5th-octave temperaments: Difference between revisions

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== 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]] * [[1029/1024|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

Line 26:

Line 27:

[[Badness]]: 0.013309

== Thunder ==

Thunder 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 [[5edo|1\5]]; slendrismic gives this a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])2 = [[8/7]] * [[1029/1024|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). This temperament gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of this temperament. In fact, this temperament combines many convergents and semiconvergents to intervals of [[5edo]] into a single, high-limit temperament:

1\5 = [[23/20]] = [[31/27]] = [[54/47]] = [[85/74]] and 2\5 = [[29/22]] = [[33/25]] = [[62/47]] = [[95/72]] = [[128/97]].

Thunder can be thought of as the [[125edo|125f]] & [[140edo|140]] temperament in the [[37-limit]] add-47 add-97, with both tunings notable in all corresponding limits.

=== 7-limit ===

7-limit Thunder also tempers out the [[4096000/4084101]] (the [[hemfiness comma]]).

[[Subgroup]]: [[7-limit|2.3.5.7]]

[[Comma list]]: [[15625/15552]], [[2100875/2097152]]

[[Optimal tuning]] ([[CTE]]): 317.059{{cent}}

{{Optimal ET sequence|legend=1| 15, 95bc, 110, 125, 140, 265, 405 }}

[[Badness]] (Dirichlet): 3.009

=== 11-limit ===

This temperament extends naturally to the 11-limit by tempering [[385/384]] = ([[147/128]])/([[63/55]]) (or equivalently [[6250/6237]]). Note that this temperament observes the comma [[441/440]], 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]].

[[Subgroup]]: [[11-limit|2.3.5.7.11]]

[[Comma list]]: [[15625/15552]], [[2100875/2097152]], [[385/384]]

[[Optimal tuning]] ([[CTE]]): 317.136{{cent}}

{{Optimal ET sequence|legend=1| 15, 95bce, 110e, 125, 140, 265e, 405ee }}

[[Badness]] (Dirichlet): 1.856

=== 13-limit ===

As this temperament 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|625/624 = S25]] and [[676/675|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|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.

[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]

[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]]

[[Optimal tuning]] ([[CTE]]): 317.136{{cent}}

[[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{{cent}}; 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 this temperament 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 [[15edo|15(ko)]], [[125edo|125f]], [[140edo|140]].

[[Subgroup]]: [[17-limit|2.3.5.7.11.13.17]]

[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]]

[[Optimal tuning]] ([[CTE]]): 317.111{{cent}}

[[Badness]] (Dirichlet): 1.493

=== 19-limit ===

As [[33/25]] and [[95/72]] are both close to the 5 EDO fourth, this temperament 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|400/399 = S20]] = ([[20/19]])/([[21/20]]), which is natural to temper given that we observe [[441/440|441/440 = S21]] as aforementioned.

[[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]

[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]], [[286/285]]

[[Optimal tuning]] ([[CTE]]): 317.091{{cent}}

[[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]]: [[23-limit|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|legend=1| 5 0 5 18 12 0 31 12 16 | 0 6 5 -3 4 14 -8 7 5 }}

[[Optimal tuning]] ([[CTE]]): 317.107{{cent}}

[[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]]: [[29-limit|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|legend=1| 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{{cent}}

[[Badness]] (Dirichlet): 1.318

=== 31-limit ===

By tempering [[3969/3968|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]]: [[31-limit|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|legend=1| 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{{cent}}

[[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]]: [[37-limit|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|legend=1| 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{{cent}}

[[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 [[8589934592/8587340257|(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|legend=1| 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{{cent}}

[[Badness]] (Dirichlet): 1.715

== Pentonismic (rank-5) ==

@@ Line 12: / Line 12: @@
 == Slendrismic ==
 : <small>''See also: [[No-fives subgroup temperaments #Slendrismic]] and [[Slendrisma]]''</small>
+In slendrismic, the period (1\5) is given a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])<sup>2</sup> = [[8/7]] * [[1029/1024|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)<sup>3</sup>. 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
@@ Line 26: / Line 27: @@
 [[Badness]]: 0.013309
+== Thunder ==
+Thunder 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 [[5edo|1\5]]; slendrismic gives this a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])<sup>2</sup> = [[8/7]] * [[1029/1024|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). This temperament gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of this temperament. In fact, this temperament combines many convergents and semiconvergents to intervals of [[5edo]] into a single, high-limit temperament:
+\5 = [[23/20]] = [[31/27]] = [[54/47]] = [[85/74]] and 2\5 = [[29/22]] = [[33/25]] = [[62/47]] = [[95/72]] = [[128/97]].
+Thunder can be thought of as the [[125edo|125f]] & [[140edo|140]] temperament in the [[37-limit]] add-47 add-97, with both tunings notable in all corresponding limits.
+=== 7-limit ===
+-limit Thunder also tempers out the [[4096000/4084101]] (the [[hemfiness comma]]).
+[[Subgroup]]: [[7-limit|2.3.5.7]]
+[[Comma list]]: [[15625/15552]], [[2100875/2097152]]
+{{Mapping|legend=1| 5 0 5 18 | 0 6 5 -3 }}
+[[Optimal tuning]] ([[CTE]]): 317.059{{cent}}
+{{Optimal ET sequence|legend=1| 15, 95bc, 110, 125, 140, 265, 405 }}
+[[Badness]] (Dirichlet): 3.009
+=== 11-limit ===
+This temperament extends naturally to the 11-limit by tempering [[385/384]] = ([[147/128]])/([[63/55]]) (or equivalently [[6250/6237]]). Note that this temperament observes the comma [[441/440]], 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]].
+{{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}
+[[Subgroup]]: [[11-limit|2.3.5.7.11]]
+[[Comma list]]: [[15625/15552]], [[2100875/2097152]], [[385/384]]
+{{Mapping|legend=1| 5 0 5 18 12 | 0 6 5 -3 4 }}
+[[Optimal tuning]] ([[CTE]]): 317.136{{cent}}
+{{Optimal ET sequence|legend=1| 15, 95bce, 110e, 125, 140, 265e, 405ee }}
+[[Badness]] (Dirichlet): 1.856
+=== 13-limit ===
+As this temperament 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]])<sup>2</sup> = [[13/9]] so that a gen above that is [[26/15]] as half of [[3/1]]. As cata tempers [[625/624|625/624 = S25]] and [[676/675|676/675 = S26 = S13/S15]] and as the [[kleisma]] is S25<sup>2</sup> * 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|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.
+{{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}
+[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
+[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]]
+{{Mapping|legend=1| 5 0 5 18 12 0 | 0 6 5 -3 4 14 }}
+[[Optimal tuning]] ([[CTE]]): 317.136{{cent}}
+{{Optimal ET sequence|legend=1| 15, 125f, 140 }}
+[[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{{cent}}; 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 this temperament 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 [[15edo|15(ko)]], [[125edo|125f]], [[140edo|140]].
+{{Optimal ET sequence|legend=1| 15, 125f, 140, 265ef, 405eef }}
+[[Subgroup]]: [[17-limit|2.3.5.7.11.13.17]]
+[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]]
+{{Mapping|legend=1| 5 0 5 18 12 0 31 | 0 6 5 -3 4 14 -8 }}
+[[Optimal tuning]] ([[CTE]]): 317.111{{cent}}
+{{Optimal ET sequence|legend=1| 15, 125f, 140 }}
+[[Badness]] (Dirichlet): 1.493
+=== 19-limit ===
+As [[33/25]] and [[95/72]] are both close to the 5 EDO fourth, this temperament extends naturally to the 19-limit by tempering [[2376/2375]] = ([[33/25]])/([[95/72]]) = ([[6/5]])<sup>3</sup> / ([[19/11]]) and thus equivalently by tempering ([[26/15]])/([[19/11]]) = [[286/285]]. This is equivalent to tempering [[400/399|400/399 = S20]] = ([[20/19]])/([[21/20]]), which is natural to temper given that we observe [[441/440|441/440 = S21]] as aforementioned.
+[[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]
+[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]], [[286/285]]
+{{Mapping|legend=1| 5 0 5 18 12 0 31 12 | 0 6 5 -3 4 14 -8 7 }}
+[[Optimal tuning]] ([[CTE]]): 317.091{{cent}}
+{{Optimal ET sequence|legend=1| 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]]: [[23-limit|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|legend=1| 5 0 5 18 12 0 31 12 16 | 0 6 5 -3 4 14 -8 7 5 }}
+[[Optimal tuning]] ([[CTE]]): 317.107{{cent}}
+{{Optimal ET sequence|legend=1| 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]]: [[29-limit|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|legend=1| 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{{cent}}
+{{Optimal ET sequence|legend=1| 15, 125f, 140 }}
+[[Badness]] (Dirichlet): 1.318
+=== 31-limit ===
+By tempering [[3969/3968|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]]: [[31-limit|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|legend=1| 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{{cent}}
+{{Optimal ET sequence|legend=1| 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]]: [[37-limit|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|legend=1| 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{{cent}}
+{{Optimal ET sequence|legend=1| 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 [[8589934592/8587340257|(128/97)<sup>5</sup> / 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|legend=1| 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{{cent}}
+{{Optimal ET sequence|legend=1| 15ko, 125f, 140 }}
+[[Badness]] (Dirichlet): 1.715
 == Pentonismic (rank-5) ==