https://xenreference.com/wiki/api.php?action=feedcontributions&feedformat=atom&user=IntharXenharmonic Reference - User contributions [en]2026-04-04T03:38:10ZUser contributionsMediaWiki 1.44.2https://xenreference.com/wiki/index.php?title=Tetracot_(temperament)&diff=5590Tetracot (temperament)2026-04-04T02:16:07Z<p>Inthar: /* Extensions */</p>
<hr />
<div>{{Infobox regtemp<br />
| Title = Tetracot<br />
| Subgroups = 2.3.5<br />
| Comma basis = [[20000/19683]] (2.3.5)<br />
| Edo join 1 = 27 | Edo join 2 = 34<br />
| Mapping = 1; 4 9<br />
| Generators = 10/9<br />
| Generators tuning = 176.1<br />
| Optimization method = CWE<br />
| MOS scales = [[6L&nbsp;1s]], [[7L&nbsp;6s]], [[7L&nbsp;13s]]<br />
| Odd limit 1 = 5 | Mistuning 1 = 3.07 | Complexity 1 = 13<br />
}}<br />
<br />
'''Tetracot''', [27 & 34] or [34 & 41], is a temperament that splits 3/2 into four flattened 10/9's.<br />
<br />
== Interval chain ==<br />
In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''.<br />
{| class="wikitable right-1 right-2"<br />
|-<br />
! #<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 176.3<br />
| 10/9<br />
|-<br />
| 2<br />
| 352.5<br />
| <br />
|-<br />
| 3<br />
| 528.8<br />
| 27/20<br />
|-<br />
| 4<br />
| 705.0<br />
| '''3/2'''<br />
|-<br />
| 5<br />
| 881.3<br />
| 5/3<br />
|-<br />
| 6<br />
| 1057.5<br />
| <br />
|-<br />
| 7<br />
| 33.8<br />
| 81/80<br />
|-<br />
| 8<br />
| 210.1<br />
| '''9/8'''<br />
|-<br />
| 9<br />
| 386.3<br />
| '''5/4'''<br />
|-<br />
| 10<br />
| 562.6<br />
|<br />
|-<br />
| 11<br />
| 738.8<br />
|<br />
|-<br />
| 12<br />
| 915.1<br />
| 27/16<br />
|-<br />
| 13<br />
| 1091.3<br />
| '''15/8'''<br />
|}<br />
<nowiki/>* in exact-5/2 tuning<br />
<br />
== Extensions ==<br />
Tetracot has a number of strong extensions, but most of them are problematic in some way. This is because the Tetracot generator is, optimally, approximately 31/28 — not easily interpretable as LCJI.<br />
* Prime 13 can be added by equating (10/9)^2 (the neutral third) with 16/13. Note that this favors a sharp 3/2 (optimally around 3.2c sharp) and a sharp 13/8 (optimally around 6.9c sharp).<br />
* Prime 11 is often added by equating 10/9 with 11/10 (thus placing 11/8 at +10 generators), but this is questionable because it produces either a very sharp 11/8 (as in 27edo and 34edo) or a flat 5/4 (as in 41edo and 48edo). An alternate extension (27p & 34), associated with 7-limit Wollemia, places 11/8 at -24 generators.<br />
* There isn't a canonical way to add prime 7. This is because 27edo and 41edo have good 7 approximations but 34edo does not. There are no less than 4 strong extensions to 2.3.5.7: Bunya (34d & 41), Monkey (34 & 41 or 41 & 48), Modus (27 & 34d), and Wollemia (27 & 34).<br />
** Monkey is notable because it's the extension tempering out [[5120/5103]], the aberschisma.<br />
** The weak extension [[Octacot (temperament)|Octacot]] (27 & 41) is more elegant; it splits the Tetracot generator into two semitones (about 88.1c) representing 21/20, thus equating three Octacot generators with 7/6 (and 11 of them with 7/4). Octacot can be extended to have prime 19 (at 17 generators) by equating 21/20 to 20/19 (equivalently, 10/9 to 21/19 or 27/20 to 19/14).<br />
<br />
== List of patent vals ==<br />
The following patent vals support 2.3.5 Tetracot. Contorted vals are not included.<br />
{| class="wikitable sortable"<br />
!Edo!!Generator tuning!!Fifth tuning<br />
|-<br />
||7||171.429||685.714<br />
|-<br />
||48||175.000||700.000<br />
|-<br />
||41||175.610||702.439<br />
|-<br />
||116||175.862||703.448<br />
|-<br />
||191||175.916||703.665<br />
|-<br />
||75||176.000||704.000<br />
|-<br />
||259||176.062||704.247<br />
|-<br />
||184||176.087||704.348<br />
|-<br />
||109||176.147||704.587<br />
|-<br />
||143||176.224||704.895<br />
|-<br />
||177||176.271||705.085<br />
|-<br />
||34||176.471||705.882<br />
|-<br />
||95||176.842||707.368<br />
|-<br />
||61||177.049||708.197<br />
|-<br />
||27||177.778||711.111<br />
|}<br />
{{Navbox regtemp}}<br />
{{Cat|temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Neutral_temperaments&diff=5589Neutral temperaments2026-04-04T02:12:32Z<p>Inthar: /* Hemififths */</p>
<hr />
<div>'''Neutral temperaments''' are any temperaments represented by the edo join 7 & 10, or any reasonable extension of such a temperament, such that the generator is a neutral third of some kind which splits 3/2 into two. They are a subset of and largely cover the ''dicot'' temperament archetype, and impose upon it the condition that the neutral third must be mapped to 2\7 and 3\10. The two most well-known neutral temperaments are the 2.3.11 (Rastmatic) and 2.3.5 (Dicot) versions.<br />
<br />
10edo is a contorted 5edo in 2.3.7, hence 7 & 10 in that subgroup represents monocot [[Archy]] temperament.<br />
<br />
== Notation ==<br />
Neutral temperaments may be notated with neutral chain-of-fifths notation. <br />
{| class="wikitable"<br />
|+<br />
!Note<br />
!24edo<br />
!Notation<br />
!2.3...<br />
!5 (Dicot)<br />
!11 (Rastmatic)<br />
!13 (Namo)<br />
!13-limit (no-fives)<br />
!13-limit<br />
|-<br />
|A<br />
|0<br />
|P1<br />
|1/1<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|At<br />
|50<br />
|sA1<br />
|<br />
|81/80<br />
|33/32<br />
|<br />
|<br />
|<br />
|-<br />
|Bb<br />
|100<br />
|m2<br />
|256/243<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Bd<br />
|150<br />
|n2<br />
|<br />
|10/9, 16/15<br />
|12/11<br />
|<br />
|<br />
|<br />
|-<br />
|B<br />
|200<br />
|M2<br />
|9/8<br />
|<br />
|<br />
|<br />
|8/7<br />
|11/10<br />
|-<br />
|C<br />
|300<br />
|m3<br />
|32/27<br />
|<br />
|<br />
|<br />
|7/6<br />
|<br />
|-<br />
|Ct<br />
|350<br />
|n3<br />
|<br />
|5/4, 6/5<br />
|11/9<br />
|16/13<br />
|<br />
|<br />
|-<br />
|C#<br />
|400<br />
|M3<br />
|81/64<br />
|<br />
|<br />
|<br />
|9/7<br />
|<br />
|-<br />
|Dd<br />
|450<br />
|sd4<br />
|<br />
|<br />
|<br />
|<br />
|14/11<br />
|<br />
|-<br />
|D<br />
|500<br />
|P4<br />
|4/3<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Dt<br />
|550<br />
|sA4<br />
|<br />
|27/20<br />
|11/8<br />
|18/13<br />
|<br />
|10/7<br />
|-<br />
|Ed<br />
|650<br />
|sd5<br />
|<br />
|40/27<br />
|16/11<br />
|13/9<br />
|<br />
|7/5<br />
|-<br />
|E<br />
|700<br />
|P5<br />
|3/2<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Et<br />
|750<br />
|sA5<br />
|<br />
|<br />
|<br />
|<br />
|11/7<br />
|<br />
|-<br />
|F<br />
|800<br />
|m6<br />
|128/81<br />
|<br />
|<br />
|<br />
|14/9<br />
|<br />
|-<br />
|Ft<br />
|850<br />
|n6<br />
|<br />
|5/3, 8/5<br />
|18/11<br />
|13/8<br />
|<br />
|<br />
|-<br />
|F#<br />
|900<br />
|M6<br />
|27/16<br />
|<br />
|<br />
|<br />
|12/7<br />
|<br />
|-<br />
|G<br />
|1000<br />
|m7<br />
|16/9<br />
|<br />
|<br />
|<br />
|7/4<br />
|20/11<br />
|-<br />
|Gt<br />
|1050<br />
|n7<br />
|<br />
|15/8, 9/5<br />
|11/6<br />
|<br />
|<br />
|<br />
|-<br />
|G#<br />
|1100<br />
|M7<br />
|243/128<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Ad<br />
|1150<br />
|sd8<br />
|<br />
|160/81<br />
|64/33<br />
|<br />
|<br />
|<br />
|-<br />
|A<br />
|1200<br />
|P8<br />
|2/1<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|}<br />
These intervals may additionally be arranged on a chart which explains their mappings to 7edo and 10edo:<br />
{| class="wikitable"<br />
! !! 0\7 !! 1\7 !! 2\7 !! 3\7 !! 4\7 !! 5\7 !! 6\7 !! 7\7<br />
|-<br />
! scope="row" | 0\10 <br />
| '''P1''' || m2|| d3|| || || || ||<br />
|-<br />
! scope="row" | 1\10 <br />
| sA1|| '''n2''' || sd3|| || || || ||<br />
|-<br />
! scope="row" | 2\10 <br />
| A1|| M2 || m3 || d4|| || || ||<br />
|-<br />
! scope="row" | 3\10 <br />
| || sA2|| '''n3''' || sd4 || || || ||<br />
|-<br />
! scope="row" | 4\10 <br />
| || A2|| M3 || '''P4''' || d5|| d6|| ||<br />
|-<br />
! scope="row" | 5\10 <br />
| || || sA3|| sA4 || sd5 || sd6|| ||<br />
|-<br />
! scope="row" | 6\10 <br />
| || || A3|| A4|| '''P5''' || m6 || d7||<br />
|-<br />
! scope="row" | 7\10 <br />
| || || || || sA5 || '''n6''' || sd7||<br />
|-<br />
! scope="row" | 8\10 <br />
| || || || || A5|| M6 || m7 ||d8<br />
|-<br />
! scope="row" | 9\10 <br />
| || || || || || sA6|| '''n7''' ||sd8<br />
|-<br />
! scope="row" | 10\10 <br />
| || || || || || A6|| M7|| '''P8'''<br />
|}<br />
<br />
== Rastmatic ==<br />
Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22.<br />
<br />
=== Etymology ===<br />
Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''Rast'' which utilizes a scale with several neutral intervals.<br />
<br />
== Hemififths ==<br />
Hemififths, 41 & 58, is the neutral temperament in the 2.3.5.7 subgroup, which tempers out 2401/2400 = S49 equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament.<br />
<br />
== Dicot ==<br />
Dicot<sup>[a]</sup>, not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out [[25/24]], the Dicot comma. The provided [[edo join]] also tempers out [[45/44]] and [[64/63]] in the 11-limit, representing the extension '''Dichotic''' and also tempering out [[55/54]]. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.<br />
<br />
Dicot makes 4:5:6 equidistant, suggesting the simplified structure of [[tertian]] harmony, the same way [[Semaphore]] does for [[chthonic harmony]]. As a result, the temperament archetype [[Neutral third scales|dicot]] is named after it.<br />
<br />
=== Etymology ===<br />
Dicot originates from the term "dicot" in botany, referring to plants with two embryonic leaves, perhaps by analogy with 3/2 being split into two generators. The name ''Dicot'' would also inspire [[Tetracot]], [[Alphatricot]], and by extension the [[ploidacot]] temperament archetype naming system as a whole.<br />
<br />
=== Tuning considerations ===<br />
A perfect ~351c tuning of the generator, while useful for understanding tertian harmony and suggested by some temperament tuning optimization systems, does not reasonably approximate either 5/4 or 6/5. The optimal tunings of Dicot are roughly bimodal, with ~360c (around 10edo) and ~343c (around 7edo) both being better tunings.<br />
<br />
== Namo ==<br />
''Namo'', ''Intertridecimal'', or ''Harmoneutral'' is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth.<br />
<br />
It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c.<br />
<br />
== Patent vals ==<br />
<br />
=== List of patent vals ===<br />
{| class="wikitable"<br />
|+<br />
!EDO<br />
!Mappings supported<br />
!Generator tuning<br />
!3/2 tuning<br />
|-<br />
|10<br />
|5, 13, 11<br />
|360.0c<br />
|720.0c<br />
|-<br />
|37<br />
|13<br />
|356.8c<br />
|713.5c<br />
|-<br />
|27<br />
|13<br />
|355.6c<br />
|711.1c<br />
|-<br />
|71<br />
|13<br />
|354.9c<br />
|709.9c<br />
|-<br />
|44<br />
|13<br />
|354.5c<br />
|709.1c<br />
|-<br />
|61<br />
|13<br />
|354.1c<br />
|708.2c<br />
|-<br />
|78<br />
|13<br />
|353.8c<br />
|707.7c<br />
|-<br />
|95<br />
|13<br />
|353.7c<br />
|707.4c<br />
|-<br />
|17<br />
|5, 13, 11<br />
|352.9c<br />
|705.9c<br />
|-<br />
|75<br />
|13<br />
|352.0c<br />
|704.0c<br />
|-<br />
|58<br />
|13, 11<br />
|351.7c<br />
|703.4c<br />
|-<br />
|41<br />
|13, 11<br />
|351.2c<br />
|702.4c<br />
|-<br />
|147<br />
|11<br />
|351.0c<br />
|702.0c<br />
|-<br />
|106<br />
|11<br />
|350.9c<br />
|701.9c<br />
|-<br />
|171<br />
|11<br />
|350.9c<br />
|701.8c<br />
|-<br />
|65<br />
|13, 11<br />
|350.8c<br />
|701.5c<br />
|-<br />
|219<br />
|11<br />
|350.7c<br />
|701.4c<br />
|-<br />
|154<br />
|11<br />
|350.6c<br />
|701.3c<br />
|-<br />
|243<br />
|11<br />
|350.62c<br />
|701.23c<br />
|-<br />
|332<br />
|11<br />
|350.60c<br />
|701.20c<br />
|-<br />
|89<br />
|11<br />
|350.56c<br />
|701.12c<br />
|-<br />
|380<br />
|11<br />
|350.53c<br />
|701.05c<br />
|-<br />
|291<br />
|11<br />
|350.52c<br />
|701.03c<br />
|-<br />
|202<br />
|11<br />
|350.50c<br />
|700.99c<br />
|-<br />
|517<br />
|11<br />
|350.48c<br />
|700.97c<br />
|-<br />
|315<br />
|11<br />
|350.48c<br />
|700.95c<br />
|-<br />
|428<br />
|11<br />
|350.47c<br />
|700.93c<br />
|-<br />
|541<br />
|11<br />
|350.46c<br />
|700.92c<br />
|-<br />
|113<br />
|11<br />
|350.44c<br />
|700.88c<br />
|-<br />
|476<br />
|11<br />
|350.42c<br />
|700.84c<br />
|-<br />
|363<br />
|11<br />
|350.41c<br />
|700.83c<br />
|-<br />
|250<br />
|11<br />
|350.40c<br />
|700.80c<br />
|-<br />
|387<br />
|11<br />
|350.39c<br />
|700.78c<br />
|-<br />
|137<br />
|11<br />
|350.36c<br />
|700.73c<br />
|-<br />
|435<br />
|11<br />
|350.34c<br />
|700.69c<br />
|-<br />
|298<br />
|11<br />
|350.34c<br />
|700.67c<br />
|-<br />
|459<br />
|11<br />
|350.33c<br />
|700.65c<br />
|-<br />
|161<br />
|11<br />
|350.31c<br />
|700.62c<br />
|-<br />
|346<br />
|11<br />
|350.29c<br />
|700.58c<br />
|-<br />
|185<br />
|11<br />
|350.27c<br />
|700.54c<br />
|-<br />
|394<br />
|11<br />
|350.25c<br />
|700.51c<br />
|-<br />
|209<br />
|11<br />
|350.24c<br />
|700.48c<br />
|-<br />
|233<br />
|11<br />
|350.21c<br />
|700.43c<br />
|-<br />
|257<br />
|11<br />
|350.19c<br />
|700.39c<br />
|-<br />
|281<br />
|11<br />
|350.18c<br />
|700.36c<br />
|-<br />
|305<br />
|11<br />
|350.16c<br />
|700.33c<br />
|-<br />
|329<br />
|11<br />
|350.15c<br />
|700.30c<br />
|-<br />
|353<br />
|11<br />
|350.14c<br />
|700.28c<br />
|-<br />
|24<br />
|13, 11<br />
|350.00c<br />
|700.00c<br />
|-<br />
|247<br />
|11<br />
|349.80c<br />
|699.60c<br />
|-<br />
|223<br />
|11<br />
|349.78c<br />
|699.55c<br />
|-<br />
|199<br />
|11<br />
|349.7c<br />
|699.5c<br />
|-<br />
|175<br />
|11<br />
|349.7c<br />
|699.4c<br />
|-<br />
|151<br />
|11<br />
|349.7c<br />
|699.3c<br />
|-<br />
|127<br />
|11<br />
|349.6c<br />
|699.2c<br />
|-<br />
|103<br />
|11<br />
|349.5c<br />
|699.0c<br />
|-<br />
|79<br />
|11<br />
|349.4c<br />
|698.7c<br />
|-<br />
|55<br />
|13, 11<br />
|349.1c<br />
|698.2c<br />
|-<br />
|31<br />
|13, 11<br />
|348.4c<br />
|696.8c<br />
|-<br />
|38<br />
|13, 11<br />
|347.4c<br />
|694.7c<br />
|-<br />
|45<br />
|13<br />
|346.7c<br />
|693.3c<br />
|-<br />
|7<br />
|5, 13, 11<br />
|342.9c<br />
|685.7c<br />
|}<br />
<br />
== Footnotes ==<br />
[a] The name ''Interpental'' has been proposed, however it currently is used by 43 & 53, a weak extension of [[Buzzard]].<br />
<br />
{{Navbox regtemp}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5588Hemifamity2026-04-04T01:59:53Z<p>Inthar: /* Structural theory */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', [[41edo|41]] & [[46edo|46]] & [[53edo|53]], is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103] = 2.3.7/5[12 & 29].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.<br />
<br />
== Structural theory ==<br />
=== Notation ===<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
=== Intervals ===<br />
<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]] (Aberschismic + Slendric)<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]] (Aberschismic + Tetracot)<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]] (Aberschismic + Schismic)<br />
* 2.3.5.7[41 & 58]: [[Hemififths]] (Aberschismic + 2401/2400)<br />
* 2.3.5.7[46 & 53]: [[Amity]] (Aberschismic + 4375/4374)<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]] (Aberschismic + Diaschismic)<br />
* 2.3.5.7[53 & 58]: [[Buzzard]] (Aberschismic + 2.3.7 Buzzard)<br />
* 2.3.5.7[87 & 99]: [[Misty]] (Aberschismic + Didacus)<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which equates 128/105 to 39/32); with edo join 41 & 46 & 53, this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]]. Inthar considers this the canonical extension of Hemifamity that adds prime 13.<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Tetracot_(temperament)&diff=5584Tetracot (temperament)2026-04-04T01:29:20Z<p>Inthar: /* Extensions */</p>
<hr />
<div>{{Infobox regtemp<br />
| Title = Tetracot<br />
| Subgroups = 2.3.5<br />
| Comma basis = [[20000/19683]] (2.3.5)<br />
| Edo join 1 = 27 | Edo join 2 = 34<br />
| Mapping = 1; 4 9<br />
| Generators = 10/9<br />
| Generators tuning = 176.1<br />
| Optimization method = CWE<br />
| MOS scales = [[6L&nbsp;1s]], [[7L&nbsp;6s]], [[7L&nbsp;13s]]<br />
| Odd limit 1 = 5 | Mistuning 1 = 3.07 | Complexity 1 = 13<br />
}}<br />
<br />
'''Tetracot''', [27 & 34] or [34 & 41], is a temperament that splits 3/2 into four flattened 10/9's.<br />
<br />
== Interval chain ==<br />
In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''.<br />
{| class="wikitable right-1 right-2"<br />
|-<br />
! #<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 176.3<br />
| 10/9<br />
|-<br />
| 2<br />
| 352.5<br />
| <br />
|-<br />
| 3<br />
| 528.8<br />
| 27/20<br />
|-<br />
| 4<br />
| 705.0<br />
| '''3/2'''<br />
|-<br />
| 5<br />
| 881.3<br />
| 5/3<br />
|-<br />
| 6<br />
| 1057.5<br />
| <br />
|-<br />
| 7<br />
| 33.8<br />
| 81/80<br />
|-<br />
| 8<br />
| 210.1<br />
| '''9/8'''<br />
|-<br />
| 9<br />
| 386.3<br />
| '''5/4'''<br />
|-<br />
| 10<br />
| 562.6<br />
|<br />
|-<br />
| 11<br />
| 738.8<br />
|<br />
|-<br />
| 12<br />
| 915.1<br />
| 27/16<br />
|-<br />
| 13<br />
| 1091.3<br />
| '''15/8'''<br />
|}<br />
<nowiki/>* in exact-5/2 tuning<br />
<br />
== Extensions ==<br />
Tetracot has a number of strong extensions, but most of them are problematic in some way. This is because the Tetracot generator is, optimally, approximately 31/28 — not easily interpretable as LCJI.<br />
* Prime 13 can be added by equating (10/9)^2 (the neutral third) with 16/13. Note that this favors a sharp 3/2 (optimally around 3.2c sharp) and a sharp 13/8 (optimally around 6.9c sharp).<br />
* Prime 11 is often added by equating 10/9 with 11/10 (thus placing 11/8 at +10 generators), but this is questionable because it produces either a very sharp 11/8 (as in 27edo and 34edo) or a flat 5/4 (as in 41edo and 48edo). An alternate extension (27p & 34), associated with 7-limit Wollemia, places 11/8 at -24 generators.<br />
* There isn't a canonical way to add prime 7. This is because 27edo and 41edo have good 7 approximations but 34edo does not. There are no less than 4 strong extensions to 2.3.5.7: Bunya (34d & 41), Monkey (34 & 41 or 41 & 48), Modus (27 & 34d), and Wollemia (27 & 34).<br />
** Monkey is notable because it's the extension tempering out [[5120/5103]], the aberschisma.<br />
** The weak extension [[Octacot]] (27 & 41) is more elegant; it splits the Tetracot generator into two semitones (about 88.1c) representing 21/20, thus equating three Octacot generators with 7/6 (and 11 of them with 7/4). Octacot can be extended to have prime 19 (at 17 generators) by equating 21/20 to 20/19 (equivalently, 10/9 to 21/19 or 27/20 to 19/14).<br />
<br />
== List of patent vals ==<br />
The following patent vals support 2.3.5 Tetracot. Contorted vals are not included.<br />
{| class="wikitable sortable"<br />
!Edo!!Generator tuning!!Fifth tuning<br />
|-<br />
||7||171.429||685.714<br />
|-<br />
||48||175.000||700.000<br />
|-<br />
||41||175.610||702.439<br />
|-<br />
||116||175.862||703.448<br />
|-<br />
||191||175.916||703.665<br />
|-<br />
||75||176.000||704.000<br />
|-<br />
||259||176.062||704.247<br />
|-<br />
||184||176.087||704.348<br />
|-<br />
||109||176.147||704.587<br />
|-<br />
||143||176.224||704.895<br />
|-<br />
||177||176.271||705.085<br />
|-<br />
||34||176.471||705.882<br />
|-<br />
||95||176.842||707.368<br />
|-<br />
||61||177.049||708.197<br />
|-<br />
||27||177.778||711.111<br />
|}<br />
{{Navbox regtemp}}<br />
{{Cat|temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5583Hemifamity2026-04-04T01:22:55Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', [[41edo|41]] & [[46edo|46]] & [[53edo|53]], is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103] = 2.3.7/5[12 & 29].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.<br />
<br />
== Structural theory ==<br />
=== Notation ===<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]] (Aberschismic + Slendric)<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]] (Aberschismic + Tetracot)<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]] (Aberschismic + Schismic)<br />
* 2.3.5.7[41 & 58]: [[Hemififths]] (Aberschismic + 2401/2400)<br />
* 2.3.5.7[46 & 53]: [[Amity]] (Aberschismic + 4375/4374)<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]] (Aberschismic + Diaschismic)<br />
* 2.3.5.7[53 & 58]: [[Buzzard]] (Aberschismic + 2.3.7 Buzzard)<br />
* 2.3.5.7[87 & 99]: [[Misty]] (Aberschismic + Didacus)<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which equates 128/105 to 39/32); with edo join 41 & 46 & 53, this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]]. Inthar considers this the canonical extension of Hemifamity that adds prime 13.<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=EDO&diff=5581EDO2026-04-04T00:59:31Z<p>Inthar: /* List of edos */</p>
<hr />
<div>An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps. It is a type of equal temperament.<br />
<br />
The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.<br />
<br />
An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties, as will one with the same number of notes as the MOS has L steps. These two edos form the boundaries of how the MOS can be tuned.<br />
<br />
The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').<br />
<br />
== Uses ==<br />
<br />
Edos are the most common type of tuning system in contemporary xenharmony. Unlike other types such as rank-2 temperaments and just intonation scales, equal temperaments allow for free modulation and transposition due to their uniform step size. That is, every n-step interval is the same as every other n-step interval. This comes at the expense of less freedom in approximating target intervals. It also encourages a less structured approach to composition where pitch shifts and interval quality changes can happen without much deeper meaning.<br />
<br />
== List of edos ==<br />
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''<br />
{| class="wikitable"<br />
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].<br />
|-<br />
!Edo<br />
!Description<br />
!First twelve steps (¢) <br />
!Fifth (¢)<br />
!Edostep interpretation<br />
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups<br />
|-<br />
| rowspan="2" |1<br />
| rowspan="2" |Equivalent to the 2-limit.<br />
| rowspan="2" |1200<br />
| rowspan="2" |1200<br />
| rowspan="2" |2/1<br />
|2<br />
|-<br />
|2<br />
|-<br />
| rowspan="2" |2<br />
| rowspan="2" |Just a 12edo tritone.<br />
| rowspan="2" |600, 1200<br />
| rowspan="2" |600<br />
| rowspan="2" |''none available in basic subgroup''<br />
|2<br />
|-<br />
|2.<3.>>5.>>7.<17<br />
|-<br />
| rowspan="2" |3<br />
| rowspan="2" |An augmented triad.<br />
| rowspan="2" |400, 800, 1200<br />
| rowspan="2" |800<br />
| rowspan="2" |5/4<br />
|2.5<br />
|-<br />
|2.>3.5.>19?<br />
|-<br />
| rowspan="2" |4<br />
| rowspan="2" |A diminished tetrad.<br />
| rowspan="2" |300, 600, 900, 1200<br />
| rowspan="2" |600<br />
| rowspan="2" |19/16<br />
|2.19<br />
|-<br />
|2.<3.<5.<7.<17<br />
|-<br />
| rowspan="2" class="thl" |[[5edo|5]]<br />
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.<br />
| rowspan="2" |240, 480, 720, 960, 1200<br />
| rowspan="2" |720<br />
| rowspan="2" |9/8, 8/7, 7/6<br />
|2.3.7<br />
|-<br />
|2.>>3.<7<br />
|-<br />
| rowspan="2" |6<br />
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo. Good approximation of 2.5.7 for its size.<br />
| rowspan="2" |200, 400, 600, 800, 1000, 1200<br />
| rowspan="2" |600, 800<br />
| rowspan="2" |9/8, 10/9, 28/25, 8/7<br />
|2.9.5<br />
|-<br />
|2.9.5.>7<br />
|-<br />
| rowspan="2" class="thl" |[[7edo|7]]<br />
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.<br />
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200<br />
| rowspan="2" |685.7<br />
| rowspan="2" |9/8, 10/9, 16/15<br />
|2.3.5.11.13<br />
|-<br />
|2.<3.<<5.>13<br />
|-<br />
| rowspan="2" |[[8edo|8]]<br />
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.<br />
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200<br />
| rowspan="2" |750<br />
| rowspan="2" |''none available in basic subgroup''<br />
|2.19<br />
|-<br />
|2.x3.x5.x7.x11.x13<br />
|-<br />
| rowspan="2" |[[9edo|9]]<br />
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group). <br />
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200<br />
| rowspan="2" |666.7<br />
| rowspan="2" |9/8, 16/15, 25/24<br />
|2.5.11<br />
|-<br />
|2.<<3.>5.<<7<br />
|-<br />
| rowspan="2" |[[10edo|10]]<br />
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].<br />
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200<br />
| rowspan="2" |720<br />
| rowspan="2" |16/15, 10/9, 81/80, 36/35<br />
|2.3.5.7.13<br />
|-<br />
|2.>>3.<7.13<br />
|-<br />
| rowspan="2" |[[11edo|11]]<br />
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].<br />
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9<br />
| rowspan="2" |654.5, 763.6<br />
| rowspan="2" |128/119, 17/16, 18/17<br />
|2.9.7.11<br />
|-<br />
|2.x3.x5.7.11<br />
|-<br />
| rowspan="2" class="thl" |[[12edo|12]]<br />
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.<br />
| rowspan="2" |{{First 12 edo intervals|edo=12}}<br />
| rowspan="2" |700<br />
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24<br />
|2.3.5.17.19<br />
|-<br />
|2.3.>5.>>7.17.19<br />
|-<br />
|[[13edo|13]]<br />
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].<br />
|{{First 12 edo intervals|edo=13}}<br />
|646.2, 738.5<br />
|17/16, 18/17, 19/18, 20/19<br />
|2.5.11.13.17.19.23<br />
|-<br />
|[[14edo|14]]<br />
|Basic [[semiquartal]].<br />
|{{First 12 edo intervals|edo=14}} <br />
|685.7<br />
|28/27, 21/20, 15/14<br />
|2.3.7.13<br />
|-<br />
| class="thl" |[[15edo|15]]<br />
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting Porcupine temperament and dubitably the 11-limit.<br />
|{{First 12 edo intervals|edo=15}}<br />
|720<br />
|81/80, 25/24, 16/15, 33/32, 36/35<br />
|2.3.5.7.11.23<br />
|-<br />
|[[16edo|16]]<br />
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].<br />
|{{First 12 edo intervals|edo=16}}<br />
|675, 750<br />
|20/19, 133/128, 26/25<br />
|2.5.7.13.19<br />
|-<br />
| class="thl" |[[17edo|17]]<br />
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.<br />
|{{First 12 edo intervals|edo=17}}<br />
|705.9<br />
|256/243, 24/23, 27/26, 33/32<br />
|2.3.13.23<br />
|-<br />
| rowspan="2" |[[18edo|18]]<br />
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric. <br />
| rowspan="2" |{{First 12 edo intervals|edo=18}}<br />
| rowspan="2" |666.6, 733.3<br />
| rowspan="2" |<br />
|2.9.5.21.13<br />
|-<br />
|2.xx3.>5.>>7.<11.<13<br />
|-<br />
| class="thl" |[[19edo|19]]<br />
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.<br />
|{{First 12 edo intervals|edo=19}}<br />
|694.7<br />
|25/24, [[diaschisma]], 36/35, 28/27<br />
|2.3.5.23<br />
|-<br />
| |20<br />
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.<br />
|{{First 12 edo intervals|edo=20}}<br />
|660, 720<br />
|<br />
|2.7.11.13.19<br />
|-<br />
| rowspan="2" |[[21edo|21]]<br />
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.<br />
| rowspan="2" |{{First 12 edo intervals|edo=21}}<br />
| rowspan="2" |685.7, 742.9<br />
| rowspan="2" |<br />
|2.3.5.7.23<br />
|-<br />
|2.x>3.x<5.7.x<11.x<13.23<br />
|-<br />
| rowspan="2" class="thl" |[[22edo|22]]<br />
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.<br />
| rowspan="2" |{{First 12 edo intervals|edo=22}}<br />
| rowspan="2" |709.1<br />
| rowspan="2" |<br />
|2.3.5.7.11.17<br />
|-<br />
|2.>3.<5.>>7.<11<br />
|-<br />
|23<br />
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.<br />
|{{First 12 edo intervals|edo=23}}<br />
|678.3<br />
|<br />
|2.x3.x5.x7.x11.13.17.23<br />
|-<br />
| div class="thl"|[[24edo|24]]<br />
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.<br />
|{{First 12 edo intervals|edo=24}}<br />
|700<br />
|<br />
|2.3.11.13.17.19<br />
|-<br />
|25<br />
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.<br />
|{{First 12 edo intervals|edo=25}}<br />
|720<br />
|<br />
|2.5.7.19<br />
|-<br />
|[[26edo|26]]<br />
|A simple tuning of Flattone. Has an absurdly accurate 7/4.<br />
|{{First 12 edo intervals|edo=26}}<br />
|692.7<br />
|<br />
|2.3.7.11.13<br />
|-<br />
|27<br />
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.<br />
|{{First 12 edo intervals|edo=27}}<br />
|711.1<br />
|<br />
|2.3.5.7.13.23<br />
|-<br />
|28<br />
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.<br />
|{{First 12 edo intervals|edo=28}}<br />
|685.7, 728.6<br />
|<br />
|2.3.5.7.11<br />
|-<br />
|[[29edo|29]]<br />
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.<br />
|{{First 12 edo intervals|edo=29}}<br />
|703.4<br />
|<br />
|2.3.7/5.11/5.13/5.19.23<br />
|-<br />
|30<br />
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.<br />
|{{First 12 edo intervals|edo=30}}<br />
|680, 720<br />
|<br />
|2.x3.x5.7.11.13<br />
|-<br />
| class="thl" |[[31edo|31]]<br />
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.<br />
|{{First 12 edo intervals|edo=31}}<br />
|696.8<br />
|<br />
|2.3.5.7.11.23<br />
|-<br />
|32<br />
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.<br />
|{{First 12 edo intervals|edo=32}}<br />
|712.5<br />
|<br />
|2.3.7.11.17.19.23<br />
|-<br />
|33<br />
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports Semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.<br />
|{{First 12 edo intervals|edo=33}}<br />
|690.9<br />
|<br />
|2.3.11.13.17.19.23<br />
|-<br />
| rowspan="2" class="thl"|[[34edo|34]]<br />
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.<br />
It is the double of 17edo, which it takes its circle of fifths from.<br />
| rowspan="2" |{{First 12 edo intervals|edo=34}}<br />
| rowspan="2" |705.9<br />
| rowspan="2" |<br />
|2.3.5.13.23<br />
|-<br />
|2.3.5.x7.13.x19.23<br />
|-<br />
|35<br />
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.<br />
|{{First 12 edo intervals|edo=35}}<br />
|685.7, 720.0<br />
|<br />
|2.5.7.11.17<br />
|-<br />
|36<br />
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.<br />
|{{First 12 edo intervals|edo=36}}<br />
|700<br />
|<br />
|2.3.7.13.17.19.23<br />
|-<br />
| rowspan="2" class="thl" |[[37edo|37]]<br />
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an Archy 3, as in Porcupine.<br />
| rowspan="2" |{{First 12 edo intervals|edo=37}}<br />
| rowspan="2" |681.1, 713.5<br />
| rowspan="2" |<br />
|2.9.5.7.11.13.17.19<br />
|-<br />
|2.<x3.5.7.11.13.17.19<br />
|-<br />
|38<br />
|19edo with neutrals. Functions as a tuning of Mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.<br />
|{{First 12 edo intervals|edo=38}}<br />
|694.7<br />
|<br />
|2.3.5.7.13.17.23<br />
|-<br />
|39<br />
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.<br />
|{{First 12 edo intervals|39|edo=39}}<br />
|707.7<br />
|<br />
|2.3.11<br />
|-<br />
|[[40edo|40]]<br />
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.<br />
|{{First 12 edo intervals|40|edo=40}}<br />
|690<br />
|<br />
|<br />
|-<br />
| class="thl" |[[41edo|41]]<br />
|The first reasonably accurate [[Hemifamity]] edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}<br />
|{{First 12 edo intervals|edo=41}}<br />
|702.4<br />
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44<br />
|2.3.5.7.11.13.19<br />
|-<br />
|42<br />
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma Archy.<br />
|{{First 12 edo intervals|edo=42}}<br />
|685.7, 714.3<br />
|<br />
|2.7.11.17.23<br />
|-<br />
|43<br />
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|Huygens]]". Like all Meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.<br />
|{{First 12 edo intervals|edo=43}}<br />
|697.7<br />
|<br />
|2.3.5.7.11.13.17<br />
|-<br />
|44<br />
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo). <br />
|{{First 12 edo intervals|edo=44}}<br />
|709.1<br />
|<br />
|2.3.5.11.13.17.19.23<br />
|-<br />
|45<br />
|A nearly optimal tuning of Flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.<br />
|{{First 12 edo intervals|edo=45}}<br />
|693.3, 720<br />
|<br />
|2.3.7.11.17.19<br />
|-<br />
| class="thl"|46<br />
|The second reasonably accurate [[Hemifamity]] edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}<br />
|{{First 12 edo intervals|edo=46}}<br />
|704.3<br />
|<br />
|2.3.5.7.11.13.17.23<br />
|-<br />
|47<br />
|The first edo with two distinct mosdiatonic scales. Supports Magic and Archy with its sharp fifth and Deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of Schismic analogue of Didacus.<br />
|{{First 12 edo intervals|edo=47}}<br />
|689.4, 714.9<br />
|<br />
|2.5.7.13.17<br />
|-<br />
|48<br />
|Four times 12edo, associated with [[Buzzard]] temperament.<br />
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300<br />
|700<br />
|<br />
|2.3.7.11.17.19.23<br />
|-<br />
|49<br />
|A nearly optimal tuning of Archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 Meantone (or, more conventionally, Didacus).<br />
|{{First 12 edo intervals|edo=49}}<br />
|710.2<br />
|<br />
|2.5.17.19<br />
|-<br />
|50<br />
|Approaches golden Meantone, and serves as a definitive tuning of Meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.<br />
|{{First 12 edo intervals|edo=50}}<br />
|696<br />
|<br />
|2.3.5.11.13.23<br />
|-<br />
|51<br />
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.<br />
|{{First 12 edo intervals|edo=51}}<br />
|705.9<br />
|<br />
|2.3.7.13<br />
|-<br />
|52<br />
|Doubles 26edo, adding a sharp Archy fifth and a more accurate 5/4 which support Porcupine temperament.<br />
|{{First 12 edo intervals|edo=52}}<br />
|692.3, 715.4<br />
|<br />
|2.5.7.11.19.23<br />
|-<br />
|class="thl"|[[53edo|53]]<br />
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).<br />
|{{First 12 edo intervals|edo=53}}<br />
|701.9<br />
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90<br />
|2.3.5.7.13.19<br />
|-<br />
|54<br />
|Double 27edo, and the sharper end of the Pajara tuning range. Can alternatively be used as a very flat Deeptone system or combining the fifths as a straddle-fifth system.<br />
|<br />
|688.9, 711.1<br />
|<br />
|2.11.13.17.23<br />
|-<br />
|55<br />
|A very sharp Meantone tuning, which is so sharp that it does not even support Septimal Meantone, and is best interpreted as Mohajira as it pertains to Meantone extensions.<br />
|<br />
|698.2<br />
|<br />
|2.3.5.11.17.23<br />
|-<br />
|56<br />
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|57<br />
|Has 19edo's 5-limit combined with better interpretations of higher limits.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|58<br />
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals. <br />
|{{First 12 edo intervals|edo=58}}<br />
|703.4<br />
|<br />
|2.3.7.17<br />
|-<br />
|59<br />
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports Porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|60<br />
|5 sets of 12edo, supporting Magic temperament and having 10edo's 7 and 13, also supporting 7-limit Compton temperament and many structures associated with 10edo and 15edo with their respective mappings.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|61<br />
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|62<br />
|Doubled 31edo, which shares its mappings through the 11-limit.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|63<br />
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.<br />
|{{First 12 edo intervals|edo=63}}<br />
|704.8<br />
|<br />
|2.3.5.7.11.13.23<br />
|-<br />
|64<br />
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of Flattone with its flat 3, 5, and 7.<br />
|{{First 12 edo intervals|edo=64}}<br />
|693.8, 712.5<br />
|<br />
|2.13.19<br />
|-<br />
|65<br />
|A non-Garibaldi Schismic system (in fact, it supports Sensi), and a straddle-7 and -13 system.<br />
|{{First 12 edo intervals|edo=65}}<br />
|701.5<br />
|<br />
|2.3.5.11.17.19<br />
|-<br />
|66<br />
|Tripled 22edo, with an improved approximation to 7 that supports Slendric. <br />
|{{First 12 edo intervals|edo=66}}<br />
|709.1, 690.9<br />
|<br />
|2.5.7.11.13.17<br />
|-<br />
|67<br />
|Approximate 1/6-comma Meantone and Slendric edo, which also supports [[Orgone]].<br />
|{{First 12 edo intervals|edo=67}}<br />
|698.5<br />
|<br />
|2.3.7.11.13.17.23<br />
|-<br />
|68<br />
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting Sensamagic. Additionally, there is a second diatonic fifth.<br />
|{{First 12 edo intervals|edo=68}}<br />
|705<br />
|<br />
|2.3.5.7.11.17.19<br />
|-<br />
|69<br />
|A nice tuning that is approximately 2/7-comma Meantone, somewhat between standard Septimal Meantone and 19edo. As a result, it is a Mohajira system (setting 7/4 to the semiflat minor seventh) but not a Septimal Meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports Archy with the sharp fifth. As a dual-fifth system, it is neogothic.<br />
|{{First 12 edo intervals|edo=69}}<br />
|695.7, 713<br />
|<br />
|2.5.7.11.13.17.19.23<br />
|-<br />
|70<br />
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a [[Hemifamity]] system, as is typical with tunings with sharpened fifths.<br />
|{{First 12 edo intervals|edo=70}}<br />
|702.9<br />
|<br />
|2.3.11.13.17<br />
|-<br />
|71<br />
|A dual-fifth system. The sharp fifth is within the Superpyth tuning range (and produces the same mapping for 5 as Superpyth), despite not supporting Archy. The flat fifth, analogously, produces Flattone's mapping for 7 and is well-tuned for Flattone, but does not support Flattone.<br />
|{{First 12 edo intervals|edo=71}}<br />
|693. 709.9<br />
|<br />
|2.5.7.13.17.23<br />
|-<br />
|72<br />
|A multiple of 12edo and a very good Miracle and Compton system. It is also the first multiple of 12 to have a second MOS diatonic.<br />
|{{First 12 edo intervals|edo=72}}<br />
|700<br />
|<br />
|2.3.5.7.11.17.19.23<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|80<br />
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.<br />
|{{First 12 edo intervals|edo=80}}<br />
|705<br />
|<br />
|2.3.5.11.13.17.19.23<br />
|-<br />
|81<br />
|A convergent to Golden Meantone, and the last one to support Meantone in the patent val.<br />
|{{First 12 edo intervals|edo=81}}<br />
|696.3<br />
|<br />
|2.9.5.11.13.17.19<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|84<br />
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].<br />
|{{First 12 edo intervals|edo=84}}<br />
|700<br />
|<br />
|2.3.5.7.13.19.23<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|87<br />
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.<br />
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.<br />
|{{First 12 edo intervals|edo=87}}<br />
|703.4<br />
|<br />
|2.3.5.7.11.13.17<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|93<br />
|The triple of 31edo. As a Meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13. <br />
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.<br />
|{{First 12 edo intervals|edo=93}}<br />
|696.8, 709.7<br />
|<br />
|2.5.7.11.13.17.19.23<br />
|-<br />
|94<br />
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of Garibaldi. <br />
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.<br />
|{{First 12 edo intervals|edo=94}}<br />
|702.1<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|99<br />
|Perhaps the strongest 7-limit edo below 100. Supports [[Hemifamity]], [[Didacus]], and [[Ennealimmal]].<br />
|{{First 12 edo intervals|edo=99}}<br />
|703.0<br />
|126/125, 225/224, kleisma<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|140<br />
|A significant edo for interval categorization, as the next resolution level up from 58edo.<br />
|{{First 12 edo intervals|edo=140}}<br />
|702.9<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|[[159edo|159]]<br />
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.<br />
|{{First 12 edo intervals|edo=159}}<br />
|701.9<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|171<br />
|Has a surgically accurate approximation of 7-limit just intonation and is at the intersection of the [[Schismic]] and [[Ennealimmal]] temperaments. It is also a [[Neutral]] temperament, as [[11/9]] is mapped to exactly half of a perfect fifth.<br />
|{{First 12 edo intervals|edo=171}}<br />
|701.8<br />
| 225/224, 5120/5103, [[kleisma]]<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|200<br />
|Notable for its extremely good approximation of 3/2, and also for being a [[Schismic]] and [[Slendric]] system with an 8/7 of exactly 234 cents.<br />
|{{First 12 edo intervals|edo=200}}<br />
|702<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|270<br />
|Notable for its very accurate approximation of the 13-limit, with all intervals in the 15-odd-limit more in-tune than out-of-tune except for 15/13 and 26/15. It also does relatively well at approximating higher prime limits.<br />
|{{First 12 edo intervals|edo=270}}<br />
|702.2<br />
|385/384, 364/363, 352/351, 351/350, 325/324, 540/539, 441/440<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|306<br />
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.<br />
|{{First 12 edo intervals|edo=306}}<br />
|702<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|311<br />
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.<br />
|{{First 12 edo intervals|edo=311}}<br />
|702.3<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|612<br />
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).<br />
|{{First 12 edo intervals|edo=612}}<br />
|702<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|665<br />
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.<br />
|{{First 12 edo intervals|edo=665}}<br />
|702<br />
|<br />
| -<br />
|}<br />
{{Cat|Core knowledge}}</div>Intharhttps://xenreference.com/wiki/index.php?title=EDO&diff=5580EDO2026-04-04T00:49:48Z<p>Inthar: </p>
<hr />
<div>An '''equal division of the octave''' ('''EDO''' or '''edo''', /ˈidoʊ/ ''EE-doh'' or /idiˈoʊ/ ''ee-dee-OH'') is a tuning system constructed by dividing the [[octave]] into a number of equal steps. It is a type of equal temperament.<br />
<br />
The dominant modern tuning system may be called 12edo (12-EDO) because it divides the octave into 12 semitones that are all the same size. It may also be called 12-tone equal temperament or 12-TET, but this is discouraged because it does not specify which interval is being equally divided.<br />
<br />
An edo with the same number of notes as a certain [[MOS]] will have crudely similar properties, as will one with the same number of notes as the MOS has L steps. These two edos form the boundaries of how the MOS can be tuned.<br />
<br />
The notation ''m''\''n'' denotes ''m'' steps of ''n''-edo, i.e. the frequency ratio 2^(''m''/''n'').<br />
<br />
== Uses ==<br />
<br />
Edos are the most common type of tuning system in contemporary xenharmony. Unlike other types such as rank-2 temperaments and just intonation scales, equal temperaments allow for free modulation and transposition due to their uniform step size. That is, every n-step interval is the same as every other n-step interval. This comes at the expense of less freedom in approximating target intervals. It also encourages a less structured approach to composition where pitch shifts and interval quality changes can happen without much deeper meaning.<br />
<br />
== List of edos ==<br />
''Do not add subgroups to edos larger than 93; these are assumed to reasonably represent all prime-limits.''<br />
{| class="wikitable"<br />
|+ Popular edos are highlighted. Temperaments are capitalized and can be found in the [[List of regular temperaments]].<br />
|-<br />
!Edo<br />
!Description<br />
!First twelve steps (¢) <br />
!Fifth (¢)<br />
!Edostep interpretation<br />
! colspan="2" |Example basic (in 2...23, primes and 9) and [[erac]] groups<br />
|-<br />
| rowspan="2" |1<br />
| rowspan="2" |Equivalent to the 2-limit.<br />
| rowspan="2" |1200<br />
| rowspan="2" |1200<br />
| rowspan="2" |2/1<br />
|2<br />
|-<br />
|2<br />
|-<br />
| rowspan="2" |2<br />
| rowspan="2" |Just a 12edo tritone.<br />
| rowspan="2" |600, 1200<br />
| rowspan="2" |600<br />
| rowspan="2" |''none available in basic subgroup''<br />
|2<br />
|-<br />
|2.<3.>>5.>>7.<17<br />
|-<br />
| rowspan="2" |3<br />
| rowspan="2" |An augmented triad.<br />
| rowspan="2" |400, 800, 1200<br />
| rowspan="2" |800<br />
| rowspan="2" |5/4<br />
|2.5<br />
|-<br />
|2.>3.5.>19?<br />
|-<br />
| rowspan="2" |4<br />
| rowspan="2" |A diminished tetrad.<br />
| rowspan="2" |300, 600, 900, 1200<br />
| rowspan="2" |600<br />
| rowspan="2" |19/16<br />
|2.19<br />
|-<br />
|2.<3.<5.<7.<17<br />
|-<br />
| rowspan="2" class="thl" |[[5edo|5]]<br />
| rowspan="2" |Equalized [[pentic]], collapsed [[diatonic]], and the smallest edo to have strong melodic properties. Good approximation of 2.3.7 for its size.<br />
| rowspan="2" |240, 480, 720, 960, 1200<br />
| rowspan="2" |720<br />
| rowspan="2" |9/8, 8/7, 7/6<br />
|2.3.7<br />
|-<br />
|2.>>3.<7<br />
|-<br />
| rowspan="2" |6<br />
| rowspan="2" |Also known as the whole-tone scale, 6edo is a subset of 12edo.<br />
| rowspan="2" |200, 400, 600, 800, 1000, 1200<br />
| rowspan="2" |600, 800<br />
| rowspan="2" |9/8, 10/9, 28/25, 8/7<br />
|2.9.5<br />
|-<br />
|2.9.5.>7<br />
|-<br />
| rowspan="2" class="thl" |[[7edo|7]]<br />
| rowspan="2" |Equalized [[diatonic]], and the first edo to (very vaguely) support diatonic functional harmony.<br />
| rowspan="2" |171.4, 342.9, 514.3, 685.7, 857.1, 1028.6, 1200<br />
| rowspan="2" |685.7<br />
| rowspan="2" |9/8, 10/9, 16/15<br />
|2.3.5.11.13<br />
|-<br />
|2.<3.<<5.>13<br />
|-<br />
| rowspan="2" |[[8edo|8]]<br />
| rowspan="2" |Notable for containing few strong consonances, but still contains in-tune ratios 12/11 and 13/10.<br />
| rowspan="2" |150, 300, 450, 600, 750, 900, 1050, 1200<br />
| rowspan="2" |750<br />
| rowspan="2" |''none available in basic subgroup''<br />
|2.19<br />
|-<br />
|2.x3.x5.x7.x11.x13<br />
|-<br />
| rowspan="2" |[[9edo|9]]<br />
| rowspan="2" |The first edo to support the [[antidiatonic]] scale, loosely resembling the pelog scale. It contains approximations to many [[Prime limit|7-limit]] intervals, but not the [[7/4]] itself (see erac group). <br />
| rowspan="2" |133.3, 266.7, 400, 533.3, 666.7, 800, 933.3, 1066.7, 1200<br />
| rowspan="2" |666.7<br />
| rowspan="2" |9/8, 16/15, 25/24<br />
|2.5.11<br />
|-<br />
|2.<<3.>5.<<7<br />
|-<br />
| rowspan="2" |[[10edo|10]]<br />
| rowspan="2" |The doubling of 5edo, useful as an interval categorization archetype and as a melodic system in its own right, supporting [[mosh]].<br />
| rowspan="2" |120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200<br />
| rowspan="2" |720<br />
| rowspan="2" |16/15, 10/9, 81/80, 36/35<br />
|2.3.5.7.13<br />
|-<br />
|2.>>3.<7.13<br />
|-<br />
| rowspan="2" |[[11edo|11]]<br />
| rowspan="2" |Basic smitonic and checkertonic. Simplest reasonable tuning of [[Orgone]].<br />
| rowspan="2" |109.1, 218.2, 327.3, 436.4, 545.5, 654.5, 763.6, 872.7, 981.8, 1090.9<br />
| rowspan="2" |654.5, 763.6<br />
| rowspan="2" |128/119, 17/16, 18/17<br />
|2.9.7.11<br />
|-<br />
|2.x3.x5.7.11<br />
|-<br />
| rowspan="2" class="thl" |[[12edo|12]]<br />
| rowspan="2" |The basic tuning of [[diatonic]], and consequently the most widespread EDO. Supports the 5-limit decently well.<br />
| rowspan="2" |{{First 12 edo intervals|edo=12}}<br />
| rowspan="2" |700<br />
| rowspan="2" |256/243, [[chromatic semitone]], 16/15, 25/24<br />
|2.3.5.17.19<br />
|-<br />
|2.3.>5.>>7.17.19<br />
|-<br />
|[[13edo|13]]<br />
|Basic [[oneirotonic]], [[archeotonic]], and [[gramitonic]].<br />
|{{First 12 edo intervals|edo=13}}<br />
|646.2, 738.5<br />
|17/16, 18/17, 19/18, 20/19<br />
|2.5.11.13.17.19.23<br />
|-<br />
|[[14edo|14]]<br />
|Basic [[semiquartal]].<br />
|{{First 12 edo intervals|edo=14}} <br />
|685.7<br />
|28/27, 21/20, 15/14<br />
|2.3.7.13<br />
|-<br />
| class="thl" |[[15edo|15]]<br />
|The basic tuning of Zarlino's [[Diatonic|intense diatonic]], a subset of pentawood which is itself a degenerate tuning of blackdye. Supporting porcupine temperament and dubitably the 11-limit.<br />
|{{First 12 edo intervals|edo=15}}<br />
|720<br />
|81/80, 25/24, 16/15, 33/32, 36/35<br />
|2.3.5.7.11.23<br />
|-<br />
|[[16edo|16]]<br />
|The most popular antidiatonic edo, which supports [[Trismegistus]] and [[Mavila]].<br />
|{{First 12 edo intervals|edo=16}}<br />
|675, 750<br />
|20/19, 133/128, 26/25<br />
|2.5.7.13.19<br />
|-<br />
| class="thl" |[[17edo|17]]<br />
|Smallest non-12 edo whose fifth is of comparable quality to 12edo's; thus, unless you're satisfied with 7edo, the first xen edo that also allows use of the MOS diatonic scale. Noted for its melodically tense third-tone, neogothic minor chords, and approximation to the 13th harmonic. The largest edo which supports a full piano range in a DAW.<br />
|{{First 12 edo intervals|edo=17}}<br />
|705.9<br />
|256/243, 24/23, 27/26, 33/32<br />
|2.3.13.23<br />
|-<br />
| rowspan="2" |[[18edo|18]]<br />
| rowspan="2" |[[Straddle-3]] version of 12edo; provides the basic version of the straddle-3 diatonic 5L1m1s as well as soft smitonic, hard oneirotonic, and basic taric. <br />
| rowspan="2" |{{First 12 edo intervals|edo=18}}<br />
| rowspan="2" |666.6, 733.3<br />
| rowspan="2" |<br />
|2.9.5.21.13<br />
|-<br />
|2.xx3.>5.>>7.<11.<13<br />
|-<br />
| class="thl" |[[19edo|19]]<br />
|A simple tuning of Meantone, with a very accurate 6/5 and a reasonably good 5/4 and 9/7. Supports Semaphore temperament.<br />
|{{First 12 edo intervals|edo=19}}<br />
|694.7<br />
|25/24, [[diaschisma]], 36/35, 28/27<br />
|2.3.5.23<br />
|-<br />
| |20<br />
|Has a balzano (2L7s) MOS scale and accurate 13:16:19 triads.<br />
|{{First 12 edo intervals|edo=20}}<br />
|660, 720<br />
|<br />
|2.7.11.13.19<br />
|-<br />
| rowspan="2" |[[21edo|21]]<br />
| rowspan="2" |Basic tuning of 7-limit Whitewood, favoring 7/4 over 5/4. Has soft (hardness 3/2) oneirotonic. Has an extremely accurate 23rd harmonic. Has a 12edo major third and a neogothic minor third, so major and minor triads sound somewhat like compressed neogothic triads.<br />
| rowspan="2" |{{First 12 edo intervals|edo=21}}<br />
| rowspan="2" |685.7, 742.9<br />
| rowspan="2" |<br />
|2.3.5.7.23<br />
|-<br />
|2.x>3.x<5.7.x<11.x<13.23<br />
|-<br />
| rowspan="2" class="thl" |[[22edo|22]]<br />
| rowspan="2" |Represents the 7-limit and 11-limit decently well, serving as the primary tuning of Pajara and also a good Superpyth tuning, especially for Archy.<br />
| rowspan="2" |{{First 12 edo intervals|edo=22}}<br />
| rowspan="2" |709.1<br />
| rowspan="2" |<br />
|2.3.5.7.11.17<br />
|-<br />
|2.>3.<5.>>7.<11<br />
|-<br />
|23<br />
|The largest edo without a diatonic, 5edo, or 7edo fifth. A straddle-3,5,7,11 edo. Has a hard armotonic and a very hard oneirotonic.<br />
|{{First 12 edo intervals|edo=23}}<br />
|678.3<br />
|<br />
|2.x3.x5.x7.x11.13.17.23<br />
|-<br />
| div class="thl"|[[24edo|24]]<br />
|Regular old quarter-tones. Good at representing neutral intervals like 11/9, and tempers artoneutral and tendoneutral thirds to the same interval.<br />
|{{First 12 edo intervals|edo=24}}<br />
|700<br />
|<br />
|2.3.11.13.17.19<br />
|-<br />
|25<br />
|A straddle-fifth tuning with a 672c fifth that supports Mavila, or that can be used as the generator for Trismegistus with the more accurate 720c fifth. Also supports Blackwood and Didacus. The largest edo which supports five octaves in a DAW without substantial modification.<br />
|{{First 12 edo intervals|edo=25}}<br />
|720<br />
|<br />
|2.5.7.19<br />
|-<br />
|[[26edo|26]]<br />
|A simple tuning of Flattone. Has an absurdly accurate 7/4.<br />
|{{First 12 edo intervals|edo=26}}<br />
|692.7<br />
|<br />
|2.3.7.11.13<br />
|-<br />
|27<br />
|A good tuning for [[Archy]] and Sensi. It has 3/2 at 16 steps.<br />
|{{First 12 edo intervals|edo=27}}<br />
|711.1<br />
|<br />
|2.3.5.7.13.23<br />
|-<br />
|28<br />
|A tuning of 7-limit Whitewood favoring 5/4 over 7/4. Has a very hard [[oneirotonic]] scale converging on Buzzard temperament.<br />
|{{First 12 edo intervals|edo=28}}<br />
|685.7, 728.6<br />
|<br />
|2.3.5.7.11<br />
|-<br />
|[[29edo|29]]<br />
|Another neogothic tuning, and the first edo to have a more accurate perfect fifth than 12edo, so it also functions as an approximation of Pythagorean tuning, and as is typical with small Pythagorean edos, Garibaldi. It has 4/3 at 12 steps.<br />
|{{First 12 edo intervals|edo=29}}<br />
|703.4<br />
|<br />
|2.3.7/5.11/5.13/5.19.23<br />
|-<br />
|30<br />
|Doubled 15edo. Due to 15edo's ~25% error on some harmonics, this becomes a straddle-3 and -5 system, which also inherits 10edo's 13/8.<br />
|{{First 12 edo intervals|edo=30}}<br />
|680, 720<br />
|<br />
|2.x3.x5.7.11.13<br />
|-<br />
| class="thl" |[[31edo|31]]<br />
|The definitive Septimal Meantone and Mohajira tuning, and the largest edo which supports four octaves in a DAW without substantial modification.<br />
|{{First 12 edo intervals|edo=31}}<br />
|696.8<br />
|<br />
|2.3.5.7.11.23<br />
|-<br />
|32<br />
|A standard tuning of [[Archy#5/4 as doubly limma-flat major third (5 & 37)|Ultrapyth]] (5 & 37); also contains 16edo as a subset allowing for the use of antidiatonic. It is a 2.x3.5 Meantone tuning; otherwise it is "okay" at most primes up to 23, similarly to 15edo for 11. It has a 5-limit zarlino scale, although it is closer to mosh than to mosdiatonic.<br />
|{{First 12 edo intervals|edo=32}}<br />
|712.5<br />
|<br />
|2.3.7.11.17.19.23<br />
|-<br />
|33<br />
|Contains a very flat perfect fifth, and as a result a near-7edo diatonic, supporting Deeptone and with a very well-tuned 13 and 11edo's 7/4 and 11/8. Supports semaphore with the flat 7/4, which can be interpreted as [[Barbados]] temperament in the patent val.<br />
|{{First 12 edo intervals|edo=33}}<br />
|690.9<br />
|<br />
|2.3.11.13.17.19.23<br />
|-<br />
| rowspan="2" class="thl"|[[34edo|34]]<br />
| rowspan="2" |An accurate medium-sized non-Meantone 5-limit edo. Supports Diaschismic, Tetracot, and Kleismic, alongside equally halving 3/2 and 4/3 and thus having both neutrals and interordinals. It can be notated with the 12-form and/or 10-form.<br />
It is the double of 17edo, which it takes its circle of fifths from.<br />
| rowspan="2" |{{First 12 edo intervals|edo=34}}<br />
| rowspan="2" |705.9<br />
| rowspan="2" |<br />
|2.3.5.13.23<br />
|-<br />
|2.3.5.x7.13.x19.23<br />
|-<br />
|35<br />
|Contains both 5edo and 7edo and is thus a direct example of a straddle-3 system.<br />
|{{First 12 edo intervals|edo=35}}<br />
|685.7, 720.0<br />
|<br />
|2.5.7.11.17<br />
|-<br />
|36<br />
|Triple 12edo, which functions as an extremely accurate [[2.3.7 subgroup|septal]] Compton and Slendric system.<br />
|{{First 12 edo intervals|edo=36}}<br />
|700<br />
|<br />
|2.3.7.13.17.19.23<br />
|-<br />
| rowspan="2" class="thl" |[[37edo|37]]<br />
| rowspan="2" |An extremely accurate no-3 (or straddle-3) 13-limit edo. Most temperaments in this subgroup have near-optimal tunings in 37edo. Can also be seen as having an archy 3, as in porcupine.<br />
| rowspan="2" |{{First 12 edo intervals|edo=37}}<br />
| rowspan="2" |681.1, 713.5<br />
| rowspan="2" |<br />
|2.9.5.7.11.13.17.19<br />
|-<br />
|2.<x3.5.7.11.13.17.19<br />
|-<br />
|38<br />
|19edo with neutrals. Functions as a tuning of mohajira, as it has a good (and consistently mapped) 11/9 despite tuning 11 poorly.<br />
|{{First 12 edo intervals|edo=38}}<br />
|694.7<br />
|<br />
|2.3.5.7.13.17.23<br />
|-<br />
|39<br />
|Is a super-Pythagorean (though not strictly Superpyth as in the temperament) diatonic system with "gothmajor" and "gothminor" thirds in-between standard septimal and neogothic thirds.<br />
|{{First 12 edo intervals|39|edo=39}}<br />
|707.7<br />
|<br />
|2.3.11<br />
|-<br />
|[[40edo|40]]<br />
|An acceptable tuning of diminished and deeptone. As a result, the 5-limit diatonic is omnidiatonic rather than zarlino or mosdiatonic. Alternatively, can be used as a straddle-3 system.<br />
|{{First 12 edo intervals|40|edo=40}}<br />
|690<br />
|<br />
|<br />
|-<br />
| class="thl" |[[41edo|41]]<br />
|The first reasonably accurate [[Hemifamity]] edo (which is also a Garibaldi edo). Used for the Kite guitar. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}<br />
|{{First 12 edo intervals|edo=41}}<br />
|702.4<br />
|81/80, 64/63, 49/48, 50/49, 55/54, 45/44<br />
|2.3.5.7.11.13.19<br />
|-<br />
|42<br />
|The largest EDO which supports three octaves in a DAW without substantial modification (considered a key cutoff for 'large EDOs' by Vector), and also the edo with the sharpest diatonic fifth, having a mosdiatonic chroma equivalent to a 12edo wholetone and being nearly 1/2-comma archy.<br />
|{{First 12 edo intervals|edo=42}}<br />
|685.7, 714.3<br />
|<br />
|2.7.11.17.23<br />
|-<br />
|43<br />
|A sharp-of-31edo Meantone tuning; its mapping of 11 is "[[Meantone|huygens]]". Like all meantone tunings that do not map 11/9 to a perfect neutral third, its 11/9 is sharp of neutral.<br />
|{{First 12 edo intervals|edo=43}}<br />
|697.7<br />
|<br />
|2.3.5.7.11.13.17<br />
|-<br />
|44<br />
|A tuning which is, very prominently, straddle-7; its other prime harmonics up to 23 are within 25% error (except for 3, which is inherited from 22edo). <br />
|{{First 12 edo intervals|edo=44}}<br />
|709.1<br />
|<br />
|2.3.5.11.13.17.19.23<br />
|-<br />
|45<br />
|A nearly optimal tuning of flattone, compromising between a good 9/7 and a reasonable interseptimal diesis. Inherits 9edo's 7/6 and has 15edo as a subset.<br />
|{{First 12 edo intervals|edo=45}}<br />
|693.3, 720<br />
|<br />
|2.3.7.11.17.19<br />
|-<br />
| class="thl"|46<br />
|The second reasonably accurate [[Hemifamity]] edo. Has a diatonic with neogothic thirds. {{Adv|One of two viably small tunings of 11-limit [[penslen]].}}<br />
|{{First 12 edo intervals|edo=46}}<br />
|704.3<br />
|<br />
|2.3.5.7.11.13.17.23<br />
|-<br />
|47<br />
|The first edo with two distinct mosdiatonic scales. Supports Magic and Archy with its sharp fifth and Deeptone with its flat fifth. Has a very accurate 9/8 as a straddle-3 system, which generates a sort of Schismic analogue of Didacus.<br />
|{{First 12 edo intervals|edo=47}}<br />
|689.4, 714.9<br />
|<br />
|2.5.7.13.17<br />
|-<br />
|48<br />
|Four times 12edo, associated with buzzard temperament.<br />
|25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300<br />
|700<br />
|<br />
|2.3.7.11.17.19.23<br />
|-<br />
|49<br />
|A nearly optimal tuning of archy which maps 5/4 to a limma-flat major third, and squeezes a 14/11 into the 2-edostep limma between 5/4 and 9/7. It also supports straddle-3 meantone (or, more conventionally, didacus).<br />
|{{First 12 edo intervals|edo=49}}<br />
|710.2<br />
|<br />
|2.5.17.19<br />
|-<br />
|50<br />
|Approaches golden meantone, and serves as a definitive tuning of meanpop. Also contains 25edo as a subset, along with 10edo, and as such has an accurate 5, 7, and 13 with the latter two divisible into 5 parts.<br />
|{{First 12 edo intervals|edo=50}}<br />
|696<br />
|<br />
|2.3.5.11.13.23<br />
|-<br />
|51<br />
|Straddle-5 and -11, with a val option mapping 6/5 to 11/9, and one mapping the 11-limit neutral thirds together with the 13-limit ones at the perfect neutral third. Has 17edo as a subset.<br />
|{{First 12 edo intervals|edo=51}}<br />
|705.9<br />
|<br />
|2.3.7.13<br />
|-<br />
|52<br />
|Doubles 26edo, adding a sharp archy fifth and a more accurate 5/4 which support porcupine temperament.<br />
|{{First 12 edo intervals|edo=52}}<br />
|692.3, 715.4<br />
|<br />
|2.5.7.11.19.23<br />
|-<br />
|class="thl"|[[53edo|53]]<br />
|Nearly identical to a circle of 53 Pythagorean fifths, serving as the most directly obvious tuning of Schismic temperament (which also functions as a Garibaldi temperament).<br />
|{{First 12 edo intervals|edo=53}}<br />
|701.9<br />
|81/80, 64/63, 50/49, 65/64, 512/507, 91/90<br />
|2.3.5.7.13.19<br />
|-<br />
|54<br />
|Double 27edo, and the sharper end of the pajara tuning range. Can alternatively be used as a very flat deeptone system or combining the fifths as a straddle-fifth system.<br />
|<br />
|688.9, 711.1<br />
|<br />
|2.11.13.17.23<br />
|-<br />
|55<br />
|A very sharp meantone tuning, which is so sharp that it does not even support septimal meantone, and is best interpreted as mohajira as it pertains to meantone extensions.<br />
|<br />
|698.2<br />
|<br />
|2.3.5.11.17.23<br />
|-<br />
|56<br />
|An edo with a diatonic scale in the "shrub" region, with a diatonic major third between neogothic and septimal major. Tempers 9/7 to 450c, however this is not actually an interordinal as it is distinguished from 21/16 by a single edostep.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|57<br />
|Has 19edo's 5-limit combined with better interpretations of higher limits.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|58<br />
|Double of 29edo, and is the first edo to support hemipythagorean harmony better than 24edo. Thus, it has perfect neutrals and interordinals, and is thus useful for defining categories of intervals. <br />
|{{First 12 edo intervals|edo=58}}<br />
|703.4<br />
|<br />
|2.3.7.17<br />
|-<br />
|59<br />
|Has the sharpest best fifth for an edo with a 2-step diatonic semitone. It supports porcupine with a flatter tuning of the generator than 22edo, but sharper than 37edo; it is in fact 22 + 37.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|60<br />
|5 sets of 12edo, supporting magic temperament and having 10edo's 7 and 13, also supporting 7-limit compton temperament and many structures associated with 10edo and 15edo with their respective mappings.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|61<br />
|Makes 8/7 - 32/27 - 6/5 - 16/13 - 5/4 - 81/64 - 21/16 equidistant.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|62<br />
|Doubled 31edo, which shares its mappings through the 11-limit.<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|63<br />
|A very good general system, as it is triple 21edo, whose harmonics are generally off by about 1/3 of a step. It is also the largest edo which supports two octaves in a DAW without substantial modification.<br />
|{{First 12 edo intervals|edo=63}}<br />
|704.8<br />
|<br />
|2.3.5.7.11.13.23<br />
|-<br />
|64<br />
|An edo whose intervals are generally far from just intonation, being straddle-3, -5, and -11. It can function as a tuning of flattone with its flat 3, 5, and 7.<br />
|{{First 12 edo intervals|edo=64}}<br />
|693.8, 712.5<br />
|<br />
|2.13.19<br />
|-<br />
|65<br />
|A non-Garibaldi Schismic system (in fact, it supports Sensi), or a straddle-7 system.<br />
|{{First 12 edo intervals|edo=65}}<br />
|701.5<br />
|<br />
|2.3.5.11.17.19<br />
|-<br />
|66<br />
|Tripled 22edo, with an improved approximation to 7 that supports Slendric. <br />
|{{First 12 edo intervals|edo=66}}<br />
|709.1, 690.9<br />
|<br />
|2.5.7.11.13.17<br />
|-<br />
|67<br />
|Approximate 1/6-comma meantone and Slendric edo, which also supports [[orgone]].<br />
|{{First 12 edo intervals|edo=67}}<br />
|698.5<br />
|<br />
|2.3.7.11.13.17.23<br />
|-<br />
|68<br />
|Doubled 34edo, which improves its approximation to 7 while retaining 34edo's structural properties; it is similar to how 34edo retains 17edo's 2.3.13 while adding 5. 5/3 is twice 9/7, supporting sensamagic. Additionally, there is a second diatonic fifth.<br />
|{{First 12 edo intervals|edo=68}}<br />
|705<br />
|<br />
|2.3.5.7.11.17.19<br />
|-<br />
|69<br />
|A nice tuning that is approximately 2/7-comma meantone, somewhat between standard septimal meantone and 19edo. As a result, it is a mohajira system (setting 7/4 to the semiflat minor seventh) but not a septimal meantone system (as the augmented sixth is interordinal). Its 7/4 is, however, reached by stacking its second-best fourth twice, which means 69edo supports archy with the sharp fifth. As a dual-fifth system, it is neogothic.<br />
|{{First 12 edo intervals|edo=69}}<br />
|695.7, 713<br />
|<br />
|2.5.7.11.13.17.19.23<br />
|-<br />
|70<br />
|Double 35edo, and thus contains a diatonic scale that is exactly in the middle of the diatonic tuning range. It is a hemifamity system, as is typical with tunings with sharpened fifths.<br />
|{{First 12 edo intervals|edo=70}}<br />
|702.9<br />
|<br />
|2.3.11.13.17<br />
|-<br />
|71<br />
|A dual-fifth system. The sharp fifth is within the superpyth tuning range (and produces the same mapping for 5 as superpyth), despite not supporting archy. The flat fifth, analogously, produces flattone's mapping for 7 and is well-tuned for flattone, but does not support flattone.<br />
|{{First 12 edo intervals|edo=71}}<br />
|693. 709.9<br />
|<br />
|2.5.7.13.17.23<br />
|-<br />
|72<br />
|A multiple of 12edo and a very good miracle and compton system. It is also the first multiple of 12 to have a second MOS diatonic.<br />
|{{First 12 edo intervals|edo=72}}<br />
|700<br />
|<br />
|2.3.5.7.11.17.19.23<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|80<br />
|Notable for its sharp tendency and high capacity for higher-limit harmony, particularly noted by Osmium.<br />
|{{First 12 edo intervals|edo=80}}<br />
|705<br />
|<br />
|2.3.5.11.13.17.19.23<br />
|-<br />
|81<br />
|A convergent to Golden Meantone, and the last one to support meantone in the patent val.<br />
|{{First 12 edo intervals|edo=81}}<br />
|696.3<br />
|<br />
|2.9.5.11.13.17.19<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|84<br />
|A tuning system notable for its large number of contorted mappings, and also for its tuning of [[Orwell]].<br />
|{{First 12 edo intervals|edo=84}}<br />
|700<br />
|<br />
|2.3.5.7.13.19.23<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|87<br />
|A good 17-limit system, which shares 29edo's 3-limit. Essentially optimal for 13-limit [[Rodan]] (41 & 46) temperament.<br />
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments.<br />
|{{First 12 edo intervals|edo=87}}<br />
|703.4<br />
|<br />
|2.3.5.7.11.13.17<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|93<br />
|The triple of 31edo. As a meantone system, it places 11/9 sharp of the neutral third, tempered together with 16/13. <br />
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the last edo whose subgroup is specified on this table.<br />
|{{First 12 edo intervals|edo=93}}<br />
|696.8, 709.7<br />
|<br />
|2.5.7.11.13.17.19.23<br />
|-<br />
|94<br />
|A Garibaldi system, being 41 + 53 and thus having a close-to-just tuning of Garibaldi. <br />
Its step size is near a significant value of approximately 13 cents where all intervals become approximated by the edo to within a reasonable degree of intonational error on free-pitch instruments. As such, it is the first edo whose subgroup is listed as "-" on the table.<br />
|{{First 12 edo intervals|edo=94}}<br />
|702.1<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|99<br />
|Perhaps the strongest 7-limit edo below 100. Supports [[Hemifamity]], [[Didacus]], and [[Ennealimmal]].<br />
|{{First 12 edo intervals|edo=99}}<br />
|703.0<br />
|126/125, 225/224, kleisma<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|140<br />
|A significant edo for interval categorization, as the next resolution level up from 58edo.<br />
|{{First 12 edo intervals|edo=140}}<br />
|702.9<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|[[159edo|159]]<br />
|Triple of 53edo, with the ability to represent intonational differences on specific intervals, and which has been extensively practiced and studied by Aura.<br />
|{{First 12 edo intervals|edo=159}}<br />
|701.9<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|171<br />
|Has a surgically accurate approximation of 7-limit just intonation and is at the intersection of the [[Schismic]] and [[Ennealimmal]] temperaments. It is also a [[Neutral]] temperament, as [[11/9]] is mapped to exactly half of a perfect fifth.<br />
|{{First 12 edo intervals|edo=171}}<br />
|701.8<br />
| 225/224, 5120/5103, [[kleisma]]<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|200<br />
|Notable for its extremely good approximation of 3/2, and also for being a [[Schismic]] and [[Slendric]] system with an 8/7 of exactly 234 cents.<br />
|{{First 12 edo intervals|edo=200}}<br />
|702<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|270<br />
|Notable for its very accurate approximation of the 13-limit, with all intervals in the 15-odd-limit more in-tune than out-of-tune except for 15/13 and 26/15. It also does relatively well at approximating higher prime limits.<br />
|{{First 12 edo intervals|edo=270}}<br />
|702.2<br />
|385/384, 364/363, 352/351, 351/350, 325/324, 540/539, 441/440<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|306<br />
|Notable for being a convergent to 3/2, and for being a multiple of 34edo (a tuning with major structural significance). Its step is the difference between a just 3/2 and 34edo's 3/2.<br />
|{{First 12 edo intervals|edo=306}}<br />
|702<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|311<br />
|An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41.<br />
|{{First 12 edo intervals|edo=311}}<br />
|702.3<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|612<br />
|Separates and accurately tunes the syntonic and Pythagorean commas, and thus also the schisma, which separates a practically just 3/2 from 12edo's approximation. Mostly notable as the double of 306edo (and thus another 34edo multiple, and consequently a 68edo multiple).<br />
|{{First 12 edo intervals|edo=612}}<br />
|702<br />
|<br />
| -<br />
|-<br />
| colspan="6" |...<br />
|-<br />
|665<br />
|Notable for being a convergent to 3/2. Tempers out the "satanic comma", so-named because it equates 666 perfect fifths (octave-reduced) to a single perfect fifth.<br />
|{{First 12 edo intervals|edo=665}}<br />
|702<br />
|<br />
| -<br />
|}<br />
{{Cat|Core knowledge}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Misty&diff=5579Misty2026-04-04T00:39:57Z<p>Inthar: </p>
<hr />
<div>'''Misty''', 12 & 87, is a 7-limit temperament with<br />
* generator 3/2 (period-reduced: 25/21)<br />
* period 1/3-octave which represents 63/50, the difference between 56/25 and 16/9.<br />
In Misty, the diesis 128/125 is split into three 126/125s (which also represent 225/224). As a result, the octave is also split into three, because by definition a 5/4 and a third of 128/125 reach 1\3. The generator is 135/128, which stacks four times to 5/4 and raises by a 400c period to 4/3. Because it is a weak extension of Didacus, 5/4 is split into two parts that stack 5 times to 7/4.<br />
<br />
Misty results from tempering out the following 2 commas:<br />
* 5120/5103, the [[aberschisma]], which equates 64/63 and 81/80<br />
* 3136/3125, the [[Didacus]] comma<br />
<br />
Notable Misty edos include [[87edo]], [[99edo]], and [[111edo]].<br />
== Theory ==<br />
=== Intervals ===<br />
Misty has the structural property of dividing the comma 81/80~64/63 into two equal kleismas 126/125~225/224.<br />
{| class="wikitable"<br />
|+<br />
!| # periods<br />
! colspan="2" |-1 (mod 3)<br />
! colspan="2" |0 (mod 3)<br />
! colspan="2" |+1 (mod 3)<br />
|-<br />
!# gens<br />
!Cents*<br />
!JI<br />
!Cents*<br />
!JI<br />
!Cents*<br />
!JI<br />
|-<br />
! -1<br />
|'''703.1'''<br />
|'''3/2'''<br />
|1103.1<br />
|189/100<br />
|303.1<br />
|25/21<br />
|-<br />
! class="thl" |0<br />
|class="thl"|800<br />
|class="thl"|100/63<br />
| class="thl" |'''0'''<br />
|class="thl"|'''1/1'''<br />
|class="thl"|400<br />
|class="thl"|63/50<br />
|-<br />
!1<br />
|896.9<br />
|42/25<br />
|96.9<br />
|135/128, 200/189<br />
|496.9<br />
|4/3<br />
|-<br />
!2<br />
|993.8<br />
|16/9<br />
|193.8<br />
|28/25<br />
|593.8<br />
|45/32<br />
|-<br />
!3<br />
|1090.6<br />
|15/8<br />
|290.6<br />
|32/27<br />
|690.6<br />
|<br />
|-<br />
!4<br />
|1187.5<br />
|125/126, 224/225<br />
|'''387.5'''<br />
|'''5/4'''<br />
|787.5<br />
|63/40, 128/81<br />
|-<br />
!5<br />
|84.4<br />
|21/20<br />
|484.4<br />
|<br />
|884.4<br />
|5/3<br />
|-<br />
!6<br />
|181.3<br />
|10/9<br />
|581.3<br />
|7/5<br />
|981.3<br />
|<br />
|-<br />
!7<br />
|278.2<br />
|75/64<br />
|678.2<br />
|40/27<br />
|1078.2<br />
|28/15<br />
|-<br />
!8<br />
|375.1<br />
|56/45<br />
|775.1<br />
|25/16<br />
|1175.1<br />
|63/64, 80/81<br />
|-<br />
!9<br />
|471.9<br />
|21/16<br />
|871.9<br />
|<br />
|71.9<br />
|25/24<br />
|-<br />
!10<br />
|568.8<br />
|50/36<br />
|'''968.8'''<br />
|'''7/4'''<br />
|168.8<br />
|<br />
|-<br />
!11<br />
|665.7<br />
|<br />
|1065.7<br />
|50/27<br />
|265.7<br />
|7/6<br />
|-<br />
!12<br />
|762.6<br />
|14/9<br />
|1162.6<br />
|49/50, 125/128<br />
|362.6<br />
|100/81<br />
|-<br />
!13<br />
|859.5<br />
|<br />
|59.5<br />
|28/27<br />
|459.5<br />
|<br />
|-<br />
!14<br />
|956.4<br />
|<br />
|156.4<br />
|35/32<br />
|556.4<br />
|<br />
|}<br />
<br />
(* exact-2/1, exact-7/4 tuning; octave-reduced)<br />
<br />
=== Derivation of 1/3-octave period ===<br />
# {{adv|128/125 {{=}} 126/125 * 64/63 is equated to 126/125 * 81/80 by the aberschisma}}<br />
# {{adv|81/80 itself {{=}} 126/125 * 225/224}}<br />
# {{adv|Didacus equates 225/224 to 126/125}}<br />
# {{adv|So we have 128/125 ~{{=}} 126/125 * 81/80 {{=}} 126/125 * 126/125 * 225/224 ~{{=}} (126/125)<sup>3</sup>}}<br />
# {{adv|Hence, 2/1 {{=}} 5/4 * 5/4 * 5/4 * 128/125 ~{{=}} (5/4 * 126/125)<sup>3</sup> {{=}} (63/50)<sup>3</sup>}}<br />
<br />
== Patent vals ==<br />
The following patent vals support 5-limit Misty, which tempers out {{monzo|26 -12 -3}}. Vals that are contorted in the 5-limit are not included.<br />
<br />
{| class="wikitable sortable"<br />
!|Edo<br />
!|7-limit extension<br />
!|Fifth<br />
|-<br />
||12||12 & 87||700.000<br />
|-<br />
||123||12 & 123||702.439<br />
|-<br />
||111||12 & 87||702.703<br />
|-<br />
||210||12 & 87||702.857<br />
|-<br />
||99||12 & 87||703.030<br />
|-<br />
||384||12 & 87||703.125<br />
|-<br />
||285||12 & 87||703.158<br />
|-<br />
||471||12 & 87||703.185<br />
|-<br />
||186||12 & 87||703.226<br />
|-<br />
||273||12 & 87||703.297<br />
|-<br />
||360||87 & 360||703.333<br />
|-<br />
||87||12 & 87, 87 & 75||703.448<br />
|-<br />
||336||87 & 75||703.571<br />
|-<br />
||249||87 & 75||703.614<br />
|-<br />
||162||87 & 75||703.704<br />
|-<br />
||237||237 & 312||703.797<br />
|-<br />
||312||237 & 312||703.846<br />
|-<br />
||75||12 & 51, 87 & 75||704.000<br />
|-<br />
||138|| ||704.348<br />
|-<br />
||63||12 & 51||704.762<br />
|-<br />
||51||12 & 51||705.882 <br />
|}<br />
{{Navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5578Hemifamity2026-04-04T00:14:08Z<p>Inthar: /* Extensions */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.<br />
<br />
== Structural theory ==<br />
=== Notation ===<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]] (Aberschismic + Slendric)<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]] (Aberschismic + Tetracot)<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]] (Aberschismic + Schismic)<br />
* 2.3.5.7[41 & 58]: [[Hemififths]] (Aberschismic + 2401/2400)<br />
* 2.3.5.7[46 & 53]: [[Amity]] (Aberschismic + 4375/4374)<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]] (Aberschismic + Diaschismic)<br />
* 2.3.5.7[53 & 58]: [[Buzzard]] (Aberschismic + 2.3.7 Buzzard)<br />
* 2.3.5.7[87 & 99]: [[Misty]] (Aberschismic + Didacus)<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which equates 128/105 to 39/32); with edo join 41 & 46 & 53, this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]]. Inthar considers this the canonical extension of Hemifamity that adds prime 13.<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5577Hemifamity2026-04-04T00:12:24Z<p>Inthar: /* Extensions */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.<br />
<br />
== Structural theory ==<br />
=== Notation ===<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]] (Aberschismic + Slendric)<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]] (Aberschismic + Tetracot)<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]] (Aberschismic + Schismic)<br />
* 2.3.5.7[41 & 58]: [[Hemififths]] (Aberschismic + 2401/2400)<br />
* 2.3.5.7[46 & 53]: [[Amity]] (Aberschismic + 4375/4374)<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]] (Aberschismic + Diaschismic)<br />
* 2.3.5.7[53 & 58]: [[Buzzard]] (Aberschismic + 2.3.7 Buzzard)<br />
* 2.3.5.7[87 & 99]: [[Misty]] (Aberschismic + Didacus)<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which equates 128/105 to 39/32); with edo join 41 & 46 & 53, this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]]. Inthar considers this the canonical extension that adds prime 13.<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5576Hemifamity2026-04-03T23:59:48Z<p>Inthar: /* Supporting rank-2 temperaments */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.<br />
<br />
== Structural theory ==<br />
=== Notation ===<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]] (Aberschismic + Slendric)<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]] (Aberschismic + Tetracot)<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]] (Aberschismic + Schismic)<br />
* 2.3.5.7[41 & 58]: [[Hemififths]] (Aberschismic + 2401/2400)<br />
* 2.3.5.7[46 & 53]: [[Amity]] (Aberschismic + 4375/4374)<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]] (Aberschismic + Diaschismic)<br />
* 2.3.5.7[53 & 58]: [[Buzzard]] (Aberschismic + 2.3.7 Buzzard)<br />
* 2.3.5.7[87 & 99]: [[Misty]] (Aberschismic + Didacus)<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); with edo join 41 & 46 & 53, this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]]. Inthar considers this the canonical extension that adds prime 13.<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Amity&diff=5575Amity2026-04-03T23:49:52Z<p>Inthar: /* Interval chain */</p>
<hr />
<div>'''Amity''' (from "acute minor third"), 2.3.5.7[46 & 53], is a complex temperament splitting 8/3 into 5 supraminor thirds about 339c-340c in size. It has a tuning that is very accurate in the 7-limit (namely 99edo).<br />
<br />
Amity is a 7-cluster temperament due to its generator being only slightly flatter than 2\7.<br />
== Interval chain ==<br />
Notes:<br />
* 243/200 is 6/5 sharpened by 81/80.<br />
* The extension to 13 given is called Hitchcock, whose 13 mapping tends to be flat due to tempering out S13.<br />
<br/><br />
* 1 gen = 128/105 ~ 243/200 ~ 39/32 (=> tempers out 5120/5103 = S8/S9 and 4096/4095 = S64)<br />
* 2 gens = 40/27<br />
* 3 gens = 9/5<br />
* 4 gens = 35/32<br />
* 5 gens = '''4/3'''<br />
* 6 gens = '''13/8''' ~ 21/13 (=> S13 is tempered out)<br />
* 7 gens = 80/81 ~ 63/64 ~ 64/65<br />
* 8 gens = 6/5<br />
* 9 gens = 48/35<br />
* 10 gens = '''16/9'''<br />
* 11 gens = 13/12 ~ 14/13<br />
* 12 gens = 21/16<br />
* 13 gens = '''8/5'''<br />
* 14 gens = 35/36<br />
* 15 gens = 32/27<br />
* 16 gens = 36/25 ~ 13/9<br />
* 17 gens = '''7/4'''<br />
* 18 gens = '''16/15'''<br />
* 19 gens = 13/10<br />
* 20 gens = 128/81<br />
* 21 gens = 20/21 ~ 24/25<br />
* 22 gens = 7/6<br />
* 23 gens = 64/45<br />
* 24 gens = 26/15<br />
* 25 gens = 256/243<br />
* 26 gens = 32/25<br />
* 27 gens = 14/9<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Amity/Patent vals]]''<br />
{{Navbox regtemp}}<br />
{{Cat|temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5573Hemifamity2026-04-03T23:45:51Z<p>Inthar: /* Extensions */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.<br />
<br />
== Structural theory ==<br />
=== Notation ===<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); with edo join 41 & 46 & 53, this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]]. Inthar considers this the canonical extension that adds prime 13.<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5572Hemifamity2026-04-03T23:19:33Z<p>Inthar: /* Extensions */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.<br />
<br />
== Structural theory ==<br />
=== Notation ===<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this has edo join 41 & 46 & 53 and is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]]. Inthar considers this the canonical extension that adds prime 13.<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity/Patent_vals&diff=5571Hemifamity/Patent vals2026-04-03T23:18:12Z<p>Inthar: </p>
<hr />
<div>The following patent vals support 2.3.5.7 [[Hemifamity]] (aka Aberschismic). Vals that are contorted in 2.3.5.7 are not included.<br />
<br />
Note on extensions:<br />
* 11-limit extensions: 41 & 46 & 53 is called Akea (tempers out 385/384), and 41 & 46 & 58 is called Pele (tempers out 896/891). The intersection of the two is Rodan (41 & 46).<br />
* 13-limit extensions: 41 & 46 & 53 tempers out 4096/4095.<br />
{| class="wikitable sortable"<br />
|-<br />
!Edo!!Extension to 11!!Extension to 13!!81/80 tuning!!3/2 tuning||5/4 tuning!!7/4 tuning<br />
|-<br />
|7||41 & 46 & 53 ||41 & 46 & 53 ||0.000||685.714||342.857||1028.571<br />
|-<br />
|12||41 & 46 & 58 ||41 & 46 & 53 ||0.000||700.000||400.000||1000.000<br />
|-<br />
|5||41 & 46 & 53, 41 & 46 & 58 || ||0.000||720.000||480.000||960.000<br />
|-<br />
|70||||||17.143||702.857||394.286||977.143<br />
|-<br />
|58||41 & 46 & 58|| ||20.690||703.448||393.103||972.414<br />
|-<br />
|111||||||21.622||702.703||389.189||972.973<br />
|-<br />
|53||41 & 46 & 53 ||41 & 46 & 53 ||22.642||701.887||384.906||973.585<br />
|-<br />
|210||||||22.857||702.857||388.571||971.429<br />
|-<br />
|157||||||22.930||703.185||389.809||970.701<br />
|-<br />
|205||||||23.415||702.439||386.341||971.707<br />
|-<br />
|152||||41 & 46 & 53 ||23.684||702.632||386.842||971.053<br />
|-<br />
|251||||||23.904||702.789||387.251||970.518<br />
|-<br />
|350||||||24.000||702.857||387.429||970.286<br />
|-<br />
|449||||||24.053||702.895||387.528||970.156<br />
|-<br />
|99||41 & 46 & 53 ||41 & 46 & 53 ||24.242||703.030||387.879||969.697<br />
|-<br />
|198||||||24.242||703.030||387.879||969.697<br />
|-<br />
|297||||||24.242||703.030||387.879||969.697<br />
|-<br />
|345||||||24.348||702.609||386.087||970.435<br />
|-<br />
|246||||41 & 46 & 53 ||24.390||702.439||385.366||970.732<br />
|-<br />
|589||||||24.448||702.886||387.097||969.779<br />
|-<br />
|147||||41 & 46 & 53 ||24.490||702.041||383.673||971.429<br />
|-<br />
|490||||||24.490||702.857||386.939||969.796<br />
|-<br />
|391||||||24.552||702.813||386.701||969.821<br />
|-<br />
|244||||||24.590||703.279||388.525||968.852<br />
|-<br />
|536||||||24.627||702.985||387.313||969.403<br />
|-<br />
|292||||||24.658||702.740||386.301||969.863<br />
|-<br />
|340||||41 & 46 & 53 ||24.706||702.353||384.706||970.588<br />
|-<br />
|437||||||24.714||702.975||387.185||969.336<br />
|-<br />
|485||||||24.742||702.680||385.979||969.897<br />
|-<br />
|630||||||24.762||702.857||386.667||969.524<br />
|-<br />
|145||41 & 46 & 58|| ||24.828||703.448||388.966||968.276<br />
|-<br />
|338||||||24.852||702.959||386.982||969.231<br />
|-<br />
|531||||||24.859||702.825||386.441||969.492<br />
|-<br />
|724||||||24.862||702.762||386.188||969.613<br />
|-<br />
|193||||41 & 46 & 53 ||24.870||702.591||385.492||969.948<br />
|-<br />
|386||||||24.870||702.591||385.492||969.948<br />
|-<br />
|577||||||24.957||702.946||386.828||969.151<br />
|-<br />
|625||||||24.960||702.720||385.920||969.600<br />
|-<br />
|48||41 & 46 & 53 ||41 & 46 & 53 ||25.000||700.000||375.000||975.000<br />
|-<br />
|432||||||25.000||702.778||386.111||969.444<br />
|-<br />
|384||||||25.000||703.125||387.500||968.750<br />
|-<br />
|671||||||25.037||702.832||386.289||969.300<br />
|-<br />
|287||||41 & 46 & 53 ||25.087||702.439||384.669||970.035<br />
|-<br />
|526||||41 & 46 & 53 ||25.095||702.662||385.551||969.582<br />
|-<br />
|239||||41 & 46 & 53 ||25.105||702.929||386.611||969.038<br />
|-<br />
|572||||||25.175||702.797||386.014||969.231<br />
|-<br />
|524||||||25.191||703.053||387.023||968.702<br />
|-<br />
|333||||41 & 46 & 53 ||25.225||702.703||385.586||969.369<br />
|-<br />
|618||||||25.243||702.913||386.408||968.932<br />
|-<br />
|285||||||25.263||703.158||387.368||968.421<br />
|-<br />
|427||||41 & 46 & 53 ||25.293||702.576||385.012||969.555<br />
|-<br />
|379||||41 & 46 & 53 ||25.330||702.902||386.280||968.865<br />
|-<br />
|473||||41 & 46 & 53 ||25.370||702.748||385.624||969.133<br />
|-<br />
|331||||||25.378||703.323||387.915||967.976<br />
|-<br />
|425||||||25.412||703.059||386.824||968.471<br />
|-<br />
|519||||||25.434||702.890||386.127||968.786<br />
|-<br />
|613||||41 & 46 & 53 ||25.449||702.773||385.644||969.005<br />
|-<br />
|471||||||25.478||703.185||387.261||968.153<br />
|-<br />
|565||||||25.487||703.009||386.549||968.496<br />
|-<br />
|94||41 & 46 & 53 ||41 & 46 & 53 ||25.532||702.128||382.979||970.213<br />
|-<br />
|234||||41 & 46 & 53 ||25.641||702.564||384.615||969.231<br />
|-<br />
|374||||41 & 46 & 53 ||25.668||702.674||385.027||968.984<br />
|-<br />
|514||||41 & 46 & 53 ||25.681||702.724||385.214||968.872<br />
|-<br />
|140||41 & 46 & 53 ||41 & 46 & 53 ||25.714||702.857||385.714||968.571<br />
|-<br />
|466||||41 & 46 & 53 ||25.751||703.004||386.266||968.240<br />
|-<br />
|326||||41 & 46 & 53 ||25.767||703.067||386.503||968.098<br />
|-<br />
|512||||||25.781||703.125||386.719||967.969<br />
|-<br />
|186||41 & 46 & 53 ||41 & 46 & 53 ||25.806||703.226||387.097||967.742<br />
|-<br />
|418||||||25.837||703.349||387.560||967.464<br />
|-<br />
|232||41 & 46 & 58 ||||25.862||703.448||387.931||967.241<br />
|-<br />
|601||||41 & 46 & 53 ||25.957||702.829||385.358||968.386<br />
|-<br />
|461||||41 & 46 & 53 ||26.030||702.820||385.249||968.330<br />
|-<br />
|507||||41 & 46 & 53 ||26.036||702.959||385.799||968.047<br />
|-<br />
|46||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||26.087||704.348||391.304||965.217<br />
|-<br />
|413||||41 & 46 & 53 ||26.150||703.148||386.441||967.554<br />
|-<br />
|367||||41 & 46 & 53 ||26.158||702.997||385.831||967.847<br />
|-<br />
|321||41 & 46 & 53 ||41 & 46 & 53 ||26.168||702.804||385.047||968.224<br />
|-<br />
|548||||41 & 46 & 53 ||26.277||702.920||385.401||967.883<br />
|-<br />
|273||41 & 46 & 53 ||41 & 46 & 53 ||26.374||703.297||386.813||967.033<br />
|-<br />
|227||41 & 46 & 53 ||41 & 46 & 53 ||26.432||703.084||385.903||967.401<br />
|-<br />
|408||41 & 46 & 53 ||41 & 46 & 53 ||26.471||702.941||385.294||967.647<br />
|-<br />
|181||41 & 46 & 53 ||41 & 46 & 53 ||26.519||702.762||384.530||967.956<br />
|-<br />
|135||41 & 46 & 53 ||41 & 46 & 53 ||26.667||702.222||382.222||968.889<br />
|-<br />
|268||41 & 46 & 53 ||41 & 46 & 53 ||26.866||702.985||385.075||967.164<br />
|-<br />
|222||41 & 46 & 53 || ||27.027||702.703||383.784||967.568<br />
|-<br />
|133||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.068||703.759||387.970||965.414<br />
|-<br />
|309||41 & 46 & 53 ||||27.184||702.913||384.466||966.990<br />
|-<br />
|87||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.586||703.448||386.207||965.517<br />
|-<br />
|128||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||28.125||703.125||384.375||965.625<br />
|-<br />
|169||41 & 46 & 53, 41 & 46 & 58|| ||28.402||702.959||383.432||965.680<br />
|-<br />
|41||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||29.268||702.439||380.488||965.854<br />
|-<br />
|39||41 & 46 & 53 ||41 & 46 & 53 ||30.769||707.692||400.000||953.846<br />
|-<br />
|34||41 & 46 & 53 ||41 & 46 & 53 ||35.294||705.882||388.235||952.941<br />
|-<br />
|29||41 & 46 & 58|| ||41.379||703.448||372.414||951.724<br />
|}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity/Patent_vals&diff=5570Hemifamity/Patent vals2026-04-03T23:17:06Z<p>Inthar: </p>
<hr />
<div>The following patent vals support 2.3.5.7 [[Hemifamity]] (aka Aberschismic). Vals that are contorted in 2.3.5.7 are not included.<br />
<br />
Note on 11-limit extensions: 41 & 46 & 53 is called Akea (tempers out 385/384), and 41 & 46 & 58 is called Pele (tempers out 896/891). The intersection of the two is Rodan (41 & 46).<br />
{| class="wikitable sortable"<br />
|-<br />
!Edo!!Extension to 11!!Extension to 13!!81/80 tuning!!3/2 tuning||5/4 tuning!!7/4 tuning<br />
|-<br />
|7||41 & 46 & 53 ||41 & 46 & 53 ||0.000||685.714||342.857||1028.571<br />
|-<br />
|12||41 & 46 & 58 ||41 & 46 & 53 ||0.000||700.000||400.000||1000.000<br />
|-<br />
|5||41 & 46 & 53, 41 & 46 & 58 || ||0.000||720.000||480.000||960.000<br />
|-<br />
|70||||||17.143||702.857||394.286||977.143<br />
|-<br />
|58||41 & 46 & 58|| ||20.690||703.448||393.103||972.414<br />
|-<br />
|111||||||21.622||702.703||389.189||972.973<br />
|-<br />
|53||41 & 46 & 53 ||41 & 46 & 53 ||22.642||701.887||384.906||973.585<br />
|-<br />
|210||||||22.857||702.857||388.571||971.429<br />
|-<br />
|157||||||22.930||703.185||389.809||970.701<br />
|-<br />
|205||||||23.415||702.439||386.341||971.707<br />
|-<br />
|152||||41 & 46 & 53 ||23.684||702.632||386.842||971.053<br />
|-<br />
|251||||||23.904||702.789||387.251||970.518<br />
|-<br />
|350||||||24.000||702.857||387.429||970.286<br />
|-<br />
|449||||||24.053||702.895||387.528||970.156<br />
|-<br />
|99||41 & 46 & 53 ||41 & 46 & 53 ||24.242||703.030||387.879||969.697<br />
|-<br />
|198||||||24.242||703.030||387.879||969.697<br />
|-<br />
|297||||||24.242||703.030||387.879||969.697<br />
|-<br />
|345||||||24.348||702.609||386.087||970.435<br />
|-<br />
|246||||41 & 46 & 53 ||24.390||702.439||385.366||970.732<br />
|-<br />
|589||||||24.448||702.886||387.097||969.779<br />
|-<br />
|147||||41 & 46 & 53 ||24.490||702.041||383.673||971.429<br />
|-<br />
|490||||||24.490||702.857||386.939||969.796<br />
|-<br />
|391||||||24.552||702.813||386.701||969.821<br />
|-<br />
|244||||||24.590||703.279||388.525||968.852<br />
|-<br />
|536||||||24.627||702.985||387.313||969.403<br />
|-<br />
|292||||||24.658||702.740||386.301||969.863<br />
|-<br />
|340||||41 & 46 & 53 ||24.706||702.353||384.706||970.588<br />
|-<br />
|437||||||24.714||702.975||387.185||969.336<br />
|-<br />
|485||||||24.742||702.680||385.979||969.897<br />
|-<br />
|630||||||24.762||702.857||386.667||969.524<br />
|-<br />
|145||41 & 46 & 58|| ||24.828||703.448||388.966||968.276<br />
|-<br />
|338||||||24.852||702.959||386.982||969.231<br />
|-<br />
|531||||||24.859||702.825||386.441||969.492<br />
|-<br />
|724||||||24.862||702.762||386.188||969.613<br />
|-<br />
|193||||41 & 46 & 53 ||24.870||702.591||385.492||969.948<br />
|-<br />
|386||||||24.870||702.591||385.492||969.948<br />
|-<br />
|577||||||24.957||702.946||386.828||969.151<br />
|-<br />
|625||||||24.960||702.720||385.920||969.600<br />
|-<br />
|48||41 & 46 & 53 ||41 & 46 & 53 ||25.000||700.000||375.000||975.000<br />
|-<br />
|432||||||25.000||702.778||386.111||969.444<br />
|-<br />
|384||||||25.000||703.125||387.500||968.750<br />
|-<br />
|671||||||25.037||702.832||386.289||969.300<br />
|-<br />
|287||||41 & 46 & 53 ||25.087||702.439||384.669||970.035<br />
|-<br />
|526||||41 & 46 & 53 ||25.095||702.662||385.551||969.582<br />
|-<br />
|239||||41 & 46 & 53 ||25.105||702.929||386.611||969.038<br />
|-<br />
|572||||||25.175||702.797||386.014||969.231<br />
|-<br />
|524||||||25.191||703.053||387.023||968.702<br />
|-<br />
|333||||41 & 46 & 53 ||25.225||702.703||385.586||969.369<br />
|-<br />
|618||||||25.243||702.913||386.408||968.932<br />
|-<br />
|285||||||25.263||703.158||387.368||968.421<br />
|-<br />
|427||||41 & 46 & 53 ||25.293||702.576||385.012||969.555<br />
|-<br />
|379||||41 & 46 & 53 ||25.330||702.902||386.280||968.865<br />
|-<br />
|473||||41 & 46 & 53 ||25.370||702.748||385.624||969.133<br />
|-<br />
|331||||||25.378||703.323||387.915||967.976<br />
|-<br />
|425||||||25.412||703.059||386.824||968.471<br />
|-<br />
|519||||||25.434||702.890||386.127||968.786<br />
|-<br />
|613||||41 & 46 & 53 ||25.449||702.773||385.644||969.005<br />
|-<br />
|471||||||25.478||703.185||387.261||968.153<br />
|-<br />
|565||||||25.487||703.009||386.549||968.496<br />
|-<br />
|94||41 & 46 & 53 ||41 & 46 & 53 ||25.532||702.128||382.979||970.213<br />
|-<br />
|234||||41 & 46 & 53 ||25.641||702.564||384.615||969.231<br />
|-<br />
|374||||41 & 46 & 53 ||25.668||702.674||385.027||968.984<br />
|-<br />
|514||||41 & 46 & 53 ||25.681||702.724||385.214||968.872<br />
|-<br />
|140||41 & 46 & 53 ||41 & 46 & 53 ||25.714||702.857||385.714||968.571<br />
|-<br />
|466||||41 & 46 & 53 ||25.751||703.004||386.266||968.240<br />
|-<br />
|326||||41 & 46 & 53 ||25.767||703.067||386.503||968.098<br />
|-<br />
|512||||||25.781||703.125||386.719||967.969<br />
|-<br />
|186||41 & 46 & 53 ||41 & 46 & 53 ||25.806||703.226||387.097||967.742<br />
|-<br />
|418||||||25.837||703.349||387.560||967.464<br />
|-<br />
|232||41 & 46 & 58 ||||25.862||703.448||387.931||967.241<br />
|-<br />
|601||||41 & 46 & 53 ||25.957||702.829||385.358||968.386<br />
|-<br />
|461||||41 & 46 & 53 ||26.030||702.820||385.249||968.330<br />
|-<br />
|507||||41 & 46 & 53 ||26.036||702.959||385.799||968.047<br />
|-<br />
|46||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||26.087||704.348||391.304||965.217<br />
|-<br />
|413||||41 & 46 & 53 ||26.150||703.148||386.441||967.554<br />
|-<br />
|367||||41 & 46 & 53 ||26.158||702.997||385.831||967.847<br />
|-<br />
|321||41 & 46 & 53 ||41 & 46 & 53 ||26.168||702.804||385.047||968.224<br />
|-<br />
|548||||41 & 46 & 53 ||26.277||702.920||385.401||967.883<br />
|-<br />
|273||41 & 46 & 53 ||41 & 46 & 53 ||26.374||703.297||386.813||967.033<br />
|-<br />
|227||41 & 46 & 53 ||41 & 46 & 53 ||26.432||703.084||385.903||967.401<br />
|-<br />
|408||41 & 46 & 53 ||41 & 46 & 53 ||26.471||702.941||385.294||967.647<br />
|-<br />
|181||41 & 46 & 53 ||41 & 46 & 53 ||26.519||702.762||384.530||967.956<br />
|-<br />
|135||41 & 46 & 53 ||41 & 46 & 53 ||26.667||702.222||382.222||968.889<br />
|-<br />
|268||41 & 46 & 53 ||41 & 46 & 53 ||26.866||702.985||385.075||967.164<br />
|-<br />
|222||41 & 46 & 53 || ||27.027||702.703||383.784||967.568<br />
|-<br />
|133||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.068||703.759||387.970||965.414<br />
|-<br />
|309||41 & 46 & 53 ||||27.184||702.913||384.466||966.990<br />
|-<br />
|87||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.586||703.448||386.207||965.517<br />
|-<br />
|128||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||28.125||703.125||384.375||965.625<br />
|-<br />
|169||41 & 46 & 53, 41 & 46 & 58|| ||28.402||702.959||383.432||965.680<br />
|-<br />
|41||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||29.268||702.439||380.488||965.854<br />
|-<br />
|39||41 & 46 & 53 ||41 & 46 & 53 ||30.769||707.692||400.000||953.846<br />
|-<br />
|34||41 & 46 & 53 ||41 & 46 & 53 ||35.294||705.882||388.235||952.941<br />
|-<br />
|29||41 & 46 & 58|| ||41.379||703.448||372.414||951.724<br />
|}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity/Patent_vals&diff=5569Hemifamity/Patent vals2026-04-03T23:15:02Z<p>Inthar: </p>
<hr />
<div>The following patent vals support 2.3.5.7 [[Hemifamity]] (aka Aberschismic). Vals that are contorted in 2.3.5.7 are not included.<br />
<br />
Note on 11-limit extensions: 41 & 46 & 53 is called Akea (tempers out 385/384), and 41 & 46 & 58 is called Pele (tempers out 896/891). The intersection of the two is Rodan (41 & 46).<br />
{| class="wikitable sortable"<br />
|-<br />
!Edo!!Extension to 11!!Extension to 13!!81/80 tuning!!3/2 tuning||5/4 tuning!!7/4 tuning<br />
|-<br />
|7||41 & 46 & 53 ||41 & 46 & 53 ||0.000||685.714||342.857||1028.571<br />
|-<br />
|12||41 & 46 & 58 ||41 & 46 & 53 ||0.000||700.000||400.000||1000.000<br />
|-<br />
|5||41 & 46 & 53, 41 & 46 & 58 || ||0.000||720.000||480.000||960.000<br />
|-<br />
|70||||||17.143||702.857||394.286||977.143<br />
|-<br />
|58||41 & 46 & 58 ||20.690||703.448||393.103||972.414<br />
|-<br />
|111||||||21.622||702.703||389.189||972.973<br />
|-<br />
|53||41 & 46 & 53 ||41 & 46 & 53 ||22.642||701.887||384.906||973.585<br />
|-<br />
|210||||||22.857||702.857||388.571||971.429<br />
|-<br />
|157||||||22.930||703.185||389.809||970.701<br />
|-<br />
|205|||||23.415||702.439||386.341||971.707<br />
|-<br />
|152||||41 & 46 & 53 ||23.684||702.632||386.842||971.053<br />
|-<br />
|251||||||23.904||702.789||387.251||970.518<br />
|-<br />
|350||||||24.000||702.857||387.429||970.286<br />
|-<br />
|449||||||24.053||702.895||387.528||970.156<br />
|-<br />
|99||41 & 46 & 53 ||41 & 46 & 53 ||24.242||703.030||387.879||969.697<br />
|-<br />
|198||||||24.242||703.030||387.879||969.697<br />
|-<br />
|297||||||24.242||703.030||387.879||969.697<br />
|-<br />
|345||||||24.348||702.609||386.087||970.435<br />
|-<br />
|246||||41 & 46 & 53 ||24.390||702.439||385.366||970.732<br />
|-<br />
|589||||||24.448||702.886||387.097||969.779<br />
|-<br />
|147||||41 & 46 & 53 ||24.490||702.041||383.673||971.429<br />
|-<br />
|490||||||24.490||702.857||386.939||969.796<br />
|-<br />
|391||||||24.552||702.813||386.701||969.821<br />
|-<br />
|244||||||24.590||703.279||388.525||968.852<br />
|-<br />
|536||||||24.627||702.985||387.313||969.403<br />
|-<br />
|292||||||24.658||702.740||386.301||969.863<br />
|-<br />
|340||||41 & 46 & 53 ||24.706||702.353||384.706||970.588<br />
|-<br />
|437||||||24.714||702.975||387.185||969.336<br />
|-<br />
|485||||||24.742||702.680||385.979||969.897<br />
|-<br />
|630||||||24.762||702.857||386.667||969.524<br />
|-<br />
|145||41 & 46 & 58|| ||24.828||703.448||388.966||968.276<br />
|-<br />
|338||||||24.852||702.959||386.982||969.231<br />
|-<br />
|531||||||24.859||702.825||386.441||969.492<br />
|-<br />
|724||||||24.862||702.762||386.188||969.613<br />
|-<br />
|193||||41 & 46 & 53 ||24.870||702.591||385.492||969.948<br />
|-<br />
|386||||||24.870||702.591||385.492||969.948<br />
|-<br />
|577||||||24.957||702.946||386.828||969.151<br />
|-<br />
|625||||||24.960||702.720||385.920||969.600<br />
|-<br />
|48||41 & 46 & 53 ||41 & 46 & 53 ||25.000||700.000||375.000||975.000<br />
|-<br />
|432||||||25.000||702.778||386.111||969.444<br />
|-<br />
|384||||||25.000||703.125||387.500||968.750<br />
|-<br />
|671||||||25.037||702.832||386.289||969.300<br />
|-<br />
|287||||41 & 46 & 53 ||25.087||702.439||384.669||970.035<br />
|-<br />
|526||||41 & 46 & 53 ||25.095||702.662||385.551||969.582<br />
|-<br />
|239||||41 & 46 & 53 ||25.105||702.929||386.611||969.038<br />
|-<br />
|572||||||25.175||702.797||386.014||969.231<br />
|-<br />
|524||||||25.191||703.053||387.023||968.702<br />
|-<br />
|333||||41 & 46 & 53 ||25.225||702.703||385.586||969.369<br />
|-<br />
|618||||||25.243||702.913||386.408||968.932<br />
|-<br />
|285||||||25.263||703.158||387.368||968.421<br />
|-<br />
|427||||41 & 46 & 53 ||25.293||702.576||385.012||969.555<br />
|-<br />
|379||||41 & 46 & 53 ||25.330||702.902||386.280||968.865<br />
|-<br />
|473||||41 & 46 & 53 ||25.370||702.748||385.624||969.133<br />
|-<br />
|331||||||25.378||703.323||387.915||967.976<br />
|-<br />
|425||||||25.412||703.059||386.824||968.471<br />
|-<br />
|519||||||25.434||702.890||386.127||968.786<br />
|-<br />
|613||||41 & 46 & 53 ||25.449||702.773||385.644||969.005<br />
|-<br />
|471||||||25.478||703.185||387.261||968.153<br />
|-<br />
|565||||||25.487||703.009||386.549||968.496<br />
|-<br />
|94||41 & 46 & 53 ||41 & 46 & 53 ||25.532||702.128||382.979||970.213<br />
|-<br />
|234||||||41 & 46 & 53 ||25.641||702.564||384.615||969.231<br />
|-<br />
|374||||41 & 46 & 53 ||25.668||702.674||385.027||968.984<br />
|-<br />
|514||||41 & 46 & 53 ||25.681||702.724||385.214||968.872<br />
|-<br />
|140||41 & 46 & 53 ||41 & 46 & 53 ||25.714||702.857||385.714||968.571<br />
|-<br />
|466||||41 & 46 & 53 ||25.751||703.004||386.266||968.240<br />
|-<br />
|326||||41 & 46 & 53 ||25.767||703.067||386.503||968.098<br />
|-<br />
|512||||||25.781||703.125||386.719||967.969<br />
|-<br />
|186||41 & 46 & 53 ||41 & 46 & 53 ||25.806||703.226||387.097||967.742<br />
|-<br />
|418||||||25.837||703.349||387.560||967.464<br />
|-<br />
|232||41 & 46 & 58 ||25.862||703.448||387.931||967.241<br />
|-<br />
|601||||41 & 46 & 53 ||25.957||702.829||385.358||968.386<br />
|-<br />
|461||||41 & 46 & 53 ||26.030||702.820||385.249||968.330<br />
|-<br />
|507||||41 & 46 & 53 ||26.036||702.959||385.799||968.047<br />
|-<br />
|46||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||26.087||704.348||391.304||965.217<br />
|-<br />
|413||||41 & 46 & 53 ||26.150||703.148||386.441||967.554<br />
|-<br />
|367||||41 & 46 & 53 ||26.158||702.997||385.831||967.847<br />
|-<br />
|321||41 & 46 & 53 ||41 & 46 & 53 ||26.168||702.804||385.047||968.224<br />
|-<br />
|548||||41 & 46 & 53 ||26.277||702.920||385.401||967.883<br />
|-<br />
|273||41 & 46 & 53 ||41 & 46 & 53 ||26.374||703.297||386.813||967.033<br />
|-<br />
|227||41 & 46 & 53 ||41 & 46 & 53 ||26.432||703.084||385.903||967.401<br />
|-<br />
|408||41 & 46 & 53 ||41 & 46 & 53 ||26.471||702.941||385.294||967.647<br />
|-<br />
|181||41 & 46 & 53 ||41 & 46 & 53 ||26.519||702.762||384.530||967.956<br />
|-<br />
|135||41 & 46 & 53 ||41 & 46 & 53 ||26.667||702.222||382.222||968.889<br />
|-<br />
|268||41 & 46 & 53 ||41 & 46 & 53 ||26.866||702.985||385.075||967.164<br />
|-<br />
|222||41 & 46 & 53 ||27.027||702.703||383.784||967.568<br />
|-<br />
|133||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.068||703.759||387.970||965.414<br />
|-<br />
|309||41 & 46 & 53 ||27.184||702.913||384.466||966.990<br />
|-<br />
|87||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.586||703.448||386.207||965.517<br />
|-<br />
|128||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||28.125||703.125||384.375||965.625<br />
|-<br />
|169||41 & 46 & 53, 41 & 46 & 58 ||28.402||702.959||383.432||965.680<br />
|-<br />
|41||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||29.268||702.439||380.488||965.854<br />
|-<br />
|39||41 & 46 & 53 ||41 & 46 & 53 ||30.769||707.692||400.000||953.846<br />
|-<br />
|34||41 & 46 & 53 ||41 & 46 & 53 ||35.294||705.882||388.235||952.941<br />
|-<br />
|29||41 & 46 & 58 ||41.379||703.448||372.414||951.724<br />
|}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity/Patent_vals&diff=5568Hemifamity/Patent vals2026-04-03T23:12:28Z<p>Inthar: </p>
<hr />
<div>The following patent vals support 2.3.5.7 [[Hemifamity]] (aka Aberschismic). Vals that are contorted in 2.3.5.7 are not included.<br />
<br />
Note on 11-limit extensions: 41 & 46 & 53 is called Akea (tempers out 385/384), and 41 & 46 & 58 is called Pele (tempers out 896/891). The intersection of the two is Rodan (41 & 46).<br />
{| class="wikitable sortable"<br />
|-<br />
!Edo!!Extension to 11!!Extension to 13!!81/80 tuning!!3/2 tuning||5/4 tuning!!7/4 tuning<br />
|-<br />
|7||41 & 46 & 53 ||41 & 46 & 53 ||0.000||685.714||342.857||1028.571<br />
|-<br />
|12||41 & 46 & 58 ||41 & 46 & 53 ||0.000||700.000||400.000||1000.000<br />
|-<br />
|5||41 & 46 & 53, 41 & 46 & 58 || ||0.000||720.000||480.000||960.000<br />
|-<br />
|70||||||17.143||702.857||394.286||977.143<br />
|-<br />
|58||41 & 46 & 58 ||20.690||703.448||393.103||972.414<br />
|-<br />
|111||||||21.622||702.703||389.189||972.973<br />
|-<br />
|53||41 & 46 & 53 ||41 & 46 & 53 ||22.642||701.887||384.906||973.585<br />
|-<br />
|210||||||22.857||702.857||388.571||971.429<br />
|-<br />
|157||||||22.930||703.185||389.809||970.701<br />
|-<br />
|205||||||23.415||702.439||386.341||971.707<br />
|-<br />
|152||||||41 & 46 & 53 ||23.684||702.632||386.842||971.053<br />
|-<br />
|251||||||23.904||702.789||387.251||970.518<br />
|-<br />
|350||||||24.000||702.857||387.429||970.286<br />
|-<br />
|449||||||24.053||702.895||387.528||970.156<br />
|-<br />
|99||41 & 46 & 53 ||41 & 46 & 53 ||24.242||703.030||387.879||969.697<br />
|-<br />
|198||||||24.242||703.030||387.879||969.697<br />
|-<br />
|297||||||24.242||703.030||387.879||969.697<br />
|-<br />
|345||||||24.348||702.609||386.087||970.435<br />
|-<br />
|246||||||41 & 46 & 53 ||24.390||702.439||385.366||970.732<br />
|-<br />
|589||||||24.448||702.886||387.097||969.779<br />
|-<br />
|147||||||41 & 46 & 53 ||24.490||702.041||383.673||971.429<br />
|-<br />
|490||||||24.490||702.857||386.939||969.796<br />
|-<br />
|391||||||24.552||702.813||386.701||969.821<br />
|-<br />
|244||||||24.590||703.279||388.525||968.852<br />
|-<br />
|536||||||24.627||702.985||387.313||969.403<br />
|-<br />
|292||||||24.658||702.740||386.301||969.863<br />
|-<br />
|340||||||41 & 46 & 53 ||24.706||702.353||384.706||970.588<br />
|-<br />
|437||||||24.714||702.975||387.185||969.336<br />
|-<br />
|485||||||24.742||702.680||385.979||969.897<br />
|-<br />
|630||||||24.762||702.857||386.667||969.524<br />
|-<br />
|145||41 & 46 & 58 ||24.828||703.448||388.966||968.276<br />
|-<br />
|338||||||24.852||702.959||386.982||969.231<br />
|-<br />
|531||||||24.859||702.825||386.441||969.492<br />
|-<br />
|724||||||24.862||702.762||386.188||969.613<br />
|-<br />
|193||||||41 & 46 & 53 ||24.870||702.591||385.492||969.948<br />
|-<br />
|386||||||24.870||702.591||385.492||969.948<br />
|-<br />
|577||||||24.957||702.946||386.828||969.151<br />
|-<br />
|625||||||24.960||702.720||385.920||969.600<br />
|-<br />
|48||41 & 46 & 53 ||41 & 46 & 53 ||25.000||700.000||375.000||975.000<br />
|-<br />
|432||||||25.000||702.778||386.111||969.444<br />
|-<br />
|384||||||25.000||703.125||387.500||968.750<br />
|-<br />
|671||||||25.037||702.832||386.289||969.300<br />
|-<br />
|287||||||41 & 46 & 53 ||25.087||702.439||384.669||970.035<br />
|-<br />
|526||||||41 & 46 & 53 ||25.095||702.662||385.551||969.582<br />
|-<br />
|239||||||41 & 46 & 53 ||25.105||702.929||386.611||969.038<br />
|-<br />
|572||||||25.175||702.797||386.014||969.231<br />
|-<br />
|524||||||25.191||703.053||387.023||968.702<br />
|-<br />
|333||||||41 & 46 & 53 ||25.225||702.703||385.586||969.369<br />
|-<br />
|618||||||25.243||702.913||386.408||968.932<br />
|-<br />
|285||||||25.263||703.158||387.368||968.421<br />
|-<br />
|427||||||41 & 46 & 53 ||25.293||702.576||385.012||969.555<br />
|-<br />
|379||||||41 & 46 & 53 ||25.330||702.902||386.280||968.865<br />
|-<br />
|473||||||41 & 46 & 53 ||25.370||702.748||385.624||969.133<br />
|-<br />
|331||||||25.378||703.323||387.915||967.976<br />
|-<br />
|425||||||25.412||703.059||386.824||968.471<br />
|-<br />
|519||||||25.434||702.890||386.127||968.786<br />
|-<br />
|613||||||41 & 46 & 53 ||25.449||702.773||385.644||969.005<br />
|-<br />
|471||||||25.478||703.185||387.261||968.153<br />
|-<br />
|565||||||25.487||703.009||386.549||968.496<br />
|-<br />
|94||41 & 46 & 53 ||41 & 46 & 53 ||25.532||702.128||382.979||970.213<br />
|-<br />
|234||||||41 & 46 & 53 ||25.641||702.564||384.615||969.231<br />
|-<br />
|374||||||41 & 46 & 53 ||25.668||702.674||385.027||968.984<br />
|-<br />
|514||||||41 & 46 & 53 ||25.681||702.724||385.214||968.872<br />
|-<br />
|140||41 & 46 & 53 ||41 & 46 & 53 ||25.714||702.857||385.714||968.571<br />
|-<br />
|466||||||41 & 46 & 53 ||25.751||703.004||386.266||968.240<br />
|-<br />
|326||||||41 & 46 & 53 ||25.767||703.067||386.503||968.098<br />
|-<br />
|512||||||25.781||703.125||386.719||967.969<br />
|-<br />
|186||41 & 46 & 53 ||41 & 46 & 53 ||25.806||703.226||387.097||967.742<br />
|-<br />
|418||||||25.837||703.349||387.560||967.464<br />
|-<br />
|232||41 & 46 & 58 ||25.862||703.448||387.931||967.241<br />
|-<br />
|601||||||41 & 46 & 53 ||25.957||702.829||385.358||968.386<br />
|-<br />
|461||||||41 & 46 & 53 ||26.030||702.820||385.249||968.330<br />
|-<br />
|507||||||41 & 46 & 53 ||26.036||702.959||385.799||968.047<br />
|-<br />
|46||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||26.087||704.348||391.304||965.217<br />
|-<br />
|413||||||41 & 46 & 53 ||26.150||703.148||386.441||967.554<br />
|-<br />
|367||||||41 & 46 & 53 ||26.158||702.997||385.831||967.847<br />
|-<br />
|321||41 & 46 & 53 ||41 & 46 & 53 ||26.168||702.804||385.047||968.224<br />
|-<br />
|548||||||41 & 46 & 53 ||26.277||702.920||385.401||967.883<br />
|-<br />
|273||41 & 46 & 53 ||41 & 46 & 53 ||26.374||703.297||386.813||967.033<br />
|-<br />
|227||41 & 46 & 53 ||41 & 46 & 53 ||26.432||703.084||385.903||967.401<br />
|-<br />
|408||41 & 46 & 53 ||41 & 46 & 53 ||26.471||702.941||385.294||967.647<br />
|-<br />
|181||41 & 46 & 53 ||41 & 46 & 53 ||26.519||702.762||384.530||967.956<br />
|-<br />
|135||41 & 46 & 53 ||41 & 46 & 53 ||26.667||702.222||382.222||968.889<br />
|-<br />
|268||41 & 46 & 53 ||41 & 46 & 53 ||26.866||702.985||385.075||967.164<br />
|-<br />
|222||41 & 46 & 53 ||27.027||702.703||383.784||967.568<br />
|-<br />
|133||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.068||703.759||387.970||965.414<br />
|-<br />
|309||41 & 46 & 53 ||27.184||702.913||384.466||966.990<br />
|-<br />
|87||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.586||703.448||386.207||965.517<br />
|-<br />
|128||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||28.125||703.125||384.375||965.625<br />
|-<br />
|169||41 & 46 & 53, 41 & 46 & 58 ||28.402||702.959||383.432||965.680<br />
|-<br />
|41||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||29.268||702.439||380.488||965.854<br />
|-<br />
|39||41 & 46 & 53 ||41 & 46 & 53 ||30.769||707.692||400.000||953.846<br />
|-<br />
|34||41 & 46 & 53 ||41 & 46 & 53 ||35.294||705.882||388.235||952.941<br />
|-<br />
|29||41 & 46 & 58 ||41.379||703.448||372.414||951.724<br />
|}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity/Patent_vals&diff=5567Hemifamity/Patent vals2026-04-03T23:11:27Z<p>Inthar: Add 13 extension</p>
<hr />
<div>The following patent vals support 2.3.5.7 [[Hemifamity]] (aka Aberschismic). Vals that are contorted in 2.3.5.7 are not included.<br />
<br />
Note on 11-limit extensions: 41 & 46 & 53 is called Akea (tempers out 385/384), and 41 & 46 & 58 is called Pele (tempers out 896/891). The intersection of the two is Rodan (41 & 46).<br />
{| class="wikitable sortable"<br />
|-<br />
!Edo!!Extension to 11!!Extension to 13!!81/80 tuning!!3/2 tuning||5/4 tuning!!7/4 tuning<br />
|-<br />
|7||41 & 46 & 53 ||41 & 46 & 53 ||0.000||685.714||342.857||1028.571<br />
|-<br />
|12||41 & 46 & 58 ||41 & 46 & 53 ||0.000||700.000||400.000||1000.000<br />
|-<br />
|5||41 & 46 & 53, 41 & 46 & 58 || ||0.000||720.000||480.000||960.000<br />
|-<br />
|70||||17.143||702.857||394.286||977.143<br />
|-<br />
|58||41 & 46 & 58 ||20.690||703.448||393.103||972.414<br />
|-<br />
|111||||21.622||702.703||389.189||972.973<br />
|-<br />
|53||41 & 46 & 53 ||41 & 46 & 53 ||22.642||701.887||384.906||973.585<br />
|-<br />
|210||||22.857||702.857||388.571||971.429<br />
|-<br />
|157||||22.930||703.185||389.809||970.701<br />
|-<br />
|205||||23.415||702.439||386.341||971.707<br />
|-<br />
|152||||41 & 46 & 53 ||23.684||702.632||386.842||971.053<br />
|-<br />
|251||||23.904||702.789||387.251||970.518<br />
|-<br />
|350||||24.000||702.857||387.429||970.286<br />
|-<br />
|449||||24.053||702.895||387.528||970.156<br />
|-<br />
|99||41 & 46 & 53 ||41 & 46 & 53 ||24.242||703.030||387.879||969.697<br />
|-<br />
|198||||24.242||703.030||387.879||969.697<br />
|-<br />
|297||||24.242||703.030||387.879||969.697<br />
|-<br />
|345||||24.348||702.609||386.087||970.435<br />
|-<br />
|246||||41 & 46 & 53 ||24.390||702.439||385.366||970.732<br />
|-<br />
|589||||24.448||702.886||387.097||969.779<br />
|-<br />
|147||||41 & 46 & 53 ||24.490||702.041||383.673||971.429<br />
|-<br />
|490||||24.490||702.857||386.939||969.796<br />
|-<br />
|391||||24.552||702.813||386.701||969.821<br />
|-<br />
|244||||24.590||703.279||388.525||968.852<br />
|-<br />
|536||||24.627||702.985||387.313||969.403<br />
|-<br />
|292||||24.658||702.740||386.301||969.863<br />
|-<br />
|340||||41 & 46 & 53 ||24.706||702.353||384.706||970.588<br />
|-<br />
|437||||24.714||702.975||387.185||969.336<br />
|-<br />
|485||||24.742||702.680||385.979||969.897<br />
|-<br />
|630||||24.762||702.857||386.667||969.524<br />
|-<br />
|145||41 & 46 & 58 ||24.828||703.448||388.966||968.276<br />
|-<br />
|338||||24.852||702.959||386.982||969.231<br />
|-<br />
|531||||24.859||702.825||386.441||969.492<br />
|-<br />
|724||||24.862||702.762||386.188||969.613<br />
|-<br />
|193||||41 & 46 & 53 ||24.870||702.591||385.492||969.948<br />
|-<br />
|386||||24.870||702.591||385.492||969.948<br />
|-<br />
|577||||24.957||702.946||386.828||969.151<br />
|-<br />
|625||||24.960||702.720||385.920||969.600<br />
|-<br />
|48||41 & 46 & 53 ||41 & 46 & 53 ||25.000||700.000||375.000||975.000<br />
|-<br />
|432||||25.000||702.778||386.111||969.444<br />
|-<br />
|384||||25.000||703.125||387.500||968.750<br />
|-<br />
|671||||25.037||702.832||386.289||969.300<br />
|-<br />
|287||||41 & 46 & 53 ||25.087||702.439||384.669||970.035<br />
|-<br />
|526||||41 & 46 & 53 ||25.095||702.662||385.551||969.582<br />
|-<br />
|239||||41 & 46 & 53 ||25.105||702.929||386.611||969.038<br />
|-<br />
|572||||25.175||702.797||386.014||969.231<br />
|-<br />
|524||||25.191||703.053||387.023||968.702<br />
|-<br />
|333||||41 & 46 & 53 ||25.225||702.703||385.586||969.369<br />
|-<br />
|618||||25.243||702.913||386.408||968.932<br />
|-<br />
|285||||25.263||703.158||387.368||968.421<br />
|-<br />
|427||||41 & 46 & 53 ||25.293||702.576||385.012||969.555<br />
|-<br />
|379||||41 & 46 & 53 ||25.330||702.902||386.280||968.865<br />
|-<br />
|473||||41 & 46 & 53 ||25.370||702.748||385.624||969.133<br />
|-<br />
|331||||25.378||703.323||387.915||967.976<br />
|-<br />
|425||||25.412||703.059||386.824||968.471<br />
|-<br />
|519||||25.434||702.890||386.127||968.786<br />
|-<br />
|613||||41 & 46 & 53 ||25.449||702.773||385.644||969.005<br />
|-<br />
|471||||25.478||703.185||387.261||968.153<br />
|-<br />
|565||||25.487||703.009||386.549||968.496<br />
|-<br />
|94||41 & 46 & 53 ||41 & 46 & 53 ||25.532||702.128||382.979||970.213<br />
|-<br />
|234||||41 & 46 & 53 ||25.641||702.564||384.615||969.231<br />
|-<br />
|374||||41 & 46 & 53 ||25.668||702.674||385.027||968.984<br />
|-<br />
|514||||41 & 46 & 53 ||25.681||702.724||385.214||968.872<br />
|-<br />
|140||41 & 46 & 53 ||41 & 46 & 53 ||25.714||702.857||385.714||968.571<br />
|-<br />
|466||||41 & 46 & 53 ||25.751||703.004||386.266||968.240<br />
|-<br />
|326||||41 & 46 & 53 ||25.767||703.067||386.503||968.098<br />
|-<br />
|512||||25.781||703.125||386.719||967.969<br />
|-<br />
|186||41 & 46 & 53 ||41 & 46 & 53 ||25.806||703.226||387.097||967.742<br />
|-<br />
|418||||25.837||703.349||387.560||967.464<br />
|-<br />
|232||41 & 46 & 58 ||25.862||703.448||387.931||967.241<br />
|-<br />
|601||||41 & 46 & 53 ||25.957||702.829||385.358||968.386<br />
|-<br />
|461||||41 & 46 & 53 ||26.030||702.820||385.249||968.330<br />
|-<br />
|507||||41 & 46 & 53 ||26.036||702.959||385.799||968.047<br />
|-<br />
|46||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||26.087||704.348||391.304||965.217<br />
|-<br />
|413||||41 & 46 & 53 ||26.150||703.148||386.441||967.554<br />
|-<br />
|367||||41 & 46 & 53 ||26.158||702.997||385.831||967.847<br />
|-<br />
|321||41 & 46 & 53 ||41 & 46 & 53 ||26.168||702.804||385.047||968.224<br />
|-<br />
|548||||41 & 46 & 53 ||26.277||702.920||385.401||967.883<br />
|-<br />
|273||41 & 46 & 53 ||41 & 46 & 53 ||26.374||703.297||386.813||967.033<br />
|-<br />
|227||41 & 46 & 53 ||41 & 46 & 53 ||26.432||703.084||385.903||967.401<br />
|-<br />
|408||41 & 46 & 53 ||41 & 46 & 53 ||26.471||702.941||385.294||967.647<br />
|-<br />
|181||41 & 46 & 53 ||41 & 46 & 53 ||26.519||702.762||384.530||967.956<br />
|-<br />
|135||41 & 46 & 53 ||41 & 46 & 53 ||26.667||702.222||382.222||968.889<br />
|-<br />
|268||41 & 46 & 53 ||41 & 46 & 53 ||26.866||702.985||385.075||967.164<br />
|-<br />
|222||41 & 46 & 53 ||27.027||702.703||383.784||967.568<br />
|-<br />
|133||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.068||703.759||387.970||965.414<br />
|-<br />
|309||41 & 46 & 53 ||27.184||702.913||384.466||966.990<br />
|-<br />
|87||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||27.586||703.448||386.207||965.517<br />
|-<br />
|128||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||28.125||703.125||384.375||965.625<br />
|-<br />
|169||41 & 46 & 53, 41 & 46 & 58 ||28.402||702.959||383.432||965.680<br />
|-<br />
|41||41 & 46 & 53, 41 & 46 & 58 ||41 & 46 & 53 ||29.268||702.439||380.488||965.854<br />
|-<br />
|39||41 & 46 & 53 ||41 & 46 & 53 ||30.769||707.692||400.000||953.846<br />
|-<br />
|34||41 & 46 & 53 ||41 & 46 & 53 ||35.294||705.882||388.235||952.941<br />
|-<br />
|29||41 & 46 & 58 ||41.379||703.448||372.414||951.724<br />
|}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5566Hemifamity2026-04-03T22:50:47Z<p>Inthar: /* Structural theory */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.<br />
<br />
== Structural theory ==<br />
=== Notation ===<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5565Hemifamity2026-04-03T22:50:18Z<p>Inthar: /* Names */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
The names ''Argent'' and ''Argentic'' come from the relationship to the silver ratio and the associated [[Golden_generator#Argent_tuning|argent tuning]] of MOS diatonic.<br />
<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Regular_temperament&diff=5564Regular temperament2026-04-03T22:45:47Z<p>Inthar: /* See also */</p>
<hr />
<div>A '''regular temperament''' is a [[temperament]] (an approximation to the harmonies of [[just intonation]]) that deforms JI intervals in a way consistent with stacking. That is, the (logarithmic) sum of two tempered intervals must always be the tempered version of the sum of the JI intervals. For example, in a regular temperament, if you stack the tempered version of [[9/8]], it must always produce the tempered versions of 81/64, 729/512, 6561/4096, etc. The tempered version of 6/5 always differs from the tempered version of 4/3 by the tempered version of 10/9, and so on. It follows from this that unlimited free modulation must be possible: any interval can be stacked as many times as you like, like in just intonation. In contrast, temperaments that do not meet this requirement (such as [[Irregular temperament#Well temperament|well temperaments]]) are considered ''irregular''. <br />
<br />
[[Equal temperament]]s are the regular temperament interpretations of [[EDO|equal tunings]], while at the other end any just intonation [[subgroup]] itself can also be considered a regular temperament of itself (such as [[Pythagorean tuning]]). <br />
<br />
In order to simplify JI, [[Comma|commas]] are tempered out: small (or not-so-small) differences between intervals that the regular temperament represents with the unison. For example, in [[Meantone]], [[5/4]] (the classical major third) and [[Diatonic major third|81/64]] (the pythagorean major third) are equated. Their difference in just intonation is the ratio [[81/80]], so we say that Meantone '''tempers out''' 81/80. As a consequence, due to the rules of a regular temperament, 6561/6400, 531441/512000, etc (ratios obtained by stacking 81/80) are also tempered out, but knowing that Meantone tempers out 81/80 is enough to determine this information, so it is usually left unstated.<br />
<br />
It turns out that tempering out a single comma reduces the dimensionality of an [[interval space]] by 1 (as long as that comma or any multiple of it can't be reached by stacking the other intervals you temper out); that is, if a JI subgroup can be reached by stacking any multiples of a minimum of three distinct intervals (such as 2, 3, and 5 in the [[5-limit]]), tempering out 1 comma leads to a system built out of two distinct intervals. This means that a [[rank-2 temperament]] (one with two generators, or a period and a generator) must always temper out ''p''-2 commas in a subgroup with ''p'' primes.<br />
<br />
{{adv|Mathematically, regular temperaments are commonly treated as ''abstract'' tunings where the intervals have no concrete cent values until such are assigned to the generators. This is because abstract regular temperaments, such as "Magic", are meant to abstract features that concrete regular temperaments, such as 7-limit 19edo and 7-limit 22edo, have in common; see the "Temperament joining" section.}}<br />
<br />
== Temperament joining ==<br />
Any two [[equal temperament]]s have one specific rank-2 temperament that they both support, which can be seen as "what those equal temperaments have in common". The temperament shared by ''x''-ET and ''y''-ET is denoted by ''x'' & ''y''. For example, [[Porcupine]] is 15 & 22. This means [[15edo|15-ET]] and [[22edo|22-ET]] (and their sum, [[37edo|37-ET]]) have the characteristics of Porcupine in common (note 15 & 37 or 22 & 37 are other valid denotations of Porcupine). And they do: [[6/5]] is tuned sharp, [[4/3]] and [[10/9]] are tuned flat, (4/3)^2 is [[7/4]], ([[11/10]])^3 is 4/3, the soft tuning of the [[onyx]] scale (1L 6s MOS pattern) is supported, etc. Moreover, it means that you can take it the other way: Porcupine can be ''defined'' not by what commas it tempers out, but as "what 15-ET and 22-ET have in common". Similarly, a rank-3 temperament may be defined by three equal temperaments, and so on.<br />
<br />
One slight subtlety is that we want to think of the ETs in this convention as bounding a range for a [[MOS]] shape formed by the temperament's generator. This means that, while 12 & 26 technically refers to Meantone in the 5-limit, an improper join of this sort should not generally be used, as the 12L 14s MOS pattern is not a scale of Meantone. Note that in the [[7-limit]], the 12 & 26 temperament is a [[Extension|weak extension]] of Meantone which does produce this MOS structure.<br />
<br />
Note that this presents a currently rather idiosyncratic way of conceptualizing higher-rank temperaments as, analogously to equal temperaments, arbitrary in [[subgroup]].<br />
<br />
== See also ==<br />
* [[List of regular temperaments]]<br />
<br />
{{Navbox regtemp}}<br />
{{Cat|Temperaments{{!}}*}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Argentic&diff=5563Argentic2026-04-03T22:42:47Z<p>Inthar: Redirected page to Hemifamity</p>
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<div>#redirect [[Hemifamity]]</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5562Hemifamity2026-04-03T22:42:28Z<p>Inthar: /* Name */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].<br />
== Names ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5561Hemifamity2026-04-03T22:42:21Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''' or '''Argentic''', 2.3.7/5[5120/5103].<br />
== Name ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Argent&diff=5560Argent2026-04-03T22:41:06Z<p>Inthar: Redirected page to Hemifamity</p>
<hr />
<div>#redirect [[Hemifamity]]</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5559Hemifamity2026-04-03T22:40:55Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argent''', 2.3.7/5[5120/5103].<br />
== Name ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5558Hemifamity2026-04-03T22:40:09Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
<br />
Restricted to the subgroup 2.3.7/5, it becomes the rank-2 temperament '''Argentismic''', 2.3.7/5[5120/5103].<br />
== Name ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5557Hemifamity2026-04-03T22:39:11Z<p>Inthar: /* Etymology */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Name ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamiton&diff=5556Hemifamiton2026-04-03T22:37:12Z<p>Inthar: Redirected page to Hemifamity</p>
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<div>#redirect [[Hemifamity]]</div>Intharhttps://xenreference.com/wiki/index.php?title=Aberschisma&diff=5555Aberschisma2026-04-03T22:36:47Z<p>Inthar: Redirected page to Hemifamity</p>
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<div>#redirect [[Hemifamity]]</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5554Hemifamity2026-04-03T22:36:33Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out '''5120/5103''' (known as the '''hemifamiton''' or the '''aberschisma'''), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Etymology ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=5120/5103&diff=55535120/51032026-04-03T22:36:15Z<p>Inthar: Redirected page to Hemifamity</p>
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<div>#redirect [[Hemifamity]]</div>Intharhttps://xenreference.com/wiki/index.php?title=Tetracot_(temperament)&diff=5552Tetracot (temperament)2026-04-03T22:36:06Z<p>Inthar: /* Extensions */</p>
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<div>{{Infobox regtemp<br />
| Title = Tetracot<br />
| Subgroups = 2.3.5<br />
| Comma basis = [[20000/19683]] (2.3.5)<br />
| Edo join 1 = 27 | Edo join 2 = 34<br />
| Mapping = 1; 4 9<br />
| Generators = 10/9<br />
| Generators tuning = 176.1<br />
| Optimization method = CWE<br />
| MOS scales = [[6L&nbsp;1s]], [[7L&nbsp;6s]], [[7L&nbsp;13s]]<br />
| Odd limit 1 = 5 | Mistuning 1 = 3.07 | Complexity 1 = 13<br />
}}<br />
<br />
'''Tetracot''', [27 & 34] or [34 & 41], is a temperament that splits 3/2 into four flattened 10/9's.<br />
<br />
== Interval chain ==<br />
In the following table, odd harmonics and subharmonics 1–15 are in '''bold'''.<br />
{| class="wikitable right-1 right-2"<br />
|-<br />
! #<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 176.3<br />
| 10/9<br />
|-<br />
| 2<br />
| 352.5<br />
| <br />
|-<br />
| 3<br />
| 528.8<br />
| 27/20<br />
|-<br />
| 4<br />
| 705.0<br />
| '''3/2'''<br />
|-<br />
| 5<br />
| 881.3<br />
| 5/3<br />
|-<br />
| 6<br />
| 1057.5<br />
| <br />
|-<br />
| 7<br />
| 33.8<br />
| 81/80<br />
|-<br />
| 8<br />
| 210.1<br />
| '''9/8'''<br />
|-<br />
| 9<br />
| 386.3<br />
| '''5/4'''<br />
|-<br />
| 10<br />
| 562.6<br />
|<br />
|-<br />
| 11<br />
| 738.8<br />
|<br />
|-<br />
| 12<br />
| 915.1<br />
| 27/16<br />
|-<br />
| 13<br />
| 1091.3<br />
| '''15/8'''<br />
|}<br />
<nowiki/>* in exact-5/2 tuning<br />
<br />
== Extensions ==<br />
Tetracot has a number of strong extensions, but most of them are problematic in some way. This is because the Tetracot generator is, optimally, approximately 31/28 — not easily interpretable as LCJI.<br />
* Prime 13 can be added by equating (10/9)^2 (the neutral third) with 16/13. Note that this favors a sharp 3/2 (optimally around 3.2c sharp) and a sharp 13/8 (optimally around 6.9c sharp).<br />
* Prime 11 is often added by equating 10/9 with 11/10 (thus placing 11/8 at +10 generators), but this is questionable because it produces either a very sharp 11/8 (as in 27edo and 34edo) or a flat 5/4 (as in 41edo and 48edo). An alternate extension (27p & 34), associated with 7-limit Wollemia, places 11/8 at -24 generators.<br />
* There isn't a canonical way to add prime 7. This is because 27edo and 41edo have good 7 approximations but 34edo does not. There are no less than 4 strong extensions to 2.3.5.7: Bunya (34d & 41), Monkey (34 & 41 or 41 & 48), Modus (27 & 34d), and Wollemia (27 & 34).<br />
** Moneky is notable because it's the extension tempering out [[5120/5103]], the aberschisma.<br />
** The weak extension [[Octacot]] (27 & 41) is more elegant; it splits the Tetracot generator into two semitones (about 88.1c) representing 21/20, thus equating three Octacot generators with 7/6 (and 11 of them with 7/4). Octacot can be extended to have prime 19 (at 17 generators) by equating 21/20 to 20/19 (equivalently, 10/9 to 21/19 or 27/20 to 19/14).<br />
<br />
== List of patent vals ==<br />
The following patent vals support 2.3.5 Tetracot. Contorted vals are not included.<br />
{| class="wikitable sortable"<br />
!Edo!!Generator tuning!!Fifth tuning<br />
|-<br />
||7||171.429||685.714<br />
|-<br />
||48||175.000||700.000<br />
|-<br />
||41||175.610||702.439<br />
|-<br />
||116||175.862||703.448<br />
|-<br />
||191||175.916||703.665<br />
|-<br />
||75||176.000||704.000<br />
|-<br />
||259||176.062||704.247<br />
|-<br />
||184||176.087||704.348<br />
|-<br />
||109||176.147||704.587<br />
|-<br />
||143||176.224||704.895<br />
|-<br />
||177||176.271||705.085<br />
|-<br />
||34||176.471||705.882<br />
|-<br />
||95||176.842||707.368<br />
|-<br />
||61||177.049||708.197<br />
|-<br />
||27||177.778||711.111<br />
|}<br />
{{Navbox regtemp}}<br />
{{Cat|temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5551Hemifamity2026-04-03T22:30:32Z<p>Inthar: /* Etymology */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Etymology ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
The older name ''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', rank-2 temperaments which both support this rank-3 temperament.<br />
<br />
The newer name ''Aberschismic'', coined by Tristan Bay, comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5550Hemifamity2026-04-03T22:30:00Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Etymology ==<br />
As of early 2026, the temperament has no single agreed-upon name.<br />
<br />
''Hemifamity'' is a portmanteau of ''Hemififths'' and ''Amity'', which both support this rank-3 temperament.<br />
<br />
The name ''Aberschismic'' comes from the fact that tempering out 5120/5103 manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Neutral_temperaments&diff=5549Neutral temperaments2026-04-03T22:28:01Z<p>Inthar: /* Rastmatic */</p>
<hr />
<div>'''Neutral temperaments''' are any temperaments represented by the edo join 7 & 10, or any reasonable extension of such a temperament, such that the generator is a neutral third of some kind which splits 3/2 into two. They are a subset of and largely cover the ''dicot'' temperament archetype, and impose upon it the condition that the neutral third must be mapped to 2\7 and 3\10. The two most well-known neutral temperaments are the 2.3.11 (Rastmatic) and 2.3.5 (Dicot) versions.<br />
<br />
10edo is a contorted 5edo in 2.3.7, hence 7 & 10 in that subgroup represents monocot [[Archy]] temperament.<br />
<br />
== Notation ==<br />
Neutral temperaments may be notated with neutral chain-of-fifths notation. <br />
{| class="wikitable"<br />
|+<br />
!Note<br />
!24edo<br />
!Notation<br />
!2.3...<br />
!5 (Dicot)<br />
!11 (Rastmatic)<br />
!13 (Namo)<br />
!13-limit (no-fives)<br />
!13-limit<br />
|-<br />
|A<br />
|0<br />
|P1<br />
|1/1<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|At<br />
|50<br />
|sA1<br />
|<br />
|81/80<br />
|33/32<br />
|<br />
|<br />
|<br />
|-<br />
|Bb<br />
|100<br />
|m2<br />
|256/243<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Bd<br />
|150<br />
|n2<br />
|<br />
|10/9, 16/15<br />
|12/11<br />
|<br />
|<br />
|<br />
|-<br />
|B<br />
|200<br />
|M2<br />
|9/8<br />
|<br />
|<br />
|<br />
|8/7<br />
|11/10<br />
|-<br />
|C<br />
|300<br />
|m3<br />
|32/27<br />
|<br />
|<br />
|<br />
|7/6<br />
|<br />
|-<br />
|Ct<br />
|350<br />
|n3<br />
|<br />
|5/4, 6/5<br />
|11/9<br />
|16/13<br />
|<br />
|<br />
|-<br />
|C#<br />
|400<br />
|M3<br />
|81/64<br />
|<br />
|<br />
|<br />
|9/7<br />
|<br />
|-<br />
|Dd<br />
|450<br />
|sd4<br />
|<br />
|<br />
|<br />
|<br />
|14/11<br />
|<br />
|-<br />
|D<br />
|500<br />
|P4<br />
|4/3<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Dt<br />
|550<br />
|sA4<br />
|<br />
|27/20<br />
|11/8<br />
|18/13<br />
|<br />
|10/7<br />
|-<br />
|Ed<br />
|650<br />
|sd5<br />
|<br />
|40/27<br />
|16/11<br />
|13/9<br />
|<br />
|7/5<br />
|-<br />
|E<br />
|700<br />
|P5<br />
|3/2<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Et<br />
|750<br />
|sA5<br />
|<br />
|<br />
|<br />
|<br />
|11/7<br />
|<br />
|-<br />
|F<br />
|800<br />
|m6<br />
|128/81<br />
|<br />
|<br />
|<br />
|14/9<br />
|<br />
|-<br />
|Ft<br />
|850<br />
|n6<br />
|<br />
|5/3, 8/5<br />
|18/11<br />
|13/8<br />
|<br />
|<br />
|-<br />
|F#<br />
|900<br />
|M6<br />
|27/16<br />
|<br />
|<br />
|<br />
|12/7<br />
|<br />
|-<br />
|G<br />
|1000<br />
|m7<br />
|16/9<br />
|<br />
|<br />
|<br />
|7/4<br />
|20/11<br />
|-<br />
|Gt<br />
|1050<br />
|n7<br />
|<br />
|15/8, 9/5<br />
|11/6<br />
|<br />
|<br />
|<br />
|-<br />
|G#<br />
|1100<br />
|M7<br />
|243/128<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Ad<br />
|1150<br />
|sd8<br />
|<br />
|160/81<br />
|64/33<br />
|<br />
|<br />
|<br />
|-<br />
|A<br />
|1200<br />
|P8<br />
|2/1<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|}<br />
These intervals may additionally be arranged on a chart which explains their mappings to 7edo and 10edo:<br />
{| class="wikitable"<br />
! !! 0\7 !! 1\7 !! 2\7 !! 3\7 !! 4\7 !! 5\7 !! 6\7 !! 7\7<br />
|-<br />
! scope="row" | 0\10 <br />
| '''P1''' || m2|| d3|| || || || ||<br />
|-<br />
! scope="row" | 1\10 <br />
| sA1|| '''n2''' || sd3|| || || || ||<br />
|-<br />
! scope="row" | 2\10 <br />
| A1|| M2 || m3 || d4|| || || ||<br />
|-<br />
! scope="row" | 3\10 <br />
| || sA2|| '''n3''' || sd4 || || || ||<br />
|-<br />
! scope="row" | 4\10 <br />
| || A2|| M3 || '''P4''' || d5|| d6|| ||<br />
|-<br />
! scope="row" | 5\10 <br />
| || || sA3|| sA4 || sd5 || sd6|| ||<br />
|-<br />
! scope="row" | 6\10 <br />
| || || A3|| A4|| '''P5''' || m6 || d7||<br />
|-<br />
! scope="row" | 7\10 <br />
| || || || || sA5 || '''n6''' || sd7||<br />
|-<br />
! scope="row" | 8\10 <br />
| || || || || A5|| M6 || m7 ||d8<br />
|-<br />
! scope="row" | 9\10 <br />
| || || || || || sA6|| '''n7''' ||sd8<br />
|-<br />
! scope="row" | 10\10 <br />
| || || || || || A6|| M7|| '''P8'''<br />
|}<br />
<br />
== Rastmatic ==<br />
Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22.<br />
<br />
=== Etymology ===<br />
Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''Rast'' which utilizes a scale with several neutral intervals.<br />
<br />
== Hemififths ==<br />
Hemififths, 41 & 58, is the neutral temperament in the 2.3.5.7 subgroup, which equates 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament.<br />
<br />
== Dicot ==<br />
Dicot<sup>[a]</sup>, not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out [[25/24]], the Dicot comma. The provided [[edo join]] also tempers out [[45/44]] and [[64/63]] in the 11-limit, representing the extension '''Dichotic''' and also tempering out [[55/54]]. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.<br />
<br />
Dicot makes 4:5:6 equidistant, suggesting the simplified structure of [[tertian]] harmony, the same way [[Semaphore]] does for [[chthonic harmony]]. As a result, the temperament archetype [[Neutral third scales|dicot]] is named after it.<br />
<br />
=== Etymology ===<br />
Dicot originates from the term "dicot" in botany, referring to plants with two embryonic leaves, perhaps by analogy with 3/2 being split into two generators. The name ''Dicot'' would also inspire [[Tetracot]], [[Alphatricot]], and by extension the [[ploidacot]] temperament archetype naming system as a whole.<br />
<br />
=== Tuning considerations ===<br />
A perfect ~351c tuning of the generator, while useful for understanding tertian harmony and suggested by some temperament tuning optimization systems, does not reasonably approximate either 5/4 or 6/5. The optimal tunings of Dicot are roughly bimodal, with ~360c (around 10edo) and ~343c (around 7edo) both being better tunings.<br />
<br />
== Namo ==<br />
''Namo'', ''Intertridecimal'', or ''Harmoneutral'' is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth.<br />
<br />
It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c.<br />
<br />
== Patent vals ==<br />
<br />
=== List of patent vals ===<br />
{| class="wikitable"<br />
|+<br />
!EDO<br />
!Mappings supported<br />
!Generator tuning<br />
!3/2 tuning<br />
|-<br />
|10<br />
|5, 13, 11<br />
|360.0c<br />
|720.0c<br />
|-<br />
|37<br />
|13<br />
|356.8c<br />
|713.5c<br />
|-<br />
|27<br />
|13<br />
|355.6c<br />
|711.1c<br />
|-<br />
|71<br />
|13<br />
|354.9c<br />
|709.9c<br />
|-<br />
|44<br />
|13<br />
|354.5c<br />
|709.1c<br />
|-<br />
|61<br />
|13<br />
|354.1c<br />
|708.2c<br />
|-<br />
|78<br />
|13<br />
|353.8c<br />
|707.7c<br />
|-<br />
|95<br />
|13<br />
|353.7c<br />
|707.4c<br />
|-<br />
|17<br />
|5, 13, 11<br />
|352.9c<br />
|705.9c<br />
|-<br />
|75<br />
|13<br />
|352.0c<br />
|704.0c<br />
|-<br />
|58<br />
|13, 11<br />
|351.7c<br />
|703.4c<br />
|-<br />
|41<br />
|13, 11<br />
|351.2c<br />
|702.4c<br />
|-<br />
|147<br />
|11<br />
|351.0c<br />
|702.0c<br />
|-<br />
|106<br />
|11<br />
|350.9c<br />
|701.9c<br />
|-<br />
|171<br />
|11<br />
|350.9c<br />
|701.8c<br />
|-<br />
|65<br />
|13, 11<br />
|350.8c<br />
|701.5c<br />
|-<br />
|219<br />
|11<br />
|350.7c<br />
|701.4c<br />
|-<br />
|154<br />
|11<br />
|350.6c<br />
|701.3c<br />
|-<br />
|243<br />
|11<br />
|350.62c<br />
|701.23c<br />
|-<br />
|332<br />
|11<br />
|350.60c<br />
|701.20c<br />
|-<br />
|89<br />
|11<br />
|350.56c<br />
|701.12c<br />
|-<br />
|380<br />
|11<br />
|350.53c<br />
|701.05c<br />
|-<br />
|291<br />
|11<br />
|350.52c<br />
|701.03c<br />
|-<br />
|202<br />
|11<br />
|350.50c<br />
|700.99c<br />
|-<br />
|517<br />
|11<br />
|350.48c<br />
|700.97c<br />
|-<br />
|315<br />
|11<br />
|350.48c<br />
|700.95c<br />
|-<br />
|428<br />
|11<br />
|350.47c<br />
|700.93c<br />
|-<br />
|541<br />
|11<br />
|350.46c<br />
|700.92c<br />
|-<br />
|113<br />
|11<br />
|350.44c<br />
|700.88c<br />
|-<br />
|476<br />
|11<br />
|350.42c<br />
|700.84c<br />
|-<br />
|363<br />
|11<br />
|350.41c<br />
|700.83c<br />
|-<br />
|250<br />
|11<br />
|350.40c<br />
|700.80c<br />
|-<br />
|387<br />
|11<br />
|350.39c<br />
|700.78c<br />
|-<br />
|137<br />
|11<br />
|350.36c<br />
|700.73c<br />
|-<br />
|435<br />
|11<br />
|350.34c<br />
|700.69c<br />
|-<br />
|298<br />
|11<br />
|350.34c<br />
|700.67c<br />
|-<br />
|459<br />
|11<br />
|350.33c<br />
|700.65c<br />
|-<br />
|161<br />
|11<br />
|350.31c<br />
|700.62c<br />
|-<br />
|346<br />
|11<br />
|350.29c<br />
|700.58c<br />
|-<br />
|185<br />
|11<br />
|350.27c<br />
|700.54c<br />
|-<br />
|394<br />
|11<br />
|350.25c<br />
|700.51c<br />
|-<br />
|209<br />
|11<br />
|350.24c<br />
|700.48c<br />
|-<br />
|233<br />
|11<br />
|350.21c<br />
|700.43c<br />
|-<br />
|257<br />
|11<br />
|350.19c<br />
|700.39c<br />
|-<br />
|281<br />
|11<br />
|350.18c<br />
|700.36c<br />
|-<br />
|305<br />
|11<br />
|350.16c<br />
|700.33c<br />
|-<br />
|329<br />
|11<br />
|350.15c<br />
|700.30c<br />
|-<br />
|353<br />
|11<br />
|350.14c<br />
|700.28c<br />
|-<br />
|24<br />
|13, 11<br />
|350.00c<br />
|700.00c<br />
|-<br />
|247<br />
|11<br />
|349.80c<br />
|699.60c<br />
|-<br />
|223<br />
|11<br />
|349.78c<br />
|699.55c<br />
|-<br />
|199<br />
|11<br />
|349.7c<br />
|699.5c<br />
|-<br />
|175<br />
|11<br />
|349.7c<br />
|699.4c<br />
|-<br />
|151<br />
|11<br />
|349.7c<br />
|699.3c<br />
|-<br />
|127<br />
|11<br />
|349.6c<br />
|699.2c<br />
|-<br />
|103<br />
|11<br />
|349.5c<br />
|699.0c<br />
|-<br />
|79<br />
|11<br />
|349.4c<br />
|698.7c<br />
|-<br />
|55<br />
|13, 11<br />
|349.1c<br />
|698.2c<br />
|-<br />
|31<br />
|13, 11<br />
|348.4c<br />
|696.8c<br />
|-<br />
|38<br />
|13, 11<br />
|347.4c<br />
|694.7c<br />
|-<br />
|45<br />
|13<br />
|346.7c<br />
|693.3c<br />
|-<br />
|7<br />
|5, 13, 11<br />
|342.9c<br />
|685.7c<br />
|}<br />
<br />
== Footnotes ==<br />
[a] The name ''Interpental'' has been proposed, however it currently is used by 43 & 53, a weak extension of [[Buzzard]].<br />
<br />
{{Navbox regtemp}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemififths&diff=5548Hemififths2026-04-03T22:26:49Z<p>Inthar: Redirected page to Neutral temperaments</p>
<hr />
<div>#redirect [[Neutral temperaments]]</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5547Hemifamity2026-04-03T22:26:03Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
It manifests in [[aberrismic theory]] where the 7-limit interpretations for chromedye (5L2m5s) and whitedye (5L2m7s) interpret the aberrisma (the s step) as both 81/80 and 64/63.<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5546Hemifamity2026-04-03T22:22:28Z<p>Inthar: /* Supporting rank-2 temperaments */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot (temperament)#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5545Hemifamity2026-04-03T22:22:16Z<p>Inthar: /* Supporting rank-2 temperaments */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 48]: [[Tetracot#Extensions|Monkey]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5544Hemifamity2026-04-03T22:21:14Z<p>Inthar: /* Supporting edos */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
The following edos map both 81/80 and 64/63 to one edostep: [[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]].<br />
=== Half-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to two edosteps: [[87edo]], [[94edo]], [[99edo]], [[111edo]].<br />
=== Third-comma resolution ===<br />
The following edos map both 81/80 and 64/63 to three edosteps: [[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]].<br />
<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5543Hemifamity2026-04-03T22:20:36Z<p>Inthar: /* Comma-resolution */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma resolution ===<br />
[[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]]<br />
<br />
=== Half-comma resolution ===<br />
[[87edo]], [[94edo]], [[99edo]], [[111edo]]<br />
=== Third-comma resolution ===<br />
[[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]]<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5542Hemifamity2026-04-03T22:20:24Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Structural theory ==<br />
Aberschismic may be notated with diatonic notation and a comma symbol representing 81/80 ~ 64/63.<br />
== Supporting edos ==<br />
=== Comma-resolution ===<br />
[[41edo]], [[46edo]], [[48edo]], [[53edo]], [[58edo]]<br />
=== Half-comma resolution ===<br />
[[87edo]], [[94edo]], [[99edo]], [[111edo]]<br />
=== Third-comma resolution ===<br />
[[128edo]], [[133edo]], [[135edo]], [[140edo]], [[145edo]], [[147edo]], [[152edo]], [[157edo]]<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5541Hemifamity2026-04-03T22:12:41Z<p>Inthar: /* Extensions */</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095 (which is the square-superparticular S64); this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5540Hemifamity2026-04-03T22:11:54Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095; this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
== List of patent vals ==<br />
:''Main article: [[Hemifamity/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5539Hemifamity2026-04-03T22:10:54Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
# splits 36/35 into two equal parts each representing the "comma" 81/80 ~ 64/63<br />
== Supporting rank-2 temperaments ==<br />
* 2.3.5.7[41 & 46]: [[Rodan]]<br />
* 2.3.5.7[41 & 53]: [[Garibaldi]]<br />
* 2.3.5.7[41 & 58]: [[Hemififths]]<br />
* 2.3.5.7[46 & 53]: [[Amity]]<br />
* 2.3.5.7[46 & 58]: Septimal [[Diaschismic]]<br />
* 2.3.5.7[53 & 58]: [[Buzzard]]<br />
* 2.3.5.7[87 & 99]: [[Misty]]<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele).<br />
* Prime 13 may be added by tempering out 4096/4095; this is the most efficient extension that adds prime 13 and is supported by edos as big as [[140edo]].<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5538Hemifamity2026-04-03T22:07:08Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
<br />
== Extensions ==<br />
* Prime 11 may be added by tempering out either 385/384 (thus equating 36/35 with 33/32; this is called Akea) or 896/891 (thus equating 81/64 with 14/11; this is called Pele)<br />
* Prime 13 may be added by tempering out 4096/4095<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5537Hemifamity2026-04-03T21:56:18Z<p>Inthar: </p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7&lbrack;[[41edo|41]] & [[46edo|46]] & [[53edo|53]]&rbrack;, is a [[7-limit]] rank-3 temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Intharhttps://xenreference.com/wiki/index.php?title=Hemifamity&diff=5536Hemifamity2026-04-03T21:54:53Z<p>Inthar: Created page with "'''Hemifamity''' or '''Aberschismic''', 2.3.5.7[41 & 46 & 53], is a 7-limit temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways: # equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63 # equates the limma 256/243 with 21/20 # equates the apotome 2187/2048 with 15/14 # equates the Pythagorean augmented fourth 729/512 with 10/7 {{navbox regtemp}} {{Cat|Temperaments}}"</p>
<hr />
<div>'''Hemifamity''' or '''Aberschismic''', 2.3.5.7[41 & 46 & 53], is a 7-limit temperament that tempers out 5120/5103 (known as the hemifamiton or the aberschisma), which manifests in several different ways:<br />
# equates the 3-spine prime commas for 5 and 7, 81/80 and 64/63<br />
# equates the limma 256/243 with 21/20<br />
# equates the apotome 2187/2048 with 15/14<br />
# equates the Pythagorean augmented fourth 729/512 with 10/7<br />
<br />
{{navbox regtemp}}<br />
{{Cat|Temperaments}}</div>Inthar