User:FloraC: Difference between revisions

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A: First, unlike 12- and 17et with ambiguous major and minor thirds, 19et's thirds are close enough to 5-limit JI that interpreting them otherwise is like a force. In 12- and 17et, those intervals can represent different ratios in different keys, while in 19et they represent the same ratios better or worse in different keys, and I'm not fond of that. Second, the harmonics of 3, 5, 7, and 13 in 19-et are all flat, so there's not much room to operate. Third, the ambiguity of 4\19 and 15\19 is nice and I want them ''ambiguous in every key''.

== Quick reference ==

=== To quickly obtain TOP tuning ===

* For any temperament tempering out {{monzo| ''m''1 ''m''2 … ''m''''n'' }}, each prime ''pi'' is tuned to log2 (''pi'')(Σ''i'' = 1''n'' ''m''''i'' log2 (''pi''))/(Σ''i'' = 1''n'' |''m''''i''| log2 (''pi'')) (in 8ves).

* For ets, 3-limit TOP tuning and TE tuning are identical (needs further study in higher limits).

* For any et tempering out {{monzo| ''n'' ''m'' }}, stretch the octave by ''δ'' = (''m'' log23 + ''n'')/(|''m''| log23 + |''n''|) to obtain 3-limit TOP/TE tuning (which is my preferred tuning for most ets).

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 A: First, unlike 12- and 17et with ambiguous major and minor thirds, 19et's thirds are close enough to 5-limit JI that interpreting them otherwise is like a force. In 12- and 17et, those intervals can represent different ratios in different keys, while in 19et they represent the same ratios better or worse in different keys, and I'm not fond of that. Second, the harmonics of 3, 5, 7, and 13 in 19-et are all flat, so there's not much room to operate. Third, the ambiguity of 4\19 and 15\19 is nice and I want them ''ambiguous in every key''.
+== Quick reference ==
+=== To quickly obtain TOP tuning ===
+* For any temperament tempering out {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>''n''</sub> }}, each prime ''p<sub>i</sub>'' is tuned to log<sub>2</sub> (''p<sub>i</sub>'')(&Sigma;<sub>''i'' = 1</sub><sup>''n''</sup> ''m''<sub>''i''</sub> log<sub>2</sub> (''p<sub>i</sub>''))/(&Sigma;<sub>''i'' = 1</sub><sup>''n''</sup> |''m''<sub>''i''</sub>| log<sub>2</sub> (''p<sub>i</sub>'')) (in 8ves).
+* For ets, 3-limit TOP tuning and TE tuning are identical (needs further study in higher limits).
+* For any et tempering out {{monzo| ''n'' ''m'' }}, stretch the octave by ''&delta;'' = (''m'' log<sub>2</sub>3 + ''n'')/(|''m''| log<sub>2</sub>3 + |''n''|) to obtain 3-limit TOP/TE tuning (which is my preferred tuning for most ets).
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