5th-octave temperaments
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5edo is the smallest xenharmonic system, as 1edo, 2edo, 3edo and 4edo are all subsets of 12edo.
The most notable 5th-octave family is limmic temperaments – tempering out 256/243 and associates 3\5 to 3/2 as well as 1\5 to 9/8, producing temperaments like blackwood. Equally notable among small equal divisions are the cloudy temperaments – identifying 8/7 with one step of 5edo.
Other families of 5-limit 5th-octave commas are:
- Pental temperaments - tempers out the [-28 25 -5⟩ comma which improves the 3/2 mapping for 5edo, producing a temperament with 3/2 as a generator and 1\5 as a period.
- Quintosec temperaments
- Trisedodge temperaments
Slendrismic
- See also: No-fives subgroup temperaments #Slendrismic and Slendrisma
In slendrismic, the period (1\5) is given a very accurate interpretation of 147/128 = (3/2)/(8/7)2 = 8/7 * S7/S8, which is a significant interval as it is the "harmonic 5edostep" in that it's a rooted (/2^n) interval that approximates 1\5 very well. The generator is 1029/1024, the difference between 8/7 and 147/128 and therefore between 3/2 and (8/7)3. The temperament is named for the very "slender" generator as well as as a pun on "slendric" (which it shouldn't be confused with). One can consider this as a microtemperament counterpart to cloudy, which equates them.
Subgroup: 2.3.7
Comma list: 68719476736/68641485507
Mapping: [⟨5 0 18], ⟨0 2 -1]]
- Mapping generators: ~147/128 = 1＼5, ~262144/151263
Optimal tuning (CTE): ~8/7 = 230.9930 (or ~1029/1024 = 9.0080)
Optimal ET sequence: 130, 135, 265, 400, 1065, 1465, 1865
Badness: 0.013309
Thunderclysmic
Thunderclysmic is a weak extension of slendrismic (above), rainy and cata, with a generator of a slightly sharp ~6/5 (befitting of any kleismic temperament), three of which making 26/15~19/11. More interesting though is that the period is 1\5; slendrismic gives this a very accurate interpretation of 147/128 = (3/2)/(8/7)2 = 8/7 * S7/S8 which is a significant interval as it is the "harmonic 5edostep" (in that it's a rooted (/2^n) interval in the 2.3.7 subgroup that approximates 1\5 very well). Thunderclysmic gives a wealth of interpretations to 5edo intervals, which are available everywhere due to 1\5 = 240 ¢ being the period of Thunderclysmic. In fact, Thunderclysmic combines many convergents and semiconvergents to intervals of 5edo into a single, high-limit temperament:
1\5 = 23/20 = 31/27 = 85/74 = 54/47 (which Thunderclysmic also equates with 63/50), and 2\5 = 33/25 = 95/72 = 29/22 = 62/47 = 128/97 (which Thunderclysmic also equates with 37/28 and 120/91).
Thunderclysmic can be thought of as the 125f & 140 temperament in the 37-limit add-47 add-97, with both tunings notable in all corresponding limits.
It is a complex temperament, however note that it is especially strong and elegant in the no-13's no-17's no-31's 37-limit add-97, on which its mapping is especially efficient, so that the gen range is -3 to 7 for a(n inclusive) span of 11 notes per 1\5 period, so that a 5*11=55 note-per-octave (multiperiod) MOS may be used.
7-limit
7-limit Thunderclysmic also tempers out the 4096000/4084101 (the hemfiness comma).
Comma list: 15625/15552, 2100875/2097152
Mapping: [⟨5 0 5 18], ⟨0 6 5 -3]]
Optimal tuning (CTE): 317.059 ¢
Optimal ET sequence: 15, 95bc, 110, 125, 140, 265, 405
Badness (Dirichlet): 3.009
11-limit
Thunderclysmic extends naturally to the 11-limit by tempering 385/384 = (147/128)/(63/55) (or equivalently 6250/6237). Note that Thunderclysmic observes the comma 441/440 = (21/20)/(22/21) = S21, as if it didn't, we would have 63/55 also equated with 8/7, leading to the 15edo tuning which tempers the cloudy comma. In the 11-limit, the 5 EDO fourth is interpreted as 33/25.
Comma list: 15625/15552, 2100875/2097152, 385/384
Mapping: [⟨5 0 5 18 12], ⟨0 6 5 -3 4]]
Optimal tuning (CTE): 317.136 ¢
Optimal ET sequence: 15, 95bce, 110e, 125, 140, 265e, 405ee
Badness (Dirichlet): 1.856
13-limit
As Thunderclysmic is a weak extension of cata, it is naturally at least 13-limit. Cata admits a fairly simple mapping of prime 13 via (6/5)2 = 13/9 so that a gen above that is 26/15 as half of 3/1. As cata tempers 625/624 = S25 and 676/675 = S26 = S13/S15 and as the kleisma is S252 * S26, this replaces the kleisma in the comma list so that we now move it to the end (as both are 13-limit). For simplicity, we show 325/324 = S25 * S26 and the more structurally important aforementioned comma 676/675, omitting 625/624. It also tempers 1001/1000 and 1716/1715 in the 13-limit.
Comma list: 2100875/2097152, 385/384, 325/324, 676/675
Mapping: [⟨5 0 5 18 12 0], ⟨0 6 5 -3 4 14]]
Optimal tuning (CTE): 317.136 ¢
Optimal ET sequence: 15, 125f, 140, 405eef
Badness (Dirichlet): 1.458
17-limit
In the 17-limit, the 5 EDO fifth is interpreted as 85/56 = 561/560 * 50/33, so that 17/16 is reached at 11 periods minus 8 gens at approx. 103 ¢; equating it with 16/15 might seem natural but is not the route taken due to the precision affording observing their difference, 256/255. (If you do want to equate 17/16 with 16/15, you get the 15 & 155 temperament instead, for which the main tuning is 155edo, but you very much do pay for it; notice the errors!) Interestingly, the CTE tuning of the 17-limit of Thunderclysmic is practically the same as that of the 29-limit (up to a thousandth of a cent), which is also notable as being where this temperament exhibits the lowest Dirichlet badness. It's also the smallest prime limit where the vals larger than 140edo haven't disappeared from the optimal ET sequence, as from the 19-limit and onwards the optimal ET sequence is always 15(ko), 125f, 140.
Comma list: 2100875/2097152, 385/384, 325/324, 676/675, 561/560
Mapping: [⟨5 0 5 18 12 0 31], ⟨0 6 5 -3 4 14 -8]]
Optimal tuning (CTE): 317.111 ¢
Optimal ET sequence: 15, 125f, 140, 265ef, 405eef
Badness (Dirichlet): 1.493
19-limit
As 33/25 and 95/72 are both close to the 5 EDO fourth, Thunderclysmic extends naturally to the 19-limit by tempering 2376/2375 = (33/25)/(95/72) = (6/5)3 / (19/11) and thus equivalently by tempering (26/15)/(19/11) = 286/285. This is equivalent to tempering 400/399 = S20 = (20/19)/(21/20), which is natural to temper given that we observe 441/440 = S21 as aforementioned.
Comma list: 2100875/2097152, 385/384, 325/324, 676/675, 561/560, 286/285
Mapping: [⟨5 0 5 18 12 0 31 12], ⟨0 6 5 -3 4 14 -8 7]]
Optimal tuning (CTE): 317.091 ¢
Optimal ET sequence: 15, 125f, 140
Badness (Dirichlet): 1.507
23-limit
By tempering 736/735 = (46/45)/(49/48) we can equate 23/20 with 147/128 = 1\5 to extend to the 23-limit. This is equivalent to tempering 253/252 = (23/21)/(12/11).
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 2100875/2097152, 385/384, 325/324, 676/675, 561/560, 286/285, 736/735
Mapping: [⟨5 0 5 18 12 0 31 12 16], ⟨0 6 5 -3 4 14 -8 7 5]]
Optimal tuning (CTE): 317.107 ¢
Optimal ET sequence: 15, 125f, 140
Badness (Dirichlet): 1.424
29-limit
By tempering (33/25)/(29/22) = 726/725 we give another (slightly simpler) interpretation to the 5 EDO fourth to extend to the 29-limit. This is equivalent to tempering 2640/2639 = (120/91)/(29/22), which reveals that another 13-limit interpretation of the 5 EDO fourth is 120/91.
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 2100875/2097152, 385/384, 325/324, 676/675, 561/560, 286/285, 736/735, 726/725
Mapping: [⟨5 0 5 18 12 0 31 12 16 19], ⟨0 6 5 -3 4 14 -8 7 5 4]]
Optimal tuning (CTE): 317.111 ¢
Optimal ET sequence: 15, 125f, 140
Badness (Dirichlet): 1.318
31-limit
By tempering 3969/3968 = S63 = (147/128)/(31/27), we give another interpretation to 1\5. This is the most complex mapping in this temperament, as reaching 27 requires 18 gens because reaching 3 requires 6 gens (as per kleismic).
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Comma list: 2100875/2097152, 385/384, 325/324, 676/675, 561/560, 286/285, 736/735, 726/725, 3969/3968
Mapping: [⟨5 0 5 18 12 0 31 12 16 19 1], ⟨0 6 5 -3 4 14 -8 7 5 4 18]]
Optimal tuning (CTE): 317.073 ¢
Optimal ET sequence: 15k, 125f, 140
Badness (Dirichlet): 1.501
37-limit
By tempering 407/406 = (37/28)/(29/22), we give another interpretation to the 5 EDO fourth. This is equivalent to equating 15/13 with 37/32 by tempering 481/480.
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
Comma list: 2100875/2097152, 385/384, 325/324, 676/675, 561/560, 286/285, 736/735, 726/725, 3969/3968, 481/480
Mapping: [⟨5 0 5 18 12 0 31 12 16 19 1 30], ⟨0 6 5 -3 4 14 -8 7 5 4 18 -3]]
Optimal tuning (CTE): 317.068 ¢
Optimal ET sequence: 15k, 125f, 140
Badness (Dirichlet): 1.537
37-limit add-47 add-97
To the 37-limit, we add equivalences 1\5 = 54/47 (tempering S48 = (48/47)/(49/48) = 2304/2303) and 3\5 = 97/64 (tempering (128/97)5 / 4 = 8589934592/8587340257), but this can be expressed using a less long ratio by describing it as tempering S96 = 9216/9215 = (97/64)/(144/95), from which we can observe 144/95 as another accurate interpretation of the 5 EDO fifth.
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.47.97
Comma list: 2100875/2097152, 385/384, 325/324, 676/675, 561/560, 286/285, 736/735, 726/725, 3969/3968, 481/480, 2304/2303, 9216/9215
Mapping: [⟨5 0 5 18 12 0 31 12 16 19 1 30 4 33], ⟨0 6 5 -3 4 14 -8 7 5 4 18 -3 18 0]]
Optimal tuning (CTE): 317.053 ¢
Optimal ET sequence: 15ko, 125f, 140
Badness (Dirichlet): 1.715
Pentonismic (rank-5)
Subgroup: 2.3.5.7.11.13
Comma list: 281974669312/281950621875
Mapping: [⟨5 0 0 0 0 24], ⟨0 1 0 0 0 -1], ⟨0 0 1 0 0 -1], ⟨0 0 0 1 0 1], ⟨0 0 0 0 1 0]]
- Mapping generators: ~224/195 = 1＼5, ~3, ~5, ~7, ~11
Supporting ETs: 10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585
Quint
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what blacksmith does.
Subgroup: 2.3.5.7
Comma list: 16/15, 27/25
Mapping: [⟨5 8 12 0], ⟨0 0 0 1]]
- Mapping generators: ~9/8, ~7
Wedgie: ⟨⟨ 0 0 5 0 8 12 ]]
- POTE: ~9/8 = 1\5, ~7/4 = 1017.903 (~21/20 = 57.903)
- CTE: ~9/8 = 1\5, ~7/4 = 968.8259 (~63/64 = 8.8259)
Badness: 0.048312