Straddle primes

2026-10-11T12:51:33Z

Hotcrystal0:

{{proposed}}

'''Straddle primes''' are functional primes in a temperament that straddle their respective just interval by having at least one flat and one sharp approximation available. Dual-fifth systems are usually straddle-3.

The ''straddle ratio'' ''m'':''n'' denotes the ratio of the flatter approximation (''m'') to the sharper approximation (''n''). A straddle ratio of approximately 1:1 is called ''equistraddle''.

== Motivation ==

Straddle primes were created for the purpose of error cancellation and flexibility of tempering. For example:
* [[37edo]] has a remarkably accurate 19-limit, with the exception of prime 3. It works, but it doesn't stack predictably. In order to represent intervals involving prime 3 more accurately, powers of 3 can involve stacking the more accurate sharp 3 with an occasional flat 3 to offset error.
* [[86edo]] is larger and has two diatonic fifths, Meantone and Archy. This gives it two diatonic scales with different qualities useful as a base for quasi-diatonic ternary scales. It also clearly straddles the whole 11-limit, providing both the option to choose a version of a prime that results in a better approximation of a ratio, and doubling the probability of encountering a target prime in a given scale. In practice, this probability is lower than double due to preferring approximations with good error cancellation.

With a sufficiently large edo, it's possible to have accurate approximations of primes while also using versions altered by an edostep to generate more interesting scales or result from stacking a different interval. This may be seen as better due to the lower error in using a different JI ratio to represent an altered prime, which begs the question, why use straddle primes at all? The answer is twofold:
* Straddling primes without using a more accurate approximation in the middle widens the smallest step size a tuning must have to represent the straddle. 37edo would have to be tripled to 111edo in order to get an accurate 3, which is totally different in character and size. Larger tuning systems are less practically convenient and make decisions harder.
* Using the same ratio to describe two slightly different intervals drastically simplifies the ratios produced by stacking it, and less care is needed to ensure that error cancellation works out mathematically.

Straddle-primes differ significantly from the typical way ratios are understood in RTT. They do not work for everyone, but their high level of flexibility makes more unusual approaches possible.

{{UserTag|g_|Ground|7766ff|I have a philosophy where I limit edo size to the minimum required to do whatever I'm trying to do, in order to make composition easier. When using a tuning with two diatonic fifths, the smallest interval I actually need always ending up being the difference between them. 3.5¢ is an often mentioned value for the just-noticeable difference (JND), but I find that it can be much larger depending on context. My lower bound for the typical size of an [[Aberrisma|aberrisma]], around 25¢, is the limit where it stops sounding like a clear melodic step in many contexts of the music I write, like a fast melody competing with other instruments, slightly detuned synths, and reverb. This also makes error less noticeable, limiting the effect of extreme concordance. Most of the time in my music, there's hardly a difference between a just 4:5:6 and one tempered with Meantone. In more ideal contexts where notes are more clear, my lower bound is about half that size, not coincidentally the step size of the largest edos I use. Even then, being audible enough as a step is a genuine concern.}}

== Eracs ==

'''Eracs''' (short for error accidentals) are symbols that indicate how much a tempered interval is flat or sharp relative to others in a subgroup. They act as variables representing small pitch differences and have no set size. They are the standard notation for groups involving straddle primes.

Eracs provide a more complete picture of error cancellation than the standard notation of non-prime intervals. For example, 11edo almost perfectly misses primes 3 and 5 present 22edo, which still allows them to cancel out for an accurate 5/3 and 15. 11edo's [[JI group]] might be 2.5/3.15.7.11 in standard notation, which is a subgroup of the erac group 2.x3.x5.7.11. This becomes even more important in edos like 23 or 29, where several low primes are critically inaccurate and representing all of the error cancellation in standard notation creates a very long JI group.

Example erac groups for edos can be found in [[EDO#List of Edos]].

=== Symbols ===
{| class="wikitable"
|+Erac Meanings (intervals represented by an underscore)
|-
|<_
|Flat by a set arbitrary amount.
|-
|>_
|Sharp by a set arbitrary amount.
|-
|x_
|Critically flat/sharp. Shorthand for <_.>_.
|-
|<x_ and >x_
|Shorthand for <<_.>_ and <_.>>_ respectively.
|-
|{ and }
|Partial eracs, indicating error that may be ignored.
|-
|~_
|Approximate, tempered equivalent. A widely accepted symbol used in this context to indicate that no eracs apply.
|}

Eracs are placed before their respective numbers. This is for readability and to remove ambiguity with the denominator, as eracs on the denominator have an inverse effect on the size of a tempered ratio. For example, >5/3 is sharp of 5/3, but 5/>3 is flat and may simplify to <5/3.

=== Erac simplification ===

Deciding how eracs should be assigned to elements of a subgroup is not an exact science. In addition to the amount of precision the composer finds most useful, the exact ways the errors cancel are important to consider.

Assume a 2.3.5 subgroup with a slightly inaccurate 3 which is still good enough to not need eracs, but a straddle 5 that's somewhere between 1:1 and 2:1 error. The direction errors tend in determines the best erac subgroup. The following calculations assume that the most important part of 2.3.5 is 4:5:6 and its inversions and retroversions, with the accuracy of 15 being less of a concern.

<pre>
Just:
3/2 702 0
5/4 386 0
5/3 884 0
</pre>
<pre>
Effectively just flat 3, straddle-5 tends flat:
3/2 699 -3
<5/4 377 -9
>5/4 392 +6
<5/3 878 -6
>5/3 893 +9
</pre>
Here, it makes the most sense to call the subgroup 2.3.<5.>5. The 3 is flat, but not by enough to be worth keeping track of in many cases. Even though the sharp 5 is more accurate, the slightly flat 3 adds to this error and makes the sharp 5/3 less accurate than the flat one. Therefore on average, the two versions of 5 result in about equal errors.

<pre>
Effectively just sharp 3, straddle-5 tends flat:
3/2 705 +3
<5/4 377 -9
>5/4 392 +6
<5/3 872 -12
>5/3 887 +3
</pre>
This is the same as above, but now the 3 is slightly sharp. In this case, the subgroup should be 2.3.<<5.>5. The 3 cancels out some error on the sharp 5, so the sharp 5 results in about half the error of the flat 5 on average.
</pre>

== Straddle temperaments ==

Straddle temperaments use inaccuracy to distort intervals in interesting ways while maintaining their lattice structure.

Temperament optimization algorithms can be built around eracs to create optimal error cancelation, such as [https://spoogly.website/tools/gto.html GTO]. Straddle-prime temperaments have the potential to replace exotemperaments by properly representing the large error in approximating certain primes, providing less misleading mappings with systematic accuracy. Straddle-prime temperaments may also be found as (strong or weak) extensions of subgroup temperaments with prime powers (such as 2.9.21).

Examples of straddle-prime temperaments include:
* Straddle-3 Meantone (sometimes called dual-3 Meantone or Greentone), where >3/2 * <3/2 = 9/4, and (9/4)^2 = 5/1. This is supported by 18edo and 37edo.
* [[Monarch|Monarch and Interarch]]
* [[Straddle cluster temperaments]]

== SRTT ==

'''SRTT''', short for Straddle Regular Temperament Theory, is a generalization of straddle temperaments with eracs that uses variables to represent small alterations of primes. Using the variable ''r'' to represent the erac alteration, the erac subgroup 2.<3.>3.5.7 may be written as 2.3/r.3r.5.7 or simplified to 2.3r.5.7.r^2. If error cancellation is not assumed, more variables may be used, such as 3m/2 for a Meantone fifth and 3a/2 for an Archy fifth as found in [[Monarch]].

=== Cluster temperaments ===

In temperaments with a generator very close to some equal temperament, intervals can be optimized by altering the generator in either direction. This flips the direction of eracs, which may be confusing. SRTT alterations may be positive or negative, and may also be stacked. For example, 4\5 is flat lf 7/4. Buzzard and Hemiseven are two well-known 5edo cluster temperaments, with the Buzzard generator flat of 2\5 and the Hemiseven generator sharp of 2\5.

a and b are arbitrary. It doesn't matter whether they are multiplied or divided, just that both instances of a have the same operation and both instances of b have different operations.
* -3 generators: 7/4 a b
* 2 generatora: 7/4 a/b
At 5edo, b is assumed to be about 1, so a is less than 1 and accounts for the flat tuning of 7/4.
* In Buzzard, the accurate 7/4 is at -3 generators. This is where a and b are about reciprocal, cancelling their errors via multiplication.
* In Hemiseven, the accurate 7/4 is at 2 generators. This is where a and b are about equal, cancelling their errors via division.
* This also means that each cycle of 5 generators ends up at a pitch that differs by a ratio of (7/4 a b)/(7/4 a/b) = b^2.
In temperaments further from optimal, the size of the offset a becomes less significant relative to the ratio between b and 1/b. This is the concept behind straddle cluster temperaments.

== Straddle-2 ==

Because edos have a justly tuned 2, straddling it requires a '''triple-straddle''', that is a '''central''' tuning close to the intended ratio and two '''fringe''' tunings on either side. These terms have not been standardized yet.

Triple-diatonic edos (those with three diatonic generators) may be interpreted as triple-straddle-3, having three viable tunings of 3/1. This is less common than standard straddle-3 because the central 3 is too accurate to really straddle 3 at all, creating a lack of balance. This can extend to triple-straddle-2, which works better than standard straddle-2 because all edos have a central 2. If this were used, it would be more intuitive to triple-straddle both 2 and 3, both to pair versions of primes with error in the same direction and allow 3/1 or 3/2 to be the "anchor consonance" allowing 2/1 to be further off. Then it becomes possible to change the octave relative to the fifth to get scales otherwise not possible. With a just 3/2 generator, a period of a sharp octave resembles Meantone and a flat octave resembles Archy.

SRTT math, using n for the sharp octave that makes Meantone (letter after m) and b for the flat octave that makes Archy (letter after a):
<pre>Just-3/2 tunings:

(2n)^-2 (3/2))^4 = 5/4, n = sqrt(81/80), 2n = 1210.753¢
(2b)^-2 (3/2))^4 = 9/7, b = sqrt(63/64), 2b = 1186.368¢

Just-4/3 tunings are also viable be replacing 3/2 in these equations with 2x / (4/3) = 3x/2, although doing this flips Meantone and Archy.

These are two viable octaves that differ by 24¢, within the typical size range of an aberrisma. It may be incredibly jarring to use them together, but also interesting.

This is distinct from something like (2n)^-2 (3/(2n))^4 = 5/4 or 81/(2n)^6 = 5/4, 2n = 1203.584¢. The result is an undertempered octave because the unaltered generator is assumed to be 3/1 (equivalent to 3/(2n)) rather than 3/2. This is still great for a more even distribution of error though.</pre>

Edos large enough to triple-straddle 2 are especially useful for containing stuctures from adjacent edos stretched or compressed. For example, [[82edo]] lacks Meantone and Archy due to triple-straddling 3, but both are possible by using an alternate octave as shown above. This extends to other structures, such as containing a compressed 27edo inherited from 81edo, known to be more optimal for various ratios, as well as compressed Meantone diasem, whose flat 3 benefits from the flat 2. When triple-straddle-2 is used, it generally sounds better to use pure octaves whenever possible to prevent the stretch or compression from accumulating over the full pitch range.

== Erac-derived temperaments ==

These were a concept derived by [[User:2^67-1|Cole]] to be able to quickly create conventional temperaments.

His procedure:

* For a flat harmonic and a sharp harmonic:
** take the flat harmonic and take it to the power of how many eracs the sharp harmonic has
** take the sharp harmonic and take it to the power of how many eracs the flat harmonic has
** multiply them together
** take it to the power of how many steps there are in an EDO
** octave-reduce (if the interval is greater than 600c octave reduced, take the inverse)

{{Cat|Atypical ratios
Notation}}

User:Hotcrystal0/Sandbox

2026-09-18T20:27:34Z

Hotcrystal0:

Schismic

2026-09-15T20:32:06Z

Hotcrystal0: argent tuning

{{Infobox regtemp|Mapping=1; 1 -8 -3|Subgroup=2.3.5, 2.3.5.19|Edo join 1=12|Edo join 2=29|Generators=3/2|Comma basis=32805/32768 (2.3.5); 361/360, 513/512 (2.3.5.19)|Odd limit 1=5|Complexity 1=12|Generators tuning=701.7|Subgroups=2.3.5, 2.3.5.19|Mistuning 1=0.217|Optimization method=CWE|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}
[[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'''[a], 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 ==
[a] 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}}

2026-05-22T20:29:20Z

Hotcrystal0: Redirected page to Meantone#7/4 as the diminished seventh (19 & 26)

#REDIRECT [[Meantone#7/4 as the diminished seventh (19 & 26)]]