15, 16, 21, 31, 37, 206cdf, 243cdf, 280cdf, 317cddf, 354ccddff
Badness (Sintel): 0.227
Manwe
Subgroup: 2.3.5.7.11
Comma list: 176/175, 1331/1323
Mapping: [⟨1 0 0 12 8], ⟨0 1 0 3 3], ⟨0 0 1 -6 -4]]
- mapping generators: ~2, ~3, ~5
- WE: ~2 = 1199.3600 ¢, ~3/2 = 702.6947 ¢, ~5/4 = 389.3118 ¢
- error map: ⟨-0.640 +0.100 +1.718 +1.467 -2.401]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5916 ¢, ~5/4 = 389.3756 ¢
- error map: ⟨0.000 +0.637 +3.062 +2.696 -1.045]
Optimal ET sequence: 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd
Badness (Sintel): 0.816
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 1331/1323
Mapping: [⟨1 0 0 12 8 13], ⟨0 1 0 3 3 0], ⟨0 0 1 -6 -4 -4]]
Optimal tunings:
- WE: ~2 = 1199.4350 ¢, ~3/2 = 702.5061 ¢, ~5/4 = 389.2327 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.7030 ¢, ~5/4 = 389.4303 ¢
Optimal ET sequence: 31, 46, 65d, 77, 80, 111, 268cd
Badness (Sintel): 1.07
Augenic
- Not to be confused with Augene.
Named by Xenllium in 2026, augenic is closely related to augene. It tempers out the augmented comma but with a free generator for 7, and then extends it to the 11-limit through the identity 128/125 = (56/55)⋅(176/175).
Subgroup: 2.3.5.7.11
Comma list: 56/55, 128/125
Mapping: [⟨3 0 7 0 2], ⟨0 1 0 0 0], ⟨0 0 0 1 1]]
- mapping generators: ~5/4, ~3, ~7
- WE: ~5/4 = 398.9239 ¢, ~3/2 = 705.1447 ¢, ~7/4 = 969.1106 ¢
- error map: ⟨-3.228 -0.039 +6.153 -6.172 +9.184]
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 705.3488 ¢, ~7/4 = 968.4397 ¢
- error map: ⟨0.000 +3.394 +13.686 -0.386 +17.122]
Optimal ET sequence: 6, 9, 12, 15, 24, 27e, 51ce, 63cee *
Badness (Sintel): 0.735