Ripple family: Difference between revisions
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The '''ripple family''' tempers out the ripple | {{Technical data page}} | ||
The '''ripple family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ripple comma]] ([[ratio]]: 6561/6250, {{monzo|legend=1| -1 8 -5 }}), which equates a stack of five [[27/25]]'s with [[4/3]]. | |||
= Ripple = | == Ripple == | ||
The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. The [[ploidacot]] of ripple is omega-pentacot. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered fourth reached at 5 semitones. Interestingly, in optimal tunings, the major third of ~5/4 does not tend to be damaged much sharpwards as one might expect from the equivalence, and is in practice sometimes even flat, so that prime 3 takes on practically the whole damage of the 5-limit equivalence, for which it has the advantage of being the simplest so still having a good chance at psychoacoustic viability. As a result though, the mapping of ~9/8 is often very flat, so that ripple can in practice be thought of as a [[dual-fifth temperament]] unless you use tunings close to [[12edo]]. | |||
Reasonable [[patent val]] tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]]. | |||
[[Subgroup]]: 2.3.5 | |||
[[Comma list]]: 6561/6250 | |||
{{Mapping|legend=1| 1 2 3 | 0 -5 -8 }} | |||
: mapping generators: ~2, ~27/25 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.2636{{c}}, ~27/25 = 100.8602{{c}} | |||
: [[error map]]: {{val| +0.264 -5.729 +7.596 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 100.7982{{c}} | |||
: [[error map]]: {{val| 0.000 -5.946 +7.300 }} | |||
[[Tuning ranges]]: | |||
* [[5-odd-limit]] [[diamond monotone]]: [92.308, 109.091] (1\13 to 1\11) | |||
* 5-odd-limit [[diamond tradeoff]]: [99.609, 105.214] | |||
{{ | {{Optimal ET sequence|legend=1| 11c, 12, 71b, 83b }} | ||
Badness: | [[Badness]] (Sintel): 3.26 | ||
=== Overview to extensions === | |||
The second comma of the comma list defines which 7-limit family member we are looking at: | |||
* Septimal ripple adds [[126/125]]; | |||
* Rip adds [[36/35]]; | |||
Both use the same nominal generator as ripple. | |||
For weak extensions, we have hemiripple and cohemiripple. Hemiripple adds [[49/48]], spliting the semitone generator in two. Cohemiripple adds [[245/243]], spliting the [[octave complement]] of the semitone generator in two. | |||
== Septimal ripple == | |||
{{See also| Dual-fifth temperaments }} | |||
Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~(21/16)⋅(4/3) = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy. | |||
''If you are looking for the former canonical extension, see [[#Rip]].'' | |||
[[Subgroup]]: 2.3.5.7 | |||
Comma list: | [[Comma list]]: [[126/125]], [[405/392]] | ||
{{Mapping|legend=1| 1 2 3 4 | 0 -5 -8 -14 }} | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1201.7546{{c}}, ~15/14 = 102.1309{{c}} | |||
: error map: {{val| +1.755 -9.100 +1.903 +8.360 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 101.7772{{c}} | |||
: error map: {{val| 0.000 -10.841 -0.531 +6.294 }} | |||
{{ | {{Optimal ET sequence|legend=1| 11cd, 12, 35, 47 }} | ||
Badness: | [[Badness]] (Sintel): 1.52 | ||
=== 11-limit === | |||
A notable [[patent val]] tuning of 11-limit ripple not appearing in the optimal ET sequence is [[47edo]]. | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: [[45/44]], [[99/98]], [[126/125]] | |||
Mapping: {{mapping| 1 2 3 4 5 | 0 -5 -8 -14 -18 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1202.5973{{c}}, ~15/14 = 102.7900{{c}} | |||
: error map: {{val| +2.597 -10.710 -0.842 +2.504 +11.449 }} | |||
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 102.2972{{c}} | |||
: error map: {{val| 0.000 -13.441 -4.691 -0.986 +7.333 }} | |||
{{Optimal ET sequence|legend=0| 11cdee, 12, 23de, 35 }} | |||
Badness (Sintel): 1.33 | |||
== Rip == | |||
Formerly known as ''septimal ripple'', but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that [[#Ripple]] admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 36/35, 2560/2401 | |||
{{Mapping|legend=1| 1 2 3 3 | 0 -5 -8 -2 }} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1195.0347{{c}}, ~21/20 = 99.0710{{c}} | |||
: error map: {{val| -4.965 -7.240 +6.223 +18.136 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 100.1093{{c}} | |||
: error map: {{val| 0.000 -2.501 +12.812 +30.956 }} | |||
= | {{Optimal ET sequence|legend=1| 11c, 12 }} | ||
[[Badness]] (Sintel): 1.51 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 36/35, 80/77, 126/121 | |||
Mapping: {{mapping| 1 2 3 3 4 | 0 -5 -8 -2 -6 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1192.7877{{c}}, ~21/20 = 98.7876{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 100.3202{{c}} | |||
{{Optimal ET sequence|legend=0| 11c, 12 }} | |||
Badness (Sintel): 1.28 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 36/35, 40/39, 66/65, 147/143 | |||
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -5 -8 -2 -6 -3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1189.8521{{c}}, ~21/20 = 97.7384{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 99.7618{{c}} | |||
{{Optimal ET sequence|legend=0| 11c, 12f, 37ccddeeeeffff }} | |||
Badness (Sintel): 1.31 | |||
== Hemiripple == | |||
Hemiripple tempers out 49/48 and splits the semitone generator in two for ~36/35. Its ploidacot is omega-decacot. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 49/48, 6561/6250 | |||
{{Mapping|legend=1| 1 2 3 3 | 0 -10 -16 -5 }} | |||
: mapping generators: ~2, ~36/35 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1203.5561{{c}}, ~36/35 = 50.9765{{c}} | |||
: error map: {{val| +3.556 -4.608 +8.730 -13.040 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 50.5928{{c}} | |||
: error map: {{val| 0.000 -7.883 +4.201 -21.790 }} | |||
{{Optimal ET sequence|legend=1| 23d, 24, 47d }} | |||
[[Badness]] (Sintel): 4.43 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 49/48, 121/120, 567/550 | Comma list: 49/48, 121/120, 567/550 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 4 | 0 -10 -16 -5 -13 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1203.5344{{c}}, ~36/35 = 50.9757{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5870{{c}} | |||
{{Optimal ET sequence|legend=0| 23de, 24, 47de }} | |||
Badness (Sintel): 2.21 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 49/48, 66/65, 121/120, 351/350 | Comma list: 49/48, 66/65, 121/120, 351/350 | ||
Mapping: | Mapping: {{mapping| 1 2 3 3 4 4 | 0 -10 -16 -5 -13 -7 }} | ||
{{ | Optimal tunings: | ||
* WE: ~2 = 1202.0936{{c}}, ~36/35 = 50.7232{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5048{{c}} | |||
{{Optimal ET sequence|legend=0| 23de, 24 }} | |||
Badness (Sintel): 1.93 | |||
[[Category: | [[Category:Temperament families]] | ||
[[Category: | [[Category:Ripple family| ]] <!-- main article --> | ||
[[Category:Ripple]] | [[Category:Ripple| ]] <!-- key article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
Latest revision as of 01:54, 28 June 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The ripple family of temperaments tempers out the ripple comma (ratio: 6561/6250, monzo: [-1 8 -5⟩), which equates a stack of five 27/25's with 4/3.
Ripple
The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. The ploidacot of ripple is omega-pentacot. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered fourth reached at 5 semitones. Interestingly, in optimal tunings, the major third of ~5/4 does not tend to be damaged much sharpwards as one might expect from the equivalence, and is in practice sometimes even flat, so that prime 3 takes on practically the whole damage of the 5-limit equivalence, for which it has the advantage of being the simplest so still having a good chance at psychoacoustic viability. As a result though, the mapping of ~9/8 is often very flat, so that ripple can in practice be thought of as a dual-fifth temperament unless you use tunings close to 12edo.
Reasonable patent val tunings not appearing in the optimal ET sequence are 35edo and 47edo.
Subgroup: 2.3.5
Comma list: 6561/6250
Mapping: [⟨1 2 3], ⟨0 -5 -8]]
- mapping generators: ~2, ~27/25
- WE: ~2 = 1200.2636 ¢, ~27/25 = 100.8602 ¢
- error map: ⟨+0.264 -5.729 +7.596]
- CWE: ~2 = 1200.0000 ¢, ~27/25 = 100.7982 ¢
- error map: ⟨0.000 -5.946 +7.300]
- 5-odd-limit diamond monotone: [92.308, 109.091] (1\13 to 1\11)
- 5-odd-limit diamond tradeoff: [99.609, 105.214]
Optimal ET sequence: 11c, 12, 71b, 83b
Badness (Sintel): 3.26
Overview to extensions
The second comma of the comma list defines which 7-limit family member we are looking at:
Both use the same nominal generator as ripple.
For weak extensions, we have hemiripple and cohemiripple. Hemiripple adds 49/48, spliting the semitone generator in two. Cohemiripple adds 245/243, spliting the octave complement of the semitone generator in two.
Septimal ripple
Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~(21/16)⋅(4/3) = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy.
If you are looking for the former canonical extension, see #Rip.
Subgroup: 2.3.5.7
Mapping: [⟨1 2 3 4], ⟨0 -5 -8 -14]]
- WE: ~2 = 1201.7546 ¢, ~15/14 = 102.1309 ¢
- error map: ⟨+1.755 -9.100 +1.903 +8.360]
- CWE: ~2 = 1200.0000 ¢, ~15/14 = 101.7772 ¢
- error map: ⟨0.000 -10.841 -0.531 +6.294]
Optimal ET sequence: 11cd, 12, 35, 47
Badness (Sintel): 1.52
11-limit
A notable patent val tuning of 11-limit ripple not appearing in the optimal ET sequence is 47edo.
Subgroup: 2.3.5.7.11
Comma list: 45/44, 99/98, 126/125
Mapping: [⟨1 2 3 4 5], ⟨0 -5 -8 -14 -18]]
Optimal tunings:
- WE: ~2 = 1202.5973 ¢, ~15/14 = 102.7900 ¢
- error map: ⟨+2.597 -10.710 -0.842 +2.504 +11.449]
- CWE: ~2 = 1200.0000 ¢, ~15/14 = 102.2972 ¢
- error map: ⟨0.000 -13.441 -4.691 -0.986 +7.333]
Optimal ET sequence: 11cdee, 12, 23de, 35
Badness (Sintel): 1.33
Rip
Formerly known as septimal ripple, but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that #Ripple admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit.
Subgroup: 2.3.5.7
Comma list: 36/35, 2560/2401
Mapping: [⟨1 2 3 3], ⟨0 -5 -8 -2]]
- WE: ~2 = 1195.0347 ¢, ~21/20 = 99.0710 ¢
- error map: ⟨-4.965 -7.240 +6.223 +18.136]
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 100.1093 ¢
- error map: ⟨0.000 -2.501 +12.812 +30.956]
Badness (Sintel): 1.51
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 80/77, 126/121
Mapping: [⟨1 2 3 3 4], ⟨0 -5 -8 -2 -6]]
Optimal tunings:
- WE: ~2 = 1192.7877 ¢, ~21/20 = 98.7876 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 100.3202 ¢
Badness (Sintel): 1.28
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 40/39, 66/65, 147/143
Mapping: [⟨1 2 3 3 4 4], ⟨0 -5 -8 -2 -6 -3]]
Optimal tunings:
- WE: ~2 = 1189.8521 ¢, ~21/20 = 97.7384 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 99.7618 ¢
Optimal ET sequence: 11c, 12f, 37ccddeeeeffff
Badness (Sintel): 1.31
Hemiripple
Hemiripple tempers out 49/48 and splits the semitone generator in two for ~36/35. Its ploidacot is omega-decacot.
Subgroup: 2.3.5.7
Comma list: 49/48, 6561/6250
Mapping: [⟨1 2 3 3], ⟨0 -10 -16 -5]]
- mapping generators: ~2, ~36/35
- WE: ~2 = 1203.5561 ¢, ~36/35 = 50.9765 ¢
- error map: ⟨+3.556 -4.608 +8.730 -13.040]
- CWE: ~2 = 1200.0000 ¢, ~36/35 = 50.5928 ¢
- error map: ⟨0.000 -7.883 +4.201 -21.790]
Optimal ET sequence: 23d, 24, 47d
Badness (Sintel): 4.43
11-limit
Subgroup: 2.3.5.7.11
Comma list: 49/48, 121/120, 567/550
Mapping: [⟨1 2 3 3 4], ⟨0 -10 -16 -5 -13]]
Optimal tunings:
- WE: ~2 = 1203.5344 ¢, ~36/35 = 50.9757 ¢
- CWE: ~2 = 1200.0000 ¢, ~36/35 = 50.5870 ¢
Optimal ET sequence: 23de, 24, 47de
Badness (Sintel): 2.21
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 66/65, 121/120, 351/350
Mapping: [⟨1 2 3 3 4 4], ⟨0 -10 -16 -5 -13 -7]]
Optimal tunings:
- WE: ~2 = 1202.0936 ¢, ~36/35 = 50.7232 ¢
- CWE: ~2 = 1200.0000 ¢, ~36/35 = 50.5048 ¢
Badness (Sintel): 1.93