|
|
| Line 54: |
Line 54: |
|
| |
|
| A: Yes it's possible. Just one more argument than pure-octave. Issue is I haven't got a satisfactory result. | | A: Yes it's possible. Just one more argument than pure-octave. Issue is I haven't got a satisfactory result. |
|
| |
| == Quick reference ==
| |
| I call equal temperaments in Tenney-Euclidean tuning "ette".
| |
|
| |
| 3-limit TE tuning, which is my preferred tuning for most ets, is "ette3".
| |
|
| |
| Some super easy formulae for such a tuning follows.
| |
|
| |
| === 3-limit TE tuning of ets ===
| |
| {{Databox|Detail|
| |
|
| |
| Given a val A, we have Tenney-weighted val V = AW, where W is the Tenney-weighting matrix.
| |
|
| |
| If T is the Tenney-weighted tuning map, then for any et, for obvious reasons,
| |
|
| |
| [math]t_2/v_2 = t_1/v_1[/math]
| |
|
| |
| Let ''c'' be the coefficient of TE-weighted tuning map ''c'' = ''t''<sub>2</sub>/''t''<sub>1</sub> = ''v''<sub>2</sub>/''v''<sub>1</sub>
| |
|
| |
| Let ''e'' be the [[TE error]] in Breed's RMS, and J be the [[JIP]], then
| |
|
| |
| [math]e = {{!}}{{!}}T - J{{!}}{{!}}_\text {RMS} = \sqrt {\frac {(t_1 - 1)^2 + (t_2 - 1)^2)}{2} }[/math]
| |
|
| |
| Since
| |
|
| |
| [math]
| |
| (t_1 - 1)^2 + (t_2 - 1)^2 \\
| |
| = t_1^2 - 2t_1 + 1 + c^2 t_1^2 - 2c t_1 + 1 \\
| |
| = (c^2 + 1)t_1^2 - 2(c + 1)t_1 + 2
| |
| [/math]
| |
|
| |
| has minimum at
| |
|
| |
| [math]t_1 = \frac{c + 1}{c^2 + 1} = \frac {v_1 (v_1 + v_2)}{v_1^2 + v_2^2}[/math]
| |
|
| |
| and ''f'' (''x'') = sqrt (''x''/2) is a monotonously increasing function
| |
|
| |
| ''e'' has the same minimum point.
| |
|
| |
| Now substitute ''t''<sub>2</sub>/''c'' for ''t''<sub>1</sub>,
| |
|
| |
| [math]
| |
| t_i = \frac {v_i (v_1 + v_2)}{v_1^2 + v_2^2}, i = 1, 2 \\
| |
| e = \frac { {{!}}v_1 - v_2{{!}} }{\sqrt {2(v_1^2 + v_2^2)} }
| |
| [/math]
| |
|
| |
| }}
| |
|
| |
| === 3-limit TOP tuning of ets ===
| |
| {{Databox|Detail|
| |
|
| |
| This part is deduced from Paul Erlich's ''Middle Path''.
| |
|
| |
| [math]
| |
| t_i = \frac {2v_i}{v_1 + v_2}, i = 1, 2 \\
| |
| e = \frac { {{!}}v_1 - v_2{{!}} }{v_1 + v_2}
| |
| [/math]
| |
|
| |
| This ''e'' is also the amount to stretch or compress each prime.
| |
|
| |
| }}
| |
|
| |
| === General TE tuning of ets ===
| |
| {{Databox|Detail|
| |
|
| |
| This time we have a sequence c = {''c''<sub>''n''</sub>}, where
| |
|
| |
| [math]c_i = v_i/v_1, i = 1, 2, \ldots, n[/math]
| |
|
| |
| And just proceed as before,
| |
|
| |
| [math]t_1 = \frac {\sum \vec c}{\vec c^\mathsf T \vec c} = \frac {v_1 \sum V}{VV^\mathsf T}[/math]
| |
|
| |
| Substitute ''t''<sub>''i''</sub>/''c''<sub>''i''</sub> for ''t''<sub>1</sub>,
| |
|
| |
| [math]
| |
| t_i = \frac {v_i \sum V}{VV^\mathsf T}, i = 1, 2, \ldots, n \\
| |
| e = \sqrt {1 - \frac {(\sum V)^2}{n VV^\mathsf T} }
| |
| [/math]
| |
|
| |
| }}
| |
|
| |
| === Notes ===
| |
| * For the nullity-1 temperament tempering out {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>''n''</sub> }}, each prime ''q<sub>i</sub>'' is tuned to
| |
| : <math>-\operatorname {sgn} (m_i) \log_2 (q_i) \frac {\sum_j m_j \log_2 (q_j)}{\sum_j \vert m_j \vert \log_2 (q_j)}</math>
| |
| * Even for ets, TOP and TE tuning are not identical, but close.
| |
| * The relative interval error space of equal temperaments in TOP tuning seems to be linear.
| |
|
| |
|
| ------ | | ------ |