User:FloraC/Quick reference: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
No edit summary
-outdated stuff
Line 43: Line 43:
* Mode 48
* Mode 48
* Followed by a record prime gap
* Followed by a record prime gap
== Tuning equal temperaments ==
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 ===
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'' &#61; ''t''<sub>2</sub>/''t''<sub>1</sub> &#61; ''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 &#61; {{!}}{{!}}T - J{{!}}{{!}}_\text {RMS} &#61; \sqrt {\frac {(t_1 - 1)^2 + (t_2 - 1)^2)}{2} }[/math]
Since
[math]
(t_1 - 1)^2 + (t_2 - 1)^2 \\
&#61; t_1^2 - 2t_1 + 1 + c^2 t_1^2 - 2c t_1 + 1 \\
&#61; (c^2 + 1)t_1^2 - 2(c + 1)t_1 + 2
[/math]
has minimum at
[math]t_1 &#61; \frac{c + 1}{c^2 + 1} &#61; \frac {v_1 (v_1 + v_2)}{v_1^2 + v_2^2}[/math]
and ''f'' (''x'') &#61; 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 &#61; \frac {v_i (v_1 + v_2)}{v_1^2 + v_2^2}, i &#61; 1, 2 \\
e &#61; \frac { {{!}}v_1 - v_2{{!}} }{\sqrt {2(v_1^2 + v_2^2)} }
[/math]
=== 3-limit TOP tuning of ets ===
This part is deduced from Paul Erlich's ''Middle Path''.
[math]
t_i &#61; \frac {2v_i}{v_1 + v_2}, i &#61; 1, 2 \\
e &#61; \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 ===
This time we have a sequence c &#61; {''c''<sub>''n''</sub>}, where
[math]c_i &#61; v_i/v_1, i &#61; 1, 2, \ldots, n[/math]
And just proceed as before,
[math]t_1 &#61; \frac {\sum \vec c}{\vec c^\mathsf T \vec c} &#61; \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 &#61; \frac {v_i \sum V}{VV^\mathsf T}, i &#61; 1, 2, \ldots, n \\
e &#61; \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.

Revision as of 08:33, 21 August 2023

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
Retrieved from "https://en.xen.wiki/index.php?title=User:FloraC/Quick_reference&oldid=122379"