Xenharmonic Wiki

5th-octave temperaments: Difference between revisions

← Older edit
VisualWikitext
No edit summary
Overthink (talk | contribs)
autopatrolled
2,790 edits
→Quint: move to very low accuracy temperaments
 
(3 intermediate revisions by 2 users not shown)
Line 4: Line 4:


Other families of 5-limit 5th-octave commas are:
Other families of 5-limit 5th-octave commas are:
* [[Pental family|Pental temperaments]] - tempers out the {{monzo|-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.
* [[Quintile family|Quintile temperaments]] - tempers out the {{monzo|-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 family|Quintosec temperaments]]
* [[Quintosec family|Quintosec temperaments]]
* [[Trisedodge family|Trisedodge temperaments]]
* [[Trisedodge family|Trisedodge temperaments]]
Line 24: Line 24:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~147/128 = 240.0000, ~8/7 = 230.9930 (~1029/1024 = 9.0080)
* [[CTE]]: ~147/128 = 240.000{{c}}, ~8/7 = 230.992{{c}} (~1029/1024 = 9.008{{c}})
* [[POTE]]: ~147/128 = 240.0000, ~8/7 = 231.0094 (~1029/1024 = 8.9906)
* [[CWE]]: ~147/128 = 240.000{{c}}, ~8/7 = 231.004{{c}} (~1029/1024 = 8.996{{c}})


{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 935, 1335, 1735, 3070, 4805d }}
{{Optimal ET sequence|legend=1| 130, 135, 265, 400, 935, 1335, 1735, 3070, 4805d }}


[[Badness]] (Sintel): 0.456
[[Badness]] (Sintel): 0.456
[[Tp tuning #T2 tuning|RMS error]]: 0.0212 cents


== Thunderclysmic ==
== Thunderclysmic ==
Line 51: Line 49:
{{Mapping|legend=1| 5 0 5 18 | 0 6 5 -3 }}
{{Mapping|legend=1| 5 0 5 18 | 0 6 5 -3 }}


[[Optimal tuning]] ([[CTE]]): 317.059{{cent}}  
[[Optimal tuning]]s:
* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.059{{c}}
* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.046{{c}}


{{Optimal ET sequence|legend=1| 15, 95bc, 110, 125, 140, 265, 405 }}
{{Optimal ET sequence|legend=1| 15, 95bc, 110, 125, 140, 265, 405 }}
Line 62: Line 62:
[[Subgroup]]: [[11-limit|2.3.5.7.11]]
[[Subgroup]]: [[11-limit|2.3.5.7.11]]


[[Comma list]]: [[15625/15552]], [[2100875/2097152]], [[385/384]]
[[Comma list]]: [[385/384]], [[1331/1323]], [[6250/6237]]


{{Mapping|legend=1| 5 0 5 18 12 | 0 6 5 -3 4 }}
{{Mapping|legend=1| 5 0 5 18 12 | 0 6 5 -3 4 }}


[[Optimal tuning]] ([[CTE]]): 317.136{{cent}}
[[Optimal tuning]]s:
* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.107{{c}}
* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.055{{c}}


{{Optimal ET sequence|legend=1| 15, 95bce, 110e, 125, 140, 265e, 405ee }}
{{Optimal ET sequence|legend=1| 15, 95bce, 110e, 125, 140, 265e, 405ee }}
Line 77: Line 79:
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]]


[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]]
[[Comma list]]: [[325/324]], [[385/384]], [[625/624]], [[1331/1323]]


{{Mapping|legend=1| 5 0 5 18 12 0 | 0 6 5 -3 4 14 }}
{{Mapping|legend=1| 5 0 5 18 12 0 | 0 6 5 -3 4 14 }}


[[Optimal tuning]] ([[CTE]]): 317.136{{cent}}
[[Optimal tuning]] ([[CTE]]): 317.136{{cent}}
[[Optimal tuning]]s:
* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.136{{c}}
* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.086{{c}}


{{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}
{{Optimal ET sequence|legend=1| 15, 125f, 140, 405eef }}
Line 92: Line 98:
[[Subgroup]]: [[17-limit|2.3.5.7.11.13.17]]
[[Subgroup]]: [[17-limit|2.3.5.7.11.13.17]]


[[Comma list]]: [[2100875/2097152]], [[385/384]], [[325/324]], [[676/675]], [[561/560]]
[[Comma list]]: [[325/324]], [[385/384]], [[442/441]], [[625/624]], [[1331/1323]]


{{Mapping|legend=1| 5 0 5 18 12 0 31 | 0 6 5 -3 4 14 -8 }}
{{Mapping|legend=1| 5 0 5 18 12 0 31 | 0 6 5 -3 4 14 -8 }}


[[Optimal tuning]] ([[CTE]]): 317.111{{cent}}
[[Optimal tuning]]s:
* [[CTE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.111{{c}}
* [[CWE]]: ~147/128 = 240.000{{c}}, ~6/5 = 317.066{{c}}


{{Optimal ET sequence|legend=1| 15, 125f, 140, 265ef, 405eef }}
{{Optimal ET sequence|legend=1| 15, 125f, 140, 265ef, 405eef }}
Line 107: Line 115:
[[Subgroup]]: [[19-limit|2.3.5.7.11.13.17.19]]
[[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]]
[[Comma list]]: [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[442/441]], [[1331/1323]]


{{Mapping|legend=1| 5 0 5 18 12 0 31 12 | 0 6 5 -3 4 14 -8 7 }}
{{Mapping|legend=1| 5 0 5 18 12 0 31 12 | 0 6 5 -3 4 14 -8 7 }}
Line 122: Line 130:
[[Subgroup]]: [[23-limit|2.3.5.7.11.13.17.19.23]]
[[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]]
[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[442/441]], [[484/483]]


{{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 | 0 6 5 -3 4 14 -8 7 5 }}
{{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 | 0 6 5 -3 4 14 -8 7 5 }}
Line 137: Line 145:
[[Subgroup]]: [[29-limit|2.3.5.7.11.13.17.19.23.29]]
[[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]]
[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[442/441]], [[484/483]], [[552/551]]


{{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 | 0 6 5 -3 4 14 -8 7 5 4 }}
{{Mapping|legend=1| 5 0 5 18 12 0 31 12 16 19 | 0 6 5 -3 4 14 -8 7 5 4 }}
Line 152: Line 160:
[[Subgroup]]: [[31-limit|2.3.5.7.11.13.17.19.23.29.31]]
[[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]]
[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[435/434]], [[442/441]], [[484/483]], [[528/527]]


{{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 }}
{{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 }}
Line 167: Line 175:
[[Subgroup]]: [[37-limit|2.3.5.7.11.13.17.19.23.29.31.37]]
[[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]]
[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[435/434]], [[442/441]], [[481/480]], [[484/483]], [[528/527]]


{{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 }}
{{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 }}
Line 182: Line 190:
[[Subgroup]]: 2.3.5.7.11.13.17.19.23.29.31.37.47.97
[[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]]
[[Comma list]]: [[253/252]], [[286/285]], [[325/324]], [[385/384]], [[400/399]], [[435/434]], [[442/441]], [[481/480]], [[484/483]], [[528/527]], [[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 }}
{{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 }}
Line 204: Line 212:


Supporting ETs: {{Optimal ET sequence|10, 50, 80, 120, 125, 270, 2000, 2460, 3125, 3395, 5585}}
Supporting ETs: {{Optimal ET sequence|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 [[blackwood]] does.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 16/15, 27/25
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
: Mapping generators: ~9/8, ~7
[[Optimal tuning]]s:
* [[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)
{{Optimal ET sequence|legend=1| 5, 15ccd }}
[[Badness]]: 0.048312


== Obscenity ==
== Obscenity ==
Retrieved from "https://en.xen.wiki/w/5th-octave_temperaments"