User:FloraC/Quick reference: Difference between revisions
Created page with "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..." |
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== Taxonomy of tuning approaches == | |||
* Tuning rationalism | |||
** JI purism: this school recognizes that the acoustic quality of JI is of top importance. Some consider music as a platonic ideal object that cannot be approximated at all. Meanwhile, its weaker version is characterized by being maximally strict about JI approximation. | |||
*** Primodality: I don't feel entitled to define this. | |||
*** Stacking based aka lattice based: a more traditional approach to JI. They recognize both the acoustic quality and the algebraic structure of JI. | |||
** JI approximabilism: this school recognizes that the acoustic quality of JI and the algebraic structure of tuning systems are similarly important, and therefore accepts a tradeoff. | |||
*** RTT: this school encompasses stacking based JI and applies approximation for custom structures. | |||
** JI agnosticism: this school suspends the question whether the acoustic quality of JI is of importance. It tends to focus on algebraic structures such as mos scales and generalizations. | |||
** JI indifferentism: this school does not believe the acoustic quality of JI is of importance. Practice in this school is orthogonal to the influence of JI. | |||
* Tuning empiricism | |||
* Tuning stochasticism | |||
== Important prime limits == | |||
; 3-limit (rank-2): | |||
* Essential interval functions | |||
; 13-limit (rank-6): | |||
* Essential interval colors | |||
* Tonality: tonal and microtonal | |||
* Categorical characteristics: pivotal, ambitonal, and semiambitonal | |||
* Mode 8 | |||
; 23-limit (rank-9): | |||
* Limit of classical functional harmony | |||
* Limit of classical concordance | |||
* Tonality: pseudotonal and pseudomicrotonal | |||
* Categorical characteristics: pseudoambitonal | |||
* Mode 12 | |||
* Followed by a record prime gap | |||
; 31-limit (rank-11) | |||
* Mode 16 | |||
; 37-limit (rank-12) | |||
; 47-limit (rank-15) | |||
* Mode 24 | |||
; 61-limit (rank-18) | |||
* Mode 32 | |||
; 89-limit (rank-24) | |||
* Mode 48 | |||
* Followed by a record prime gap | |||
== Tuning equal temperaments == | |||
I call equal temperaments in Tenney-Euclidean tuning "ette". | I call equal temperaments in Tenney-Euclidean tuning "ette". | ||
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Some super easy formulae for such a tuning follows. | Some super easy formulae for such a tuning follows. | ||
== 3-limit TE tuning of ets == | === 3-limit TE tuning of ets === | ||
Given a val A, we have Tenney-weighted val V = AW, where W is the Tenney-weighting matrix. | Given a val A, we have Tenney-weighted val V = AW, where W is the Tenney-weighting matrix. | ||
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[/math] | [/math] | ||
== 3-limit TOP tuning of ets == | === 3-limit TOP tuning of ets === | ||
This part is deduced from Paul Erlich's ''Middle Path''. | This part is deduced from Paul Erlich's ''Middle Path''. | ||
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This ''e'' is also the amount to stretch or compress each prime. | This ''e'' is also the amount to stretch or compress each prime. | ||
== General TE tuning of ets == | === General TE tuning of ets === | ||
This time we have a sequence c = {''c''<sub>''n''</sub>}, where | This time we have a sequence c = {''c''<sub>''n''</sub>}, where | ||
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[/math] | [/math] | ||
== Notes == | === 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 | * 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> | : <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 | * Even for ets, TOP and TE tuning are not identical, but close. | ||