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Schismic

2026-09-21T15:45:41Z

FloraC: Cleanup on infobox; + category

{{Infobox regtemp
| Mapping = 1; 1 -8 -3
| Subgroups = 2.3.5, 2.3.5.19
| Edo join 1 = 12 | Edo join 2 = 29
| Generators = 3/2 | Generators tuning = 701.7 | Optimization method = CWE
| MOS scales = [[5L 2s]], [[5L 7s]], [[12L 5s]]
| Comma basis = 32805/32768 (2.3.5); <br> 361/360, 513/512 (2.3.5.19)
| Odd limit 1 = 5 | Complexity 1 = 12 | Mistuning 1 = 0.217
}}
[[File:Schismic17.png|thumb|Schismic 17-note MOS. This is a sharp tuning; the Pythagorean comma has been inflated to function as an [[aberrisma]].]]
'''Schismic''', '''Schismatic'''<sup>[a]</sup>, or '''Helmholtz''' [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.

The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out [[513/512]], which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma; [[8/5]] is split into two, with 24/19 and 19/15 being equated, tempering out 361/360. 2.3.5.19 Schismic may be called ''Nestoria''.

While Schismic can be considered a 12-form temperament, the primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]]. Functionally reasonable tunings of Schismic[17] lie broadly between 29 and 53edo and include argent tuning on the sharper end (e.g. 70edo with the non-patent flat (~377c) mapping of 5/4) and Pythagorean tuning on the flatter end.

48edo and 58edo are the first two edos to observe the schisma - that is, to not support Schismic - while tuning the fifth within the aforementioned Schismic tuning range.

== Extensions ==

=== Prime 7 ===
The most important 7-limit extension of Schismic is Garibaldi (also 12 & 29, although 41 & 53 represents its tuning range better), which equates 64/63 with the syntonic comma. This is not considered canonical due to a significant loss in accuracy (tuned best with a fifth slightly sharp of just) - that is, Garibaldi is not a microtemperament - but it is still more accurate than [[Meantone]] as well as distinguishing 5-limit, 7-limit, and Pythagorean intervals in any given interval category. Garibaldi is an intuitive way of organizing just intonation as it tempers together the defining ~20-30c commas of the 7-limit.

==== Primes 11 and 13 ====
A reasonable extension to the 11-limit assuming Garibaldi is Cassandra (41 & 53), which sets 33/32 to twice the Garibaldi comma; alternatively there is Andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 11/9 is set to the opposite neutral third to 16/13 to extend to the 13-limit.

=== Prime 17 ===
Schismic generally does not have 17; two options are setting 17/16 equal to 16/15 and 15/14 (preferring flatter schismic tunings) and tempering together 17/16 and 18/17 (which results in a weak extension including 17/12 as the semioctave).

== Intervals ==
These interpretations assume Cassandra (in Pythagorean tuning).
{| class="wikitable"
|+
! colspan="3" |Up from the unison
! colspan="3" |Down from the octave
|-
!#
!Cents
!JI
!#
!Cents
!JI
|-
|'''0'''
|'''0.00'''
|'''1/1'''
|'''0'''
|'''1,200.00'''
|'''2/1'''
|-
|'''1'''
|'''701.96'''
|'''3/2'''
|1
|498.04
|4/3
|-
|2
|203.91
|9/8
|2
|996.09
|16/9
|-
|3
|905.87
|32/19, 27/16
|'''3'''
|'''294.13'''
|'''19/16'''
|-
|4
|407.82
|19/15
|4
|792.18
|19/12
|-
|5
|1,109.78
|19/10
|5
|90.22
|19/18, 20/19, 21/20
|-
|6
|611.73
|10/7
|6
|588.27
|7/5
|-
|7
|113.69
|16/15, 15/14
|7
|1,086.31
|15/8
|-
|8
|815.64
|8/5
|'''8'''
|'''384.36'''
|'''5/4'''
|-
|9
|317.60
|5/3
|9
|882.40
|6/5
|-
|10
|1,019.55
|9/5
|10
|180.45
|10/9
|-
|11
|521.51
|27/20
|11
|678.49
|40/27
|-
|12
|23.46
|50/49
|12
|1,176.54
|49/25
|-
|13
|725.42
|32/21
|13
|474.58
|21/16
|-
|14
|227.37
|8/7
|'''14'''
|'''972.63'''
|'''7/4'''
|-
|15
|929.33
|12/7
|15
|270.67
|7/6
|-
|16
|431.28
|9/7
|16
|768.72
|14/9
|-
|17
|1,133.24
|28/27
|17
|66.76
|27/14
|-
|18
|635.19
|13/9
|18
|564.81
|18/13
|-
|19
|137.15
|13/12
|19
|1,062.85
|24/13
|-
|'''20'''
|'''839.10'''
|'''13/8'''
|20
|360.90
|16/13
|-
|21
|341.06
|11/9
|21
|858.94
|18/11
|-
|22
|1,043.01
|11/6
|22
|156.99
|12/11
|-
|'''23'''
|'''544.97'''
|'''11/8'''
|23
|655.03
|16/11
|-
|24
|46.92
|33/32
|24
|1,153.08
|64/33
|-
|25
|748.88
|54/35
|25
|451.12
|35/27
|-
|26
|250.83
|81/70
|26
|949.17
|140/81
|-
|27
|952.79
|26/15
|27
|247.21
|15/13
|-
|28
|454.74
|13/10
|28
|745.26
|20/13
|-
|29
|1,156.70
|39/20
|29
|43.30
|40/39
|}

== Patent vals ==
The following patent vals up to 272edo support Schismic.
{| class="wikitable sortable mw-collapsible"
|+
!EDO
!Generator tuning
!Extension info
|-
|12
|700.0000
|Garibaldi
|-
|101
|700.9901
|
|-
|89
|701.1236
|
|-
|77
|701.2987
|
|-
|219
|701.3699
|
|-
|142
|701.4085
|
|-
|207
|701.4493
|
|-
|272
|701.4706
|
|-
|65
|701.5385
|
|-
|248
|701.6129
|
|-
|183
|701.6393
|
|-
|118
|701.6949
|
|-
|171
|701.7544
|
|-
|224
|701.7857
|(no Schismic 19)
|-
|53
|701.8868
|Garibaldi
|-
|253
|701.9763
|(no Schismic 19)
|-
|200
|702.0000
|(no Schismic 19)
|-
|147
|702.0408
|Garibaldi
|-
|94
|702.1277
|Garibaldi
|-
|135
|702.2222
|Garibaldi
|-
|41
|702.4390
|Garibaldi
|-
|29
|703.4483
|Garibaldi
|-
|17
|705.8824
|
|}

The full list of patent vals supporting schismic is below:
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+
!EDO
!Generator tuning
!Extension info
|-
|12
|700.0000
|Garibaldi
|-
|101
|700.9901
|
|-
|89
|701.1236
|
|-
|77
|701.2987
|
|-
|219
|701.3699
|
|-
|142
|701.4085
|
|-
|207
|701.4493
|
|-
|272
|701.4706
|
|-
|65
|701.5385
|
|-
|508
|701.5748
|(no Schismic 19)
|-
|443
|701.5801
|(no Schismic 19)
|-
|378
|701.5873
|(no Schismic 19)
|-
|313
|701.5974
|(no Schismic 19)
|-
|561
|701.6043
|(no Schismic 19)
|-
|248
|701.6129
|
|-
|679
|701.6200
|(no Schismic 19)
|-
|431
|701.6241
|(no Schismic 19)
|-
|614
|701.6287
|(no Schismic 19)
|-
|797
|701.6311
|(no Schismic 19)
|-
|183
|701.6393
|
|-
|1033
|701.6457
|(no Schismic 19)
|-
|850
|701.6471
|(no Schismic 19)
|-
|667
|701.6492
|(no Schismic 19)
|-
|1151
|701.6507
|(no Schismic 19)
|-
|484
|701.6529
|(no Schismic 19)
|-
|1269
|701.6548
|(no Schismic 19)
|-
|785
|701.6561
|(no Schismic 19)
|-
|1086
|701.6575
|(no Schismic 19)
|-
|1387
|701.6583
|(no Schismic 19)
|-
|301
|701.6611
|(no Schismic 19)
|-
|1322
|701.6641
|(no Schismic 19)
|-
|1021
|701.6650
|(no Schismic 19)
|-
|720
|701.6667
|(no Schismic 19)
|-
|1139
|701.6681
|(no Schismic 19)
|-
|1558
|701.6688
|(no Schismic 19)
|-
|419
|701.6706
|(no Schismic 19)
|-
|1794
|701.6722
|(no Schismic 19)
|-
|1375
|701.6727
|(no Schismic 19)
|-
|956
|701.6736
|(no Schismic 19)
|-
|1493
|701.6745
|(no Schismic 19)
|-
|2030
|701.6749
|(no Schismic 19)
|-
|537
|701.6760
|(no Schismic 19)
|-
|1729
|701.6773
|(no Schismic 19)
|-
|1192
|701.6779
|(no Schismic 19)
|-
|1847
|701.6784
|(no Schismic 19)
|-
|655
|701.6794
|(no Schismic 19)
|-
|2083
|701.6803
|(no Schismic 19)
|-
|1428
|701.6807
|(no Schismic 19)
|-
|773
|701.6818
|(no Schismic 19)
|-
|1664
|701.6827
|(no Schismic 19)
|-
|891
|701.6835
|(no Schismic 19)
|-
|1900
|701.6842
|(no Schismic 19)
|-
|1009
|701.6848
|(no Schismic 19)
|-
|2136
|701.6854
|(no Schismic 19)
|-
|1127
|701.6859
|(no Schismic 19)
|-
|1245
|701.6867
|(no Schismic 19)
|-
|1363
|701.6875
|(no Schismic 19)
|-
|1481
|701.6880
|(no Schismic 19)
|-
|1599
|701.6886
|(no Schismic 19)
|-
|1717
|701.6890
|(no Schismic 19)
|-
|1835
|701.6894
|(no Schismic 19)
|-
|1953
|701.6897
|(no Schismic 19)
|-
|2071
|701.6900
|(no Schismic 19)
|-
|2189
|701.6903
|(no Schismic 19)
|-
|118
|701.6949
|
|-
|2295
|701.6993
|(no Schismic 19)
|-
|2177
|701.6996
|(no Schismic 19)
|-
|2059
|701.6999
|(no Schismic 19)
|-
|1941
|701.7002
|(no Schismic 19)
|-
|1823
|701.7005
|(no Schismic 19)
|-
|1705
|701.7009
|(no Schismic 19)
|-
|1587
|701.7013
|(no Schismic 19)
|-
|1469
|701.7018
|(no Schismic 19)
|-
|1351
|701.7024
|(no Schismic 19)
|-
|1233
|701.7032
|(no Schismic 19)
|-
|2348
|701.7036
|(no Schismic 19)
|-
|1115
|701.7040
|(no Schismic 19)
|-
|2112
|701.7045
|(no Schismic 19)
|-
|997
|701.7051
|(no Schismic 19)
|-
|1876
|701.7058
|(no Schismic 19)
|-
|879
|701.7065
|(no Schismic 19)
|-
|1640
|701.7073
|(no Schismic 19)
|-
|2401
|701.7076
|(no Schismic 19)
|-
|761
|701.7083
|(no Schismic 19)
|-
|2165
|701.7090
|(no Schismic 19)
|-
|1404
|701.7094
|(no Schismic 19)
|-
|2047
|701.7098
|(no Schismic 19)
|-
|643
|701.7107
|(no Schismic 19)
|-
|2454
|701.7115
|(no Schismic 19)
|-
|1811
|701.7118
|(no Schismic 19)
|-
|1168
|701.7123
|(no Schismic 19)
|-
|1693
|701.7129
|(no Schismic 19)
|-
|2218
|701.7133
|(no Schismic 19)
|-
|525
|701.7143
|(no Schismic 19)
|-
|1982
|701.7154
|(no Schismic 19)
|-
|1457
|701.7159
|(no Schismic 19)
|-
|2389
|701.7162
|(no Schismic 19)
|-
|932
|701.7167
|(no Schismic 19)
|-
|2271
|701.7173
|(no Schismic 19)
|-
|1339
|701.7177
|(no Schismic 19)
|-
|1746
|701.7182
|(no Schismic 19)
|-
|2153
|701.7185
|(no Schismic 19)
|-
|407
|701.7199
|(no Schismic 19)
|-
|2324
|701.7212
|(no Schismic 19)
|-
|1917
|701.7214
|(no Schismic 19)
|-
|1510
|701.7219
|(no Schismic 19)
|-
|1103
|701.7226
|(no Schismic 19)
|-
|1799
|701.7232
|(no Schismic 19)
|-
|2495
|701.7234
|(no Schismic 19)
|-
|696
|701.7241
|(no Schismic 19)
|-
|2377
|701.7249
|(no Schismic 19)
|-
|1681
|701.7252
|(no Schismic 19)
|-
|985
|701.7259
|(no Schismic 19)
|-
|2259
|701.7264
|(no Schismic 19)
|-
|1274
|701.7268
|(no Schismic 19)
|-
|1563
|701.7274
|(no Schismic 19)
|-
|1852
|701.7279
|(no Schismic 19)
|-
|2141
|701.7282
|(no Schismic 19)
|-
|2430
|701.7284
|(no Schismic 19)
|-
|289
|701.7301
|(no Schismic 19)
|-
|2483
|701.7318
|(no Schismic 19)
|-
|2194
|701.7320
|(no Schismic 19)
|-
|1905
|701.7323
|(no Schismic 19)
|-
|1616
|701.7327
|(no Schismic 19)
|-
|1327
|701.7332
|(no Schismic 19)
|-
|2365
|701.7336
|(no Schismic 19)
|-
|1038
|701.7341
|(no Schismic 19)
|-
|1787
|701.7348
|(no Schismic 19)
|-
|2536
|701.7350
|(no Schismic 19)
|-
|749
|701.7356
|(no Schismic 19)
|-
|2707
|701.7362
|(no Schismic 19)
|-
|1958
|701.7365
|(no Schismic 19)
|-
|1209
|701.7370
|(no Schismic 19)
|-
|1669
|701.7376
|(no Schismic 19)
|-
|2129
|701.7379
|(no Schismic 19)
|-
|2589
|701.7381
|(no Schismic 19)
|-
|460
|701.7391
|(no Schismic 19)
|-
|2471
|701.7402
|(no Schismic 19)
|-
|2011
|701.7404
|(no Schismic 19)
|-
|1551
|701.7408
|(no Schismic 19)
|-
|1091
|701.7415
|(no Schismic 19)
|-
|1722
|701.7422
|(no Schismic 19)
|-
|2353
|701.7425
|(no Schismic 19)
|-
|631
|701.7433
|(no Schismic 19)
|-
|2064
|701.7442
|(no Schismic 19)
|-
|1433
|701.7446
|(no Schismic 19)
|-
|802
|701.7456
|(no Schismic 19)
|-
|1775
|701.7465
|(no Schismic 19)
|-
|973
|701.7472
|(no Schismic 19)
|-
|1144
|701.7483
|(no Schismic 19)
|-
|1315
|701.7490
|(no Schismic 19)
|-
|1486
|701.7497
|(no Schismic 19)
|-
|1657
|701.7502
|(no Schismic 19)
|-
|1828
|701.7505
|(no Schismic 19)
|-
|171
|701.7544
|
|-
|1421
|701.7593
|(no Schismic 19)
|-
|1250
|701.7600
|(no Schismic 19)
|-
|1079
|701.7609
|(no Schismic 19)
|-
|908
|701.7621
|(no Schismic 19)
|-
|737
|701.7639
|(no Schismic 19)
|-
|1303
|701.7652
|(no Schismic 19)
|-
|566
|701.7668
|(no Schismic 19)
|-
|961
|701.7690
|(no Schismic 19)
|-
|395
|701.7722
|(no Schismic 19)
|-
|1014
|701.7751
|(no Schismic 19)
|-
|619
|701.7771
|(no Schismic 19)
|-
|843
|701.7794
|(no Schismic 19)
|-
|1067
|701.7807
|(no Schismic 19)
|-
|224
|701.7857
|(no Schismic 19)
|-
|725
|701.7931
|(no Schismic 19)
|-
|501
|701.7964
|(no Schismic 19)
|-
|778
|701.7995
|(no Schismic 19)
|-
|277
|701.8051
|(no Schismic 19)
|-
|607
|701.8122
|(no Schismic 19)
|-
|330
|701.8182
|(no Schismic 19)
|-
|383
|701.8277
|(no Schismic 19)
|-
|436
|701.8349
|(no Schismic 19)
|-
|489
|701.8405
|(no Schismic 19)
|-
|542
|701.8450
|(no Schismic 19)
|-
|53
|701.8868
|Garibaldi
|-
|253
|701.9763
|(no Schismic 19)
|-
|200
|702.0000
|(no Schismic 19)
|-
|147
|702.0408
|Garibaldi
|-
|94
|702.1277
|Garibaldi
|-
|135
|702.2222
|Garibaldi
|-
|41
|702.4390
|Garibaldi
|-
|29
|703.4483
|Garibaldi
|-
|17
|705.8824
|
|}

== Footnotes ==
<sup>[a]</sup> Despite ending in -isma, ''schisma'' is a [[Temperament naming#Comma declension categories|3rd-declension]] comma name, therefore its temperaments are ''Schismatic'' and ''Schismic'' (which, as it is a 2.3.5 comma, refer to the same temperament).

{{Navbox regtemp}}
[[Category: Temperaments]]

41edo

2026-08-14T11:33:18Z

FloraC: /* Edostep interpretations */ + 50/49; sort by size

'''41edo''', or 41 equal divisions of the octave, is an equal tuning with a step size of approximately 29 cents. It is known for its relatively good approximation of 11-limit just intonation.

== Theory ==

==== JI approximation ====
41edo is most accurately a 2.3.5.7.11 tuning, though it also has an acceptable if sharp 13th harmonic, notably widening the difference between the [[Collection of chords#Arto triad|arto]] (10:13:15) and [[Collection of chords#Tendo triad|tendo]] (1/10:1/13:1/15) triads such that they become simple [[Slendric]] divisions of the fifth. Because it is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 7-6-4-7-6-7-4. However, it also features a MOS diatonic of 7-7-3-7-7-7-3.

{{Harmonics in ED|41|31|0}}
{| class="wikitable"
|+Thirds in 41edo
!Quality
|Subminor
|'''Farminor'''
|Nearminor
|Neutral
|Nearmajor
|'''Farmajor'''
|Supermajor
|-
!Cents
|263
|'''293'''
|322
|351
|381
|'''410'''
|439
|-
!Just interpretation
|7/6
|'''13/11'''
|6/5
|11/9
|5/4
|'''14/11'''
|9/7
|-
!Steps
|9
|'''10'''
|11
|12
|13
|'''14'''
|15
|}
Thirds available in the diatonic scale generated by stacking the perfect fifth are bolded.

=== Edostep interpretations ===
One step of 41edo represents the following ratios of 2.3.5.7.11.13.19:
* 49/48 (between 7/6 and 8/7)
* 50/49 (between 7/5 and 10/7)
* 55/54 (between 27/20 and 11/8)
* 56/55 (between 5/4 and 14/11)
* 57/56 (between 7/6 and 19/16)
* 64/63 (between 7/4 and 16/9)
* 65/64 (between 16/13 and 5/4)
* 66/65 (between 13/11 and 6/5)
* 81/80 (between 5/4 and 81/64)
* 91/90 (between 9/7 and 13/10)
* 96/95 (between 19/16 and 6/5)
* 99/98 (between 14/11 and 9/7)
* 121/120 (between 12/11 and 11/10)
* 169/168 (between 14/13 and 13/12)
Two steps of 41edo represent the following ratios:
* 28/27 (between 9/8 and 7/6)
* 33/32 (between 4/3 and 11/8)
* 36/35 (between 7/4 and 9/5)
* 39/38 (between 19/16 and 39/16, or 19/15 and 13/10)

=== Derivation ===
41edo is derived by tempering out neither 81/80 nor 64/63, but rather setting them equal, and using the same step to separate 5/4 from a neutral third and 8/7 from 7/6.

=== Chords ===
41edo has three different flavors of minor and major intervals as well as neutral intervals. Its subminor and supermajor intervals approximate simpler septimal ratios such as 7/4 and 9/7, while its supraminor ("nearminor") and submajor ("nearmajor") intervals approximate classical 5-limit harmony which includes ratios like 5/4 and 9/5, and its plain major and plain minor intervals approximate classic 3-limit ratios. As a result, 41edo has nine qualities of tertian, fifth-bounded triad: tendo, supermajor, novamajor, nearmajor, neutral, nearminor, novaminor, subminor. However, 41edo lacks true interordinal intervals (to reach a tuning with both neutrals and interordinals, 58edo must be used), so as for latal fourth-bounded triads, there are only four qualities.

=== Scales ===
41edo's 5-limit intervals are not found particularly early on in the chain of fifths, with 6/5 being an augmented second and 5/4 a diminished fourth. Notably, 41edo has a 17-note chromatic scale generated by the perfect fifth, 3-3-3-1-3-3-1-3-3-3-1-3-3-1-3-3-1, in which the classical (~5/4) major and classical (~6/5) minor thirds span the same number of scale steps, giving the scale familiar chord qualities.

11-limit [[penslen]] is available in 41edo (because of Slendric and 385/384 tempering): 2-1-5-1-2-5-1-2-5-1-2-1-5-2-1-5.

=== Regular temperaments ===
41edo shares Schismic (and its extension Garibaldi, and thus Marvel and Aberschismic) with 29edo, Slendric (and its extension Miracle) with 31edo, Tetracot with 34edo, and Magic with 22edo. Magic is especially important here as it forms the fret layout and main string tuning for the Kite guitar.

It also contains a slightly-stretched version of equal Bohlen-Pierce tuning (where the perfect twelfth of 3/1 is split into 13 equal parts) via every fifth step. If used in a linear temperament as the generator, this temperament is called Bohpier.

== Notation ==
Since 41edo has a perfect fifth which is split exactly in half, semisharps and semiflats (as in [[Diatonic notation|neutral diatonic notation]]) can be used to notate it. A useful addition is ups and downs, which naturally reflect 41edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor. (In fact, "up" can be declared equivalent to "super"/"supra" and "down" equivalent to "sub"; preferable to Unque and also indicative of the flat tuning of 5/4 in the temperament.)

== Practice ==
41edo is used by the musician and conlanger Lamplight as a standard tuning for their ''Shasavic'' theory of music.

=== Kite's work ===
There is a large amount of evidence suggesting that Kite's body of work is implicitly designed with 41edo in mind.

Most notably, the Kite Guitar itself uses 41edo, and is advertised as a general-purpose just-intonation-esque instrument, notably treating 41edo implicitly as the solution to compromising high-accuracy JI with equal temperament. The Kite Guitar manages the high density of notes by only having frets for every other note of 41edo, and tuning the strings in a way that allows one string to fill in the gaps of an adjacent string.

On another note, the ''color notation'' system seems specifically well-designed for 41edo, as a tuning of tetracot with high accuracy in the 2.3.5.7.11 subgroup. Colors can unambiguously be assigned to 41edo interval qualities without any of the extra overhead that comes with managing color notation for just intonation or larger edos, but while feeling like it uses the notation to its full potential. The qualities in order, from subminor to supermajor, are ''zo'' (7/), minor ''wa'' (3), ''gu'' (/5), ''(i)la'' (11)'', yo'' (5/), major ''wa'' (3), and ''ru'' (/7). This also serves to explain the unusual nature of "la" as "lavender", with the 11/ and /11 colors conflated due to the fact that in 41edo, rastmic tempering is assumed.

Additionally, ups and downs seems to work particularly well for the combination of undecimal garibaldi, slendric, and rastmic tempering, which is a characteristic of specifically 41edo, due to allowing the arrow to refer to a large number of different structural commas that are in the size range.

== Multiples ==
=== 82edo ===
82 is the first edo to have three diatonic fifths, and thus may be seen as the logical extension of [[Monarch]], which uses two diatonic fifths. It is a notable [[Hemiseven]] tuning, especially for its superset in which 7/4 is split into six parts.
{{Harmonics in ED|82|37|0}}

{{Navbox EDO}}
{{Cat|Edos}}

Template:Navbox regtemp

2026-08-14T11:31:26Z

FloraC: Fixes

<br />{{Navbox
| name = Navbox regtemp
| Title = [[Regular temperament]]s
| Is Collapsible = true
| Header 1 = Rank-2
| Header 2 = Acot
| Data 2 = [[Blackwood]] (1/5-octave) • [[7edo#Whitewood temperament|Whitewood]] (1/7-octave) • [[12edo#Compton temperament|Compton]] (1/12-octave)
| Header 3 = Monocot
| Data 3 = [[Meantone]] • [[Schismic]] • [[Gentle tuning|Gentle-fifth temperaments]] • [[Archy]]
| Header 4 = Complexity 2
| Data 4 = [[Diaschismic]] (diploid monocot) • [[Pajara]] (diploid monocot) • [[Injera]] (diploid monocot) • [[Rastmatic]] (dicot) • [[Mohajira]] (dicot) • [[Intertridecimal (temperament)|Intertridecimal]] (dicot) • [[Interseptimal (temperament)|Interseptimal]] (alpha-dicot)
| Header 5 = Complexity 3
| Data 5 = [[Augmented (temperament)|Augmented]] (triploid) • [[Misty]] (triploid) • [[Slendric]] (tricot) • [[Porcupine]] (omega-tricot)
| Header 6 = Complexity 4
| Data 6 = [[Diminished (temperament)|Diminished]] (tetraploid) • [[Tetracot (temperament)|Tetracot]] (tetracot) • [[Buzzard]] (alpha-tetracot) • [[Squares]] (beta-tetracot) • [[Negri]] (omega-tetracot)
| Header 7 = Complexity 5-6
| Data 7 = [[Magic]] (alpha-pentacot) • [[Amity]] (gamma-pentacot) • [[Miracle]] (hexacot) • [[Kleismic]] (alpha-hexacot) • [[Hemiseven]] (gamma-hexacot) • [[Hedgehog]] (diploid alpha-tricot)
| Header 8 = Higher complexity
| Data 8 = [[Orwell]] (alpha-heptacot) • [[Sensi]] (beta-heptacot) • [[Octacot (temperament)|Octacot]] (octacot) • [[Wurschmidt]] (beta-octacot) • [[Carlos Alpha|Valentine]] (enneacot) • [[Ammonite]] (epsilon-enneacot) • [[Myna]] (beta-decacot) • [[Ennealimmal]] (enneaploid dicot)
| Header 9 = Straddle-3
| Data 9 = [[A-Team]] (alter-tricot) • [[Machine]] (alter-monocot) • [[Paddle]] (alter-dicot)
| Header 10 = No-3
| Data 10 = [[Mabilic]] (alpha-triseph) • [[Orgone]] (trimech) • [[Didacus]] (diseph) • [[Sidewalk]] (epsilon-heptaseph)
| Header 11 = No-2
| Data 11 = [[Sensamagic]] (monogem)
| Header 12 = Exotemperaments
| Data 12 = [[Dicot (temperament)|Dicot]] • [[Mavila]] • [[Father]]
| Header 13 = Higher-rank
| Header 14 = Rank-3
| Data 14 = [[Aberschismic]] • [[Marvel]] • [[Parapyth]] • [[Pentagoth]] • [[Intergan]]
| Header 15 = Straddle Rank-3
| Data 15 = [[Hibernal]] • [[Monarch]]
}}

Category:Message boxes

2026-08-11T09:30:25Z

FloraC: Created page with "Category:Templates"

[[Category:Templates]]

Template:UserTag

2026-08-11T09:29:29Z

FloraC: + category

<includeonly><div style="
display: flex;
align-items: flex-start;
gap: 0.5em;
">[[User:{{{2}}}|<span style="
/* be careful with line breaks because any one of them could end up displaying in the parser */
--s: color-mix(in srgb, #{{{3}}} 50%, #000 50%); /* use color-mix instead of rgba to avoid visible overlapping */
display: inline-block;
text-align: center;
padding: 0.2em;
min-width: 1.5em;
height: 1.5em;
color: #fff;
box-shadow:
inset 1px 1px 0 rgba(255,255,255,0.5),
inset -1px -1px 0 rgba(0,0,0,0.5);
text-shadow:
-1px -1px 2px var(--s),
1px -1px 2px var(--s),
-1px 1px 2px var(--s),
1px 1px 2px var(--s);
background: #{{{3}}};
">{{{1}}}</span>]]<div style="
color: color-mix(in srgb, #{{{3}}} 33%, currentColor 67%);
">{{{4}}}</div></div>[[Category:Pages with usertags]]</includeonly><noinclude>
UserTag should be used for passages that can't be written objectively without losing meaning, often indicating a first-person perspective with personal experience. Their usage is highly encouraged because they allow a range of perspectives to be expressed in any article. They may be removed to consolidate information that follows consensus into the standard format.

Articles entirely written from a specific person's perspective should use [[Template: UTBox]].

<pre>43edo is a good tuning of the antidiatonic 17:20:25 chord in Vengeance temperament, which is notably simple and contains 5/4. It also has the 17:21:25 +1+1 chord.

{{UserTag|g_|Ground|7766ff|43edo is my favorite tuning for quasi-antidiatonic music because of how pure its approximation of 17:20:21:25 sounds, uniquely stabilized by a simple fifth, 5/4, and +1+1. This is my antidiatonic equivalent to the quasi-diatonic 7-limit. It can't be beaten.
Here's a line break, but I'm done talking.
}}</pre>

# User tag, ideally one or two letters
# Username for the link to the user page
# Hex color without #
# Text block assigned to the tag

[[Category:Templates]]
</noinclude>

Buzzard

2026-07-23T10:21:35Z

FloraC: /* Extensions */ math error? Described here was an undocumented low-accuracy 11-limit extension

{{Infobox regtemp
| Title = Buzzard
| Subgroups = 2.3.7, 2.3.5.7, 2.3.5.7.11.13
| Comma basis = [[65536/64827]] (2.3.7); <br>[[1728/1715]], [[5120/5103]] (7-limit);<br>[[176/175]], [[351/350]], [[540/539]], [[676/675]]<br>(13-limit)
| Edo join 1 = 53 | Edo join 2 = 58
| Mapping = 1; 4 21 -3 39 27
| Generators = 21/16
| Generators tuning = 475.7
| Optimization method = CWE
| MOS scales = 3L 2s
| Ploidacot = alpha-tetracot
| Pergen =
| Color name =
| Odd limit 1 = 2.3.7 9 | Mistuning 1 = 3.42 | Complexity 1 = 13
| Odd limit 2 = 15 | Mistuning 2 = 4.09 | Complexity 2 = 43
}}
'''Buzzard''', 2.3.7[{{e|48}} & {{e|53}}], is a 2.3.7 temperament that splits 3/1 into four sharpened 21/16 subfourths. It can be considered the "simplest 2.3.7 temperament after [[Slendric]]" since it equates two (not one) 64/63's with one 49/48, exaggerating the difference between 21/16 and 64/49. This 64/63 interval is primarily used as an inflection comma, as it separates the simple 3-limit intervals from 7-limit consonances; as a result of this "spine and comma" arrangement of the consonances, the MOS scales generated by Buzzard are not always the primary type of scale considered in the temperament.

Since Buzzard sharpens 147/128, equating it to an exact semifourth at -2 generators, it can be extended to 2.3.7.13/5 by equating 147/128 to 15/13, thus tempering out the huntsma, 640/637.

== List of patent vals ==
{| class="wikitable"
|+
!Edo
!Fifth tuning
!7/4 tuning
!Generator tuning
|-
|23
|678.26
|991.30
|469.57
|-
|28
|685.71
|985.71
|471.43
|-
|33
|690.91
|981.82
|472.73
|-
|38
|694.74
|978.95
|473.68
|-
|43
|697.67
|976.74
|474.42
|-
|48
|700.00
|975.00
|475.00
|-
|101
|700.99
|974.26
|475.25
|-
|53
|701.89
|973.58
|475.47
|-
|111
|702.70
|972.97
|475.68
|-
|58
|703.45
|972.41
|475.86
|-
|179
|703.91
|972.07
|475.98
|-
|121
|704.13
|971.90
|476.03
|-
|184
|704.35
|971.74
|476.09
|-
|63
|704.76
|971.43
|476.19
|-
|131
|705.34
|970.99
|476.34
|-
|68
|705.88
|970.59
|476.47
|-
|73
|706.85
|969.86
|476.71
|-
|78
|707.69
|969.23
|476.92
|-
|83
|708.43
|968.67
|477.11
|-
|5
|720.00
|960.00
|480.00
|}

== Extensions ==
While 13/5 has an obvious mapping via the 15/13 semifourth at -2 generators, 13 and 5 themselves are less obvious, as is 11.

=== Aberschismic Buzzard ===
The most common extension of Buzzard is Aberschismic -- that is, tempering out 5120/5103. This equates the 64/63 inflection comma with 81/80 and 105/104, allowing the 5-limit and 7-limit consonances to each be reached by one inflection away from a 3-limit consonance. This also provides a corresponding mapping of prime 13 as the difference between 15/8 and 15/13.

==== Valinorsmic Buzzard ====
The "canonical" mapping of prime 11 is to equate the inflection comma with 100/99; this equally spaces the sequence 21/16 - 4/3 - 27/20 - 15/11, and maps 13/11 to the diatonic minor third, right in the middle of 6/5 and 7/6.

==== Telepathic Buzzard ("Buteo") ====
An alternative mapping of prime 11 is to conflate the inflection comma with 55/54 instead. This equally spaces the 7/6 - 32/27 - 6/5 - 11/9 sequence, as well as 21/16 - 4/3 - 27/20 - 11/8. This tuning is sometimes called Buteo, to distinguish from the Valinormic tuning that is considered the canonical extension of Buzzard.

=== Sensamagic Buzzard ("Lemongrass") ===
An alternative way to reach prime 5 is to equate 49/48 with 81/80, thus equating the sequence 5/4 - 80/63 - 81/64 - 9/7. Prime 13 is still the difference between 15/8 and 15/13, but in this case the tuning of 15/8 is much flatter than in Aberschismic Buzzard, thus providing a sharper tuning of 13/8. This extension is sometimes called Lemongrass, to distinguish from the Aberschismic tuning that is considered canonical.

== Scales ==
{{Wip}}{{Navbox regtemp}}
{{Cat| temperaments}}

EDO page skeleton

2026-07-20T09:01:51Z

FloraC: FloraC moved page EDO page skeleton to Xenharmonic Reference:EDO page skeleton: Wiki organization topic

#REDIRECT [[Xenharmonic Reference:EDO page skeleton]]

XenReference:EDO page skeleton

2026-07-20T09:01:51Z

FloraC: FloraC moved page EDO page skeleton to Xenharmonic Reference:EDO page skeleton: Wiki organization topic

This page is to be used as a template for making EDO pages.

'''Xedo''', or X equal divisions of the octave (sometimes called '''X-TET''' or '''X-tone equal temperament'''), is the [[Equal temperament|equal tuning]] featuring steps of (1200/X) = Y [[Cent|cents]] exactly, X of which stack to the perfect octave [[2/1]].

[Overview of tuning properties for key intervals]

== General theory ==

=== Derivation ===
[Derivation by equal division of various key intervals or of interval qualities]

=== JI approximation ===
[Overview of viable vals, tuning tendencies, and accurate/structurally interesting subgroups. Always cover the patent val alongside any notable non-patent vals.]{{Harmonics in ED|0|prime}}

=== Edostep interpretations ===
[Cover these in a list]

=== Intervals and notation ===
{| class="wikitable"
|+
! rowspan="2" |Edostep
! rowspan="2" |Cents
! colspan="2" |Intervals
! colspan="3" |Notation
! colspan="4" |Bame (optional)
|-
!Group 1
!Additional groups
!Notation 1 (usually ups and downs)
!Notation 2 (usually neutral diatonic notation, if different))
!Additional notations
!System 1 (usually ADIN)
!System 2 (usually ups and downs, if meaningfully different)
!System 3 (usually neutral diatonic, if different)
!Additional systems
|-
|0
|XXX
|XX/XX
|XX/XX
|D
|D
|J
|...
|...
|...
|...
|-
|...
|...
|...
|...
|...
|...
|...
|
|
|
|
|-
|
|
|
|
|
|
|
|
|
|
|
|-
|
|
|
|
|
|
|
|
|
|
|
|}
[Add solfege in a separate table, if applicable]

== Tempering properties ==

=== Tempered commas ===
[List commas with S-expressions and examples of what they equate]

=== Arithmetic progressions ===

=== Notable structural chains ===
[Generator chains]

== Compositional theory ==

=== Tertian structure ===
[Describe]
{| class="wikitable"
|+Thirds in Xedo
!Quality ([[ADIN]])
|'''Mosdiatonic Quality'''
|Quality
|-
!Cents
|'''XXX'''
|XXX
|-
!Just interpretation
|'''X/X'''
|X/X
|-
!Steps
|'''X'''
|X
|}
Diatonic thirds are bolded.

=== Scales ===
[if edo is composite, link to subset edos and discuss scales of subset edos there]

[List scales and scale descriptions including structure (generators if applicable), notable intervals available, associated temperaments, relationships to other scales, and logic for derivation.]

[Explain more complex scale theory topics here]

==== Tables of scales ====

=== Harmony ===
[explain notable JI and DR chords. TRY TO PROVIDE COHERENT CHORD SYSTEMS RATHER THAN JUST LISTING RANDOM CHORDS WITH NO RELATION. Explain tunings of familiar chords]

[Explain systems of harmony here, this is how you put the chords together to make music]

[Sections such as "functional harmony", "modal harmony", etc - varies based on the edo and the personal composition style]

[Use individual voices maybe?]

==== Tables of chords ====

== Instruments ==
[Put instruments and isomorphic layouts here]

== Supersets and subsets ==

== Comparisons to other tuning systems ==

== Music in Xedo ==

== See also ==

Module:Harmonics in ED

2026-07-16T16:59:15Z

FloraC: Use HTML newline for proper line spacing

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
local p_cols = {"prime2", "prime3", "prime5", "prime7", "prime11", "prime13", "prime17", "prime19", "prime23", "prime29", "prime31", "prime37", "prime41", "prime43", "prime47", "prime53", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime"}

local odds = {3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57}
local o_cols = {"prime3", "prime5", "prime7", "higherPrime", "prime11", "prime13", "higherPrime", "prime17", "prime19", "higherPrime", "prime23", "higherPrime", "higherPrime", "prime29", "prime31", "higherPrime", "higherPrime", "prime37", "higherPrime", "prime41", "prime43", "higherPrime", "prime47", "higherPrime", "higherPrime", "prime53", "higherPrime", "higherPrime"}

local function steps(et, harm) -- steps of harmonic
return math.floor((math.log(harm) / math.log(2) * et) + 0.5)
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = math.log(harm) / math.log(2) * et
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
abs_err = math.abs(err)
if(abs_err == 0) then
return "acc0"
end
if(abs_err < 0.03125) then
return "acc1"
end
return "acc" .. math.floor(abs_err * 12) + 2
end

function p.table(frame) -- making the table itself
local et = frame.args["et"]
local nowiki = frame.args["nowiki"]

local type = frame.args["type"]
if (type == "0") then
type = "prime";
end

local harms = primes;
local cols = p_cols;
local p_lim = frame.args["p_lim"]
if (p_lim == "odd" or p_lim == "prime") then
type = p_lim;
p_lim = "default";
end

if (type == "prime") then
if (p_lim == "default") then
p_lim = 31;
end
elseif (type == "odd") then
if (p_lim == "default") then
p_lim = 25;
end
harms = odds;
cols = o_cols;
end
local lim = math.floor(p_lim)
local p_len = #harms

local tab = "{| class=\"wikitable center-all\"\n"
if (tonumber(nowiki) == 1) then
tab = "<nowiki>" .. tab end
tab = tab .. "|+Approximation of " .. type .. " harmonics in "
tab = tab .. et .. "edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! class=\""
tab = tab .. cols[i] .. "\" |" .. harms[i] .. "\n"
if harms[i] >= lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| "
if rel_err(et, harms[i]) > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", abs_err(et, harms[i])) .. "\n"
if harms[i] >= lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, harms[i])
tab = tab .. "| class=\"" .. rel_col(er) .. "\" | "
if er > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if harms[i] >= lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps<br>(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, harms[i]) .. "<br>(" .. steps_re(et, harms[i]) .. ")\n"
if harms[i] >= lim then break end
end
tab = tab .. "|}"
if (tonumber(nowiki) == 1) then
tab = tab .. "</nowiki>" end
return tab
end

return p

Module:Harmonics in ED

2026-07-16T16:54:59Z

FloraC: Centering

local p = {}
local primes = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
local p_cols = {"prime2", "prime3", "prime5", "prime7", "prime11", "prime13", "prime17", "prime19", "prime23", "prime29", "prime31", "prime37", "prime41", "prime43", "prime47", "prime53", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime", "higherPrime"}

local odds = {3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57}
local o_cols = {"prime3", "prime5", "prime7", "higherPrime", "prime11", "prime13", "higherPrime", "prime17", "prime19", "higherPrime", "prime23", "higherPrime", "higherPrime", "prime29", "prime31", "higherPrime", "higherPrime", "prime37", "higherPrime", "prime41", "prime43", "higherPrime", "prime47", "higherPrime", "higherPrime", "prime53", "higherPrime", "higherPrime"}

local function steps(et, harm) -- steps of harmonic
return math.floor((math.log(harm) / math.log(2) * et) + 0.5)
end

local function steps_re(et, harm) -- steps of reduced harmonic
return steps(et, harm) - (math.floor(math.log(harm) / math.log(2)) * et)
end

local function rel_err(et, harm) -- relative error of harmonic
st = math.log(harm) / math.log(2) * et
return math.floor(st + 0.5) - st
end

local function abs_err(et, harm) -- absolute error of harmonic
return 1200 / et * rel_err(et, harm)
end

local function rel_col(err) -- color used for relative error
abs_err = math.abs(err)
if(abs_err == 0) then
return "acc0"
end
if(abs_err < 0.03125) then
return "acc1"
end
return "acc" .. math.floor(abs_err * 12) + 2
end

function p.table(frame) -- making the table itself
local et = frame.args["et"]
local nowiki = frame.args["nowiki"]

local type = frame.args["type"]
if (type == "0") then
type = "prime";
end

local harms = primes;
local cols = p_cols;
local p_lim = frame.args["p_lim"]
if (p_lim == "odd" or p_lim == "prime") then
type = p_lim;
p_lim = "default";
end

if (type == "prime") then
if (p_lim == "default") then
p_lim = 31;
end
elseif (type == "odd") then
if (p_lim == "default") then
p_lim = 25;
end
harms = odds;
cols = o_cols;
end
local lim = math.floor(p_lim)
local p_len = #harms

local tab = "{| class=\"wikitable center-all\"\n"
if (tonumber(nowiki) == 1) then
tab = "<nowiki>" .. tab end
tab = tab .. "|+Approximation of " .. type .. " harmonics in "
tab = tab .. et .. "edo\n"
tab = tab .. "! colspan=\"2\" |Harmonic\n"
for i = 1,p_len do
tab = tab .. "! class=\""
tab = tab .. cols[i] .. "\" |" .. harms[i] .. "\n"
if harms[i] >= lim then break end
end
tab = tab .. "|-\n! rowspan=\"2\" |Error\n!Absolute (¢)\n"
for i = 1,p_len do
tab = tab .. "| "
if rel_err(et, harms[i]) > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", abs_err(et, harms[i])) .. "\n"
if harms[i] >= lim then break end
end
tab = tab .. "|-\n!Relative (%)\n"
for i = 1,p_len do
local er = rel_err(et, harms[i])
tab = tab .. "| class=\"" .. rel_col(er) .. "\" | "
if er > 0 then tab = tab .. "+" end
tab = tab .. string.format("%.1f", er * 100) .. "\n"
if harms[i] >= lim then break end
end
tab = tab .. "|-\n! colspan=\"2\" |Steps\n(reduced)\n"
for i = 1,p_len do
tab = tab .. "|" .. steps(et, harms[i]) .. "\n(" .. steps_re(et, harms[i]) .. ")\n"
if harms[i] >= lim then break end
end
tab = tab .. "|}"
if (tonumber(nowiki) == 1) then
tab = tab .. "</nowiki>" end
return tab
end

return p

Template:Navbox regtemp

2026-07-15T03:33:38Z

FloraC: Correct ploidacot

<br />{{Navbox
| name = Navbox regtemp
| Title = [[Regular temperament]]s
| Is Collapsible = true
| Header 1 = Rank-2
| Header 2 = Acot
| Data 2 = [[Blackwood]] (1/5-octave) • [[7edo#Whitewood temperament|Whitewood]] (1/7-octave) • [[12edo#Compton temperament|Compton]] (1/12-octave)
| Header 3 = Monocot
| Data 3 = [[Meantone]] • [[Schismic]] • [[Gentle tuning|Gentle-fifth temperaments]] • [[Archy]]
| Header 4 = Complexity 2
| Data 4 = [[Diaschismic]] (diploid monocot) • [[Pajara]] (diploid monocot) • [[Injera]] (diploid monocot) • [[Rastmatic]] (dicot) • [[Mohajira]] (dicot) • [[Intertridecimal (temperament)|Intertridecimal]] (dicot) • [[Interseptimal (temperament)|Interseptimal]] (alpha-dicot)
| Header 5 = Complexity 3
| Data 5 = [[Augmented (temperament)|Augmented]] (triploid) • [[Misty]] (triploid) • [[Slendric]] (tricot) • [[Porcupine]] (omega-tricot) • [[Hedgehog]] (diploid alpha-tricot)
| Header 6 = Complexity 4
| Data 6 = [[Diminished (temperament)|Diminished]] (tetraploid) • [[Tetracot (temperament)|Tetracot]] (tetracot) • [[Buzzard]] (alpha-tetracot) • [[Squares]] (beta-tetracot) • [[Negri]] (omega-tetracot)
| Header 7 = Complexity 5-6
| Data 7 = [[Magic]] (alpha-pentacot) • [[Amity]] (gamma-pentacot) • [[Kleismic]] (alpha-hexacot) • [[Miracle]] (hexacot)
| Header 8 = Higher complexity
| Data 8 = [[Orwell]] (alpha-heptacot) • [[Sensi]] (beta-heptacot) • [[Octacot (temperament)|Octacot]] (octacot) • [[Wurschmidt]] (beta-octacot) • [[Carlos Alpha|Valentine]] (enneacot) • [[Ammonite]] (epsilon-enneacot) • [[Myna]] (beta-decacot) • [[Ennealimmal]] (enneaploid dicot)
| Header 9 = Straddle-3
| Data 9 = [[A-Team]] (alter-tricot) • [[Machine]] (alter-monocot)
| Header 10 = No-3
| Data 10 = [[Mabilic]] (alpha-triseph) • [[Orgone]] (trimech) • [[Didacus]] (diseph) • [[Sidewalk]] (epsilon-heptaseph)
| Header 11 = No-octaves
| Data 11 = [[Sensamagic]] (monogem)
| Header 12 = Exotemperaments
| Data 12 = [[Dicot (temperament)|Dicot]] • [[Mavila]] • [[Father]]
| Header 13 = Higher-rank
| Header 14 = Rank-3
| Data 14 = [[Aberschismic]] • [[Marvel]] • [[Parapyth]] • [[Pentagoth]] • [[Hibernal]] • [[Intergan]]
}}

Essentially tempered chord

2026-05-24T06:44:20Z

FloraC: Redirected page to Collection of chords#Essentially tempered chords

#redirect [[Collection of chords #Essentially tempered chords]]

Collection of chords

2026-05-24T06:43:39Z

FloraC: /* Essentially tempered chords */ + acronym. This should be a redirect target

[[File:Chords.png|thumb|428x428px|Way too many chords.]]
This page covers chords with fewer than 5 pitch classes. At 5 pitch classes or above, the "chord" becomes more treatable as a scale, and may be found at [[Collection of scales]].

Chords here are [[rotation]]-agnostic unless otherwise specified; the provided interval rulers are in 58edo except for the essentially tempered chords.

== Triads ==
A '''triad''' is a chord containing 3 pitch classes. That is, C-E-G is a triad, but C-E-C is not a triad because even though the Cs may be separated by an octave, they are the same pitch class. Some theorists restrict the word 'triad' to exclusively refer to ''tertian'' triads, composed of two stacked thirds.

=== Fifth-bounded triads ===
<hr>These triads are bounded by a perfect fifth.

==== Suspended triad ====
{{Interval ruler|58|0, 500, 700|1200|2}}{{Interval ruler|58|0, 200, 700|4=2}}

Intervals: '''Unison - Perfect Fourth - Perfect Fifth, Unison - Major Second - Perfect Fifth'''

Just tunings: Usually 6:8:9; 8:9:12

The suspended triad refers to two triads, sus4 and sus2, which are often each other's inversions. The sus4 triad is usually tuned with the perfect fourth and fifth octave complements, so it is the inversion of a corresponding sus2 triad. Sus triads are the simplest kind of triad in Pythagorean tuning, however they are considerably less concordant than other triads, especially with flatter tunings of the fifth (and thus sharper tunings of the fourth, and flatter tunings of the second).

==== Tendo and arto triads ====

<hr>

===== Tendo triad =====
{{Interval ruler|58|0, 450, 700|4=2}}

Intervals: '''Unison - Ultramajor Third - Perfect Fifth'''

Just tunings: 16:21:24, 10:13:15

===== Arto triad =====
{{Interval ruler|58|0, 250, 700|4=2}}

Intervals: '''Unison - Inframinor Third - Perfect Fifth'''

Just tunings: 14:16:21, 26:30:39, 20:23:30<hr>
The tendo triad is generally similarly consonant to the sus4 triad. The intervals themselves are more complex, but the smaller interval is also wider, making the triad feel more open.

The arto triad is the "minor" counterpart of the tendo triad, and serves a similar role in harmony to a standard minor chord, allowing arto and tendo triads to, in a sense, fully replace minor and major ones and form their own system of harmony.

The tendo and arto triads may be used in place of standard major and minor triads to form a tonality system called ''extraclassical tonality,'' which can be expressed in a number of types of tuning systems.

Notably, the tendo triad's third is separated from that of the arto triad by over 200 cents, meaning both can be played at the same time and avoid clashes, similarly to a suspended chord. This phenomenon is called ''cross-tonality'' and it makes 5-, 8-, 12- or 9-form scales wherein both thirds coexist over the same root useful to play in. A simple interpretation of a cross-tonal chord is 20:23:26:30, which suggests the tempering out of 300/299.

Cross-tonal chords are functionally unresolved and ambiguous similar to suspended chords, and cross-tonality also allows notes from arto and tendo scales to be played simultaneously without clashing.

Arto/tendo cross-tonality is a major contributor to the "omniconsonant" sound of 5edo.

* In the 34edo system, tendo differs from diatonic major by a sextula; 34edo is 2.3.5.13.17 so arto and tendo harmony becomes one of the primary kinds of xenharmony that retains the fifth-based structure.
** In hemipythagorean in general, arto and tendo thirds have a canonical tuning value, being half of the perfect fourth and major sixth respectively.
* Assuming the diatonic major and minor thirds are close to pythagorean (such as in 12edo or 41edo), tendo is equivalent to semiaugmented.
* Assuming meantone thirds close to golden meantone, tendo and arto are at the augmented and diminished thirds.
* In sharp archy tunings (sharper than 27edo), the supermajor and subminor thirds approach tendo and arto.
* In slendric, the equal divisions of the fifth may be used as particularly wide arto and tendo tunings.

==== Supermajor and subminor triads ====

<hr>

===== Supermajor triad =====
{{Interval ruler|58|0, 430, 700|4=2}}

Intervals: '''Unison - Supermajor Third - Perfect Fifth'''

Just tunings: 14:18:21

===== Subminor triad =====
{{Interval ruler|58|0, 270, 700|4=2}}

Intervals: '''Unison - Subminor Third - Perfect Fifth'''

Just tunings: 6:7:9

<hr>

The supermajor triad contains a supermajor third, and is usually seen as a 7-limit triad containing the interval 9/7. It is interestingly the *less* stable of the two basic 2.3.7 triads, despite being major, giving 2.3.7 harmony a unique sound.

This type of triad is usually thought of in the 7-limit, where it is significantly more stable than its major counterpart, owing to its much simpler position in the harmonic series. This essentially makes the 2.3.7 subgroup work somewhat backwards melodically compared to the 5-limit - melodically similar chords are stable in 2.3.7 and unstable in 2.3.5.

==== Major and minor triads ====
<hr>

===== Major triad =====
{{Interval ruler|58|0, 390, 700|4=2}}

Intervals: '''Unison - Major Third - Perfect Fifth'''

Just tunings: 4:5:6, 64:81:96, 22:28:33

===== Minor triad =====
{{Interval ruler|58|0, 310, 700|4=2}}

Intervals: '''Unison - Minor Third - Perfect Fifth'''

Just tunings: 10:12:15, 16:19:24, 54:64:81, 22:26:33

<hr>

"Major" and "minor" as chord types tend to encompass both nearmajor/minor and farmajor/minor, partially due to the prevalence of meantone tuning and 12edo itself causing the functional conflation of 5/4 with farmajor thirds. The 58edo intervals shown in the diagram sit between the two categories.

The major triad is one of the two standard varieties of triad. It has a stable, rooted sound coming from its simple enumeration and low placement in the harmonic series. It is represented in 12edo and in the standard tuning of the diatonic scale.

The minor triad is the other of the two standard triads, appearing in and giving its name to the minor scale. While it is conventionally less stable than the major triad, the influence of 16:19:24 in 12edo, according to some, gives it a sense of stability. It is represented in 12edo and in the standard tuning of the diatonic scale.

==== Neutral triad ====
{{Interval ruler|58|0, 350, 700|4=2}}

Intervals: '''Unison - Neutral Third - Perfect Fifth'''

Just tunings: 18:22:27, 26:32:39, 2:sqrt6:3

The neutral triad is associated with the 11-limit and 13-limit. It exists between major and minor triads, and often wants to resolve to either one, creating a tense, unstable sound similar to that of a diminished triad, or in other cases an "ambiguous" quality. While 2:sqrt6:3 isn't strictly just intonation, it is the tuning that utilizes the equal division of the perfect fifth into two identical neutral thirds, so it is in a sense as fundamental as the JI tunings for this particular case.
=== Other triads ===

==== Augmented and diminished triads ====

===== Augmented triad =====
{{Interval ruler|58|0, 390, 780|4=2}}

Intervals: '''Unison - Major Third - Augmented Fifth'''

===== Diminished triad =====
{{Interval ruler|58|0, 300, 620|4=2}}

Intervals: '''Unison - Minor Third - Diminished Fifth'''

==== Latal triads ====
''Main article: [[Chthonic harmony]]''

===== Major latal triad =====
{{Interval ruler|58|0, 270, 500|4=2}}

Intervals: '''Unison - Subminor Third - Perfect Fourth'''

===== Minor latal triad =====
{{Interval ruler|58|0, 230, 500|4=2}}

Intervals: '''Unison - Supermajor Second - Perfect Fourth'''

== Tetrads ==

==== Arto-tendo tetrad ====
{{Interval ruler|58|0, 250, 450, 700|4=2}}

Intervals: '''Unison - Inframinor Third - Ultramajor Third - Perfect Fifth'''

Just tunings: [1/1 15/13 13/10 3/2], 20:23:26:30

The combination of the arto and tendo triads. See [[#Arto triad]] and [[#Tendo triad]] above.

==== Major seventh tetrad ====
{{Interval ruler|58|0, 400, 700, 1100|4=2}}

Intervals: '''Unison - Major Third - Perfect Fifth - Major Seventh'''

Just tunings: 8:10:12:15

==== Minor seventh tetrad ====
{{Interval ruler|58|0, 300, 700, 1020|4=2}}

Intervals: '''Unison - Minor Third - Perfect Fifth - Minor Seventh'''

Just tunings: 10:12:15:18

==== Dominant tetrad ====
{{Interval ruler|58|0, 400, 700, 1000|4=2}}

Intervals: '''Unison - Major Third - Perfect Fifth - Minor Seventh'''

Just tunings: [1/1 5/4 3/2 16/9]

Contains a tritone from the third to the seventh, and is thus generally a tense chord. Found on the V of major diatonic scales, where it resolves to the tonic.

==== Harmonic tetrad ====
{{Interval ruler|58|0, 400, 700, 970|4=2}}

Intervals: '''Unison - Major Third - Perfect Fifth - Subminor Seventh'''

Just tunings: 4:5:6:7

This tetrad approximates the harmonic series and can be seen as a tetradic analogue of 4:5:6 that extends it to the 7-limit.

==== Minor harmonic tetrad ====
{{Interval ruler|58|0, 300, 700, 930|4=2}}

Intervals: '''Unison - Minor Third - Perfect Fifth - Supermajor Sixth'''

Just tunings: 70:84:105:120

Exists in JI, but in jubilic temperaments it becomes a clear "minor" counterpart to the harmonic tetrad. Below are the major and minor harmonic tetrads in 22edo.

{{Interval ruler|22|0, 370, 700, 970|4=2}}{{Interval ruler|22|0, 320, 700, 930|4=2}}

== Trines ==
A trine is a chord with three notes where the two outer notes are separated by an octave. It might not be considered a [[triad]], but rather a specific voicing of a [[dyad]]; trines are not rotation-agnostic.

Conventionally, a trine's middle note lies between an ultramajor third (~450c) and an inframinor sixth (~750c), similar to the qualities of third in a standard fifth-bounded triad generally not exceeding <250c or >450c. The 3-limit trines 3:4:6 and 2:3:4 are very important for 3-limit harmony as concordant conventional triads cannot be constructed.

Trines are also useful in Armodue theory.

=== Ultramajor third trine ===
{{Interval ruler|58|0, 450, 1200|4=2}}

Intervals: '''Unison - Ultramajor Third - Octave'''

Just tunings: 10:13:20

=== Utonal trine ===
{{Interval ruler|58|0, 500, 1200|4=2}}

Intervals: '''Unison - Perfect Fourth - Octave'''

Just tunings: 3:4:6

=== Semiaugmented fourth trine ===
{{Interval ruler|58|0, 550, 1200|4=2}}

Intervals: '''Unison - Semiaugmented Fourth - Octave'''

Just tunings: 8:11:16

=== Neutral trine ===
{{Interval ruler|58|0, 600, 1200|4=2}}

Intervals: '''Unison - Tritone - Octave'''

Just tunings: 5:7:10, 7:10:14

=== Semidiminished fifth trine ===
{{Interval ruler|58|0, 650, 1200|4=2}}

Intervals: '''Unison - Semidiminished Fifth - Octave'''

Just tunings: 8:11:16

=== Otonal trine ===
{{Interval ruler|58|0, 700, 1200|4=2}}

Intervals: '''Unison - Perfect Fifth - Octave'''

Just tunings: 2:3:4

=== Inframinor sixth trine ===
{{Interval ruler|58|0, 750, 1200|4=2}}

Intervals: '''Unison - Inframinor Sixth - Octave'''

Just tunings: 13:20:26

== Essentially tempered chords ==
An '''essentially tempered chord''', a.k.a. '''dyadically essentially tempered chord''' ('''det chord'''), is a chord whose structure is dependent on a temperament.

==== Slendric tetrad ====
{{Interval ruler|41|0, 230, 460, 700|4=2}} (41edo)

Intervals: '''Unison - Supermajor Second - Subfourth - Perfect Fifth'''

Temperament: Gamelic

Distances: 8/7, 8/7, 8/7

Just tuning: [1/1 8/7 21/16 3/2]

A special case of an arto-tendo chord where all three distances between the notes are equal and equated to 8/7, while the chord's span is a perfect fifth 3/2. This can only happen in [[Slendric]] temperament.

==== Symmetric augmented triad ====
{{Interval ruler|36|0, 400, 800, 1200|4=2}} (36edo)

Intervals: '''Unison - Major Third - Minor Sixth'''

Temperament: Augmented

Distances: 5/4, 5/4

Just tuning: [1/1 5/4 8/5]

==== Symmetric diminished tetrad ====
{{Interval ruler|36|0, 300, 600, 900, 1200|4=2}} (36edo)

Intervals: '''Unison - Minor Third - Tritone - Major Sixth'''

Temperament: Diminished

Distances: 6/5, 6/5, 6/5

Just tuning: [1/1 6/5 36/25 5/3]

Meantone

2026-05-24T06:41:29Z

FloraC: /* Chords */ dominant seventh and half-diminished are 7-limit det chords too

[[File:Meantone.png|thumb|Meantone equates four 3/2s to 5/1 and generates pentic (2L 3s) and mosdiatonic (5L 2s) scales.]]
'''Meantone''', or rarely '''Syntonic''' or '''Didymus,''' is a widespread historical [[temperament]] that forms the basis of Western music theory, where the [[Perfect fifth|fifths]] are flattened to about 696[[Cent|c]] to produce a [[diatonic major third]] tuned to roughly [[5/4]], enabling the use of 5-limit harmony in the diatonic scale. When all the fifths are tuned the same, Meantone is a [[regular temperament]], where the period is the [[octave]], the generator is 3/2, and four generators stack to reach the 5th harmonic, meaning that the '''syntonic comma''', which is the difference between Pythagorean intervals and nearby 5-limit intervals and has a ratio of 81/80, is tempered out.

As a monocot temperament, Meantone can be notated with standard [[diatonic notation]], and in fact diatonic notation works the best for Meantone as the 5-limit 4:5:6 harmonic triad becomes simply C-E-G on C, and the chromatic semitone is usually smaller than the diatonic semitone. Meantone is a 7-form temperament, and is tuned well around the golden tuning of diatonic. Unsurprisingly, 7edo supports Meantone, and so does 12edo (which is the simplest ET to do so without exotempering the 5-limit), and so the best tunings of Meantone lie in between those two extremes. [7 & 12] is thus the ''edo join'' for meantone.

== Extensions ==
The edo join [7 & 12] results in an exotempered extension called ''dominant'' where 7/4 and 9/5 are equated, and is tuned best around Pythagorean tuning.

If an unmapped (not equated to a stack of anything else, as [[Classical major third|5/4]] is in Blackwood) prime 7 is introduced as a second generator, then the result can be called Didymus.7. It is supported by 36edo's patent val. (This is not technically an extension.)

More accurate extensions of Meantone's diatonic structure to include other primes follow.

==== 7/4 as the augmented sixth (12 & 19) ====
This is the primary extension of Meantone to the 7-limit, where 7/4 is the augmented sixth (C-A#, +10 fifths). It is best tuned with the generator around 696 cents. 5/4 is 384 cents, and 7/4 is 960 cents. It is notable for being the most accurate extension, as well as containing the [[Golden sequences and tuning|golden tuning]] of the diatonic scale, and thus a melodically convenient chromatic and enharmonic scale. This means that the [[augmented diesis]] 128/125, is equated with the septimal quartertone 36/35, and the 5-limit supermajor and subminor intervals are equated with their septimal counterparts.

However, one drawback of this temperament is the large degree of complexity required to get to the 11th and 13th harmonics. In fact, there are two main options. In both cases, the [[Tridecimal neutral thirds|tridecimal neutral third]] 16/13 is conflated with the [[Undecimal neutral thirds|undecimal neutral third]] 11/9, representing a characteristic tendency to make 11/9 the sharper of the two 11-limit neutral thirds. (As a result, one might find it useful to irregularly map 11/9.)

===== 11-limit[12 & 19] =====
The 11-limit form of 12 & 19 is an exotemperament called ''meanenneadecal'', which tunes 11/8 very sharp and conflates 14/11 with 5/4 (because both 12edo and 19edo do so). More accurate extensions are below.

===== 11/8 as the double-augmented third (12 & 31) =====
This is best tuned around 697 cents, and places 11/9 as the double-augmented second (C-Dx, +16 fifths) and conflates 14/11 with [[Septimal supermajor third|9/7]] placed as the diminished fourth (C-Fb, -8 fifths). 13/8 is mapped to the double-diminished seventh (C-Bbb, -9 fifths).

===== 11/8 as the double-diminished fifth (19 & 31) =====
This is best tuned around 696 cents, and places 11/9 as the double-diminished fourth (C-Fbb, -15 fifths). 13/8 is mapped to the double-augmented fifth (C-Gxx, +15 fifths).

==== 7/4 as the diminished seventh (19 & 26) ====
This temperament, often called "Flattone", sets 7/4 equal to the diminished seventh, and is best tuned with the generator 3/2 around 693 cents, 5/4 at 372 cents, and 7/4 at 963 cents. It is a melodically intuitive extension, as it creates an [[equiheptatonic]] scale with a quartertone-sized chroma, and interval sizes tend to match with their corresponding interval categories. For example, it can be easily extended to map prime 11 to the augmented fourth (C-F#, +6 fifths) and 13 to the minor sixth (C-Ab, -4 fifths) tuned to around 558 and 828 cents respectively. 26edo is the most commonly used tuning, though it can be tuned more accurately with 45edo. It is a 7-cluster temperament, as indicated by the edo join (26 - 19 = 7).

== Chords ==
Meantone's main feature is its conflation of the standard harmonic triad 4:5:6 with the diatonic major triad P1–M3–P5, thus equating the [[Diatonic #MOS diatonic|MOS diatonic scale]] with the 5-limit tuning of [[Diatonic #Greek diatonic scales|Ancient Greek diatonic]] and allowing for 5-limit consonances to be easily accessed within a continuous circle of fifths. Modern Western music theory, which is derived in large part from meantone practice, treats triadic harmony (chords made by stacking two thirds over a root) as the basis of concordance, as the only way to fit three [[5-odd-limit]] intervals in one octave is via some permutation of 4/3, 5/4, and 6/5, which will always make some rotation or retroversion of 4:5:6.

The major and minor seventh chords in meantone diatonic can be enumerated as 8:10:12:15 and 10:12:15:18 respectively. The dominant seventh chord is 20:25:30:36, or in septimal meantone, the 1–5/4–3/2–9/5 [[collection of chords #Essentially tempered chords|essentially tempered chord]]; the half-diminished seventh chord is similarly 25:30:36:45, or in septimal meantone, the 1–6/5–10/7–9/5 essentially tempered chord. Additionally, the 5:6:7:9 chord is available as P1–m3–A4–m7.

During the late Renaissance era, septimal meantone tunings were the basis of Augmented Sixth chords. The Italian Sixth chord can be enumerated as 4:5:7, with the intervals of a root, a major third, and an augmented sixth; the German Sixth chord adds an additional interval 3/2 above the root, providing a full 4:5:6:7, whereas the French Sixth chord adds the augmented fourth of 7/5, making a 20:25:28:35 chord.

The septimal triads, 6:7:9 and 14:18:21, can additionally be found at P1-A2-P5 and P1-d4-P5 respectively. These can be further extended to the septimal seventh chords, 12:14:18:21 and 14:18:21:27, which are respectively P1-A2-P5-A6 and P1-d4-P5-d1 in septimal meantone.

5-limit Meantone also contains an essentially tempered chord, where 1-9/8-3/2-5/3-2 contains steps of 9/8, 4/3, 9/8, and 6/5. Note that in just intonation, the top interval would be 27/16, not 5/3, or the two whole tones would be different sizes (resulting in a 40/27 [[Wolf interval|wolf]] fifth).

== Tunings ==
As essentially the only temperament that is both [[regular]] and attested outside [[xenharmony]], Meantone has a number of historical tunings that today correspond to various extensions and approximate edos. Here, "comma" refers to the syntonic comma.

=== 1/11-comma Meantone ===
This tuning of Meantone is almost perfectly approximated by 12edo, having a fifth tuning of nearly exactly 700 cents and a step ratio of nearly exactly 2 (~basic). 12edo by definition lowers the fifth by 1/12 of a Pythagorean comma; setting 1/12 of a Pythagorean comma to 1/11 of a syntonic comma is done in edos such as [[34edo#612edo|612edo]].

=== 1/5-comma Meantone ===
This tuning of Meantone equalizes the error on 3/2 and 5/4, tuning the former to 697.65 cents and the latter to 390.61 cents. Equivalently, it tunes 16/15 purely. Its step ratio is 1.748 (minisoft), and consequently it is well approximated by 43edo.

=== Quarter-comma Meantone ===
This tunes the fifth 1/4-comma flat, to a size of 696.57 cents. It has a just 5/4, and a step ratio of 1.65 (quasisoft). It approximates [[31edo]], and is often (rather insultingly to the rest of 31edo) seen as the latter's primary feature. It extends to 11-limit 19 & 31.

=== Golden meantone ===
''Main article: [[Golden generator]]''

Golden meantone is the tuning of meantone wherein the large and small steps of the diatonic scale are in the golden ratio. It is the only meantone tuning which produces exclusively soft scales, and meantone's general proximity to the golden tuning captures the difficulty of representing it (and the 5-limit as a whole) within a specific form.

=== 2/7-comma Meantone ===
This is the tuning of Meantone situated roughly between 50edo's and 69edo's tunings, with a fifth of 695.81 cents and a step ratio of 1.584 (quasisoft). Consequently, it extends to Septimal Meantone, but with a rather poor approximation of 7/4. However, it tunes other septimal intervals like 9/7 and 7/6 more accurately. It tunes 25/24 purely, and can thus be considered a compromise between 1/4-comma's perfect 5/4 and 1/3-comma's perfect 6/5.

=== 1/3-comma Meantone ===
This is the tuning of the fifth to 694.78 cents, which has a just 6/5 and is extremely close to [[19edo]], having a step ratio of 1.503 (~monosoft). As a result, it does not cleanly extend to the 11-limit, although as it is slightly sharp of 19edo it does technically extend to 7-limit 12 & 19.

=== Silver flattone ===
Silver flattone is the tuning of meantone such that the step size ratios of the diatonic and enharmonic (19-note) scale steps are the same, and that that ratio is the square root of 2. Alternatively, the step size ratio found in the chromatic scale is the silver ratio, sqrt(2)+1. It is somewhat sharp for flattone, tuning 7/4 flat of 960 cents. Silver flattone is the soft counterpart of [[argent]] tuning.

=== 2/5-comma Meantone ===
This is very close to the [[45edo]] tuning of Meantone, tuning the fifth 693.35 cents and having a just 27/25 (note that 27/25 is tempered together with 16/15 in this system, resulting in a sharp minor second). As a result of the flat tuning, this extends to Flattone, rather than to Septimal Meantone. Its step ratio is 1.401 (parasoft) and is thus close to silver flattone.

=== 1/2-comma Meantone ===
This is close to 33edo's diatonic tuning, which is not Meantone. As a result, it can be considered the lower bound of Meantone's tuning, where the tone is tuned to a just 10/9. It tunes the fifth to 691.2 cents. Its step ratio is 1.26 (ultrasoft).

=== (Half Comma) Cleantone ===
Cleantone is Hans-Peter Deutsch's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 (in fact, any interval in the JI group (3/2).(5/4)) justly, but 2/1 complements of these intervals are detuned.

=== Lucy Tuning ([[88edo|88edo]]) ===
This is nearly indistinguishable from the 88edo tuning of Meantone, with a fifth (600 + 300/π cents) just 0.038 cents higher than 88edo's 695.55-ish fifth. Its major third (1200/π = 381.97 cents) is flat 4.3 cents, but closer than 1/3 Meantone's. The proper extension to the 7-limit (and possible 11-limit) is Mothra (8/7 = 200+100/π = 231.83 cents), which yields a 7/4 flat by only 0.659 cents.

=== [[31edo]] ===

=== [[19edo]] ===

=== [[12edo]] ===

== List of patent vals ==
{| class="wikitable"
|+
!EDO
!7-limit strong extensions
!11-limit strong extensions
!Generator tuning
|-
|5
|Dominant
|
|720c
|-
|12
|Dominant, Septimal Meantone
|[12 & 31], [12 & 19]
|700c
|-
|67
|
|
|698.5c
|-
|55
|
|
|698.2c
|-
|98
|
|
|698c
|-
|43
|Septimal Meantone
|[12 & 31]
|697.7c
|-
|117
|
|[12 & 31]
|697.4c
|-
|74
|Septimal Meantone
|[12 & 31]
|697.3c
|-
|105
|Septimal Meantone
|[12 & 31]
|697.1c
|-
|31
|Septimal Meantone
|[12 & 31], [19 & 31]
|696.8c
|-
|81
|Septimal Meantone
|[19 & 31]
|696.3c
|-
|50
|Septimal Meantone
|[19 & 31]
|696c
|-
|69
|
|[19 & 31]
|695.7c
|-
|88
|
|[26 & 31]
|695.5c
|-
|19
|Septimal Meantone, Flattone
|[12 & 19], [19 & 31], (Flattone)
|694.7c
|-
|45
|Flattone
|(Flattone)
|693.3c
|-
|26
|Flattone
|(Flattone)
|692.3c
|-
|7
|Dominant, Flattone
|(Flattone)
|685.7c
|}

{{Navbox regtemp}}

{{Cat|Temperaments}}

Ups and downs notation

2026-05-12T18:37:38Z

FloraC: Dunno what else to add honestly

'''Ups and downs notation''' refers broadly to a number of notations devised for notating [[equal temperament]]s or similar systems with up and down arrows (^ and v) to indicate alterations smaller than a semitone.

One early adopter of this is '''Kite's ups and downs''' ('''KUD''') '''notation''', which notates any edo up to a certain extent by letting the arrows stand for a single edostep and combining them with standard [[diatonic notation]]. More commonly used is the '''Stein–Zimmermann–Gould''' ('''SZG''') '''notation''', which is a system built on a similar philosophy with more conventional-looking symbols, and is the main system covered on this page.

== Basic premise ==
Standard diatonic notation covers the [[chain of fifths]]. If present, neutral diatonic notation covers half-chroma alterations. In SZG, its use alternates between edos of odd and even sharpnesses i.e. the number of steps the [[chromatic semitone]] is mapped to. For sharp-2 systems such as 17edo, 24edo and 31edo, its use is sufficient to notate the full edo and no arrows are needed.

Ups (^) and downs (v) raise and lower the pitch of a note by a small amount, most commonly one edostep in at the very least all edos up to and including 67edo. Above that point, it may be useful in some systems to let an up or down accidental stand for more than one edostep. For instance, in 94edo, it is best to let the arrows stand for the garibaldi comma. In those systems, lifts (/) and drops (\) may be used to stand for a single edostep. Ups/downs and lifts/drops are always placed before the note/degree/interval name.

Interval arithmetic is preserved, with ups and downs added in. For instance, an upmajor third on an upminor third is a double-up ("dup") perfect fifth. Note that the same terminology is used for the up and down accidentals as for the qualities they define, so for example an "upflat three" degree makes an "upminor third" interval with the unison. This becomes potentially ambiguous when the quality of the interval is not specified, but in most edos this is not a concern.

Note that any arrow or Pythagorean accidental cancels any previous ones.

== Kite's chord names ==
Kite has specified a naming standard for chords in his notation.

In this standard, alterations are enclosed in parentheses, additions never are. Alterations always come last in the chord name. In edos with 7edo's fifth, every interval is perfect, with no major or minor; this serves to resolve the ambiguity introduced by using up/down for both qualities and accidentals. Do not use lowercase roman numerals for chords; a "v" symbol for down might be confused with the "v" symbol in Roman numerals.

Ups and downs apply to chord qualities as well as to intervals, so that "Cv" is a downmajor chord and "C^m" is an upminor chord. Ups and downs may appear in bass figures as well. Note that in general, ups and downs also apply to the sixth, seventh, and eleventh if placed before the normal chord quality, so that "C^m7" is a minor seventh chord with both the third and the seventh raised by a step. If a distance other than a perfect fifth is warranted, the added seventh is placed after a comma punctuation mark (,), like with added intervals in general.

Ups and downs notation

2026-05-12T18:23:13Z

FloraC: Rework on intro & mark Kite's idiosyncrasies

'''Ups and downs notation''' refers broadly to a number of notations devised for notating [[equal temperament]]s or similar systems with up and down arrows (^ and v) to indicate alterations smaller than a semitone.

One early adopter of this is '''Kite's ups and downs''' ('''KUD''') '''notation''', which notates any edo up to a certain extent by letting the arrows stand for a single edostep and combining them with standard [[diatonic notation]]. More commonly used is the '''Stein–Zimmermann–Gould''' ('''SZG''') '''notation''', which is a system built on a similar philosophy with more conventional-looking symbols, and is the main system covered on this page.

== Basic premise ==
Standard diatonic notation covers the chain of fifths. If present, neutral diatonic notation covers half-chroma alterations. (This is sufficient for covering systems such as 17edo and 31edo.)

Ups (^) and downs (v) raise and lower the pitch of a note by 1 EDOstep respectively, in at the very least all edos up to and including 67edo. Above that point, it may be useful in some systems to allow an up or down accidental to stand for more than 1 EDOstep. For instance, in 94edo, it is best to let the arrows stand for the garibaldi comma. In those systems, lifts (/) and drops (\) may be used to stand for a single EDOstep. Ups/downs and lifts/drops are always placed before the note/degree/interval name.

Interval arithmetic is preserved, with ups and downs added in. For instance, an upmajor third on an upminor third is a double-up ("dup") perfect fifth. Note that the same terminology is used for the up and down accidentals as for the qualities they define, so for example an "upflat three" degree makes an "upminor third" interval with the unison. This becomes potentially ambiguous when the quality of the interval is not specified, but in most EDOs this is not a concern.

Note that any arrow or Pythagorean accidental cancels any previous ones.

== Kite's chord names ==
Kite has specified a naming standard for chords in his notation.

In this standard, alterations are enclosed in parentheses, additions never are. Alterations always come last in the chord name. In edos with 7edo's fifth, every interval is perfect, with no major or minor; this serves to resolve the ambiguity introduced by using up/down for both qualities and accidentals. Do not use lowercase roman numerals for chords; a "v" symbol for down might be confused with the "v" symbol in Roman numerals.

Ups and downs apply to chord qualities as well as to intervals, so that "Cv" is a downmajor chord and "C^m" is an upminor chord. Ups and downs may appear in bass figures as well. Note that in general, ups and downs also apply to the sixth, seventh, and eleventh if placed before the normal chord quality, so that "C^m7" is a minor seventh chord with both the third and the seventh raised by a step. If a distance other than a perfect fifth is warranted, the added seventh is placed after a comma punctuation mark (,), like with added intervals in general.

User:FloraC

2026-04-05T09:17:47Z

FloraC: Created page with "Hi, you might know me from the [https://en.xen.wiki/w/User:FloraC Xenharmonic Wiki]. I'm just here to see what this is about."

Hi, you might know me from the [https://en.xen.wiki/w/User:FloraC Xenharmonic Wiki]. I'm just here to see what this is about.