User:FloraC/Quick reference: Difference between revisions

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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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I call equal temperaments in Tenney-Euclidean tuning "ette".  
== 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 a.k.a. 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


3-limit TE tuning, which is my preferred tuning for most ets, is "ette3".
== Important prime limits ==
; 2-limit (rank-1):
* Essential equivalence
* Completes the harmonic series for the first octave


Some super easy formulae for such a tuning follows.
; 3-limit (rank-2):
* Essential interval functions
* Completes the harmonic series for the first 2 octaves
* Rank is a highly composite number


== 3-limit TE tuning of ets ==
; 5-limit (rank-3):
Given a val A, we have Tenney-weighted val V = AW, where W is the Tenney-weighting matrix.
* Completes the harmonic series for the first 2 octaves and a fifth


If T is the Tenney-weighted tuning map, then for any et, for obvious reasons,
; 7-limit (rank-4):
* Tonality: tonal
* Categorical characteristics: pivotal and semiambitonal
* Completes the harmonic series for the first 3 octaves
* Rank is a highly composite number


[math]t_2/v_2 = t_1/v_1[/math]
; 11-limit (rank-5):
* Completes the harmonic series for the first 3 octaves and a fifth


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>
; 13-limit (rank-6):
* Essential interval colors
* Tonality: microtonal
* Categorical characteristics: ambitonal and semiambitonal
* Completes the harmonic series for the first 4 octaves
* Rank is a highly composite number


Let ''e'' be the [[TE error]] in Breed's RMS, and J be the [[JIP]], then
; 23-limit (rank-9):
* Limit of classical functional harmony
* Limit of classical concordance
* Tonality: pseudotonal and pseudomicrotonal
* Categorical characteristics: pseudoambitonal
* Completes the harmonic series for the first 4 octaves and a fifth
* Followed by a record prime gap


[math]e &#61; {{!}}{{!}}T - J{{!}}{{!}}_\text {RMS} &#61; \sqrt {\frac {(t_1 - 1)^2 + (t_2 - 1)^2)}{2} }[/math]
; 31-limit (rank-11)
* Completes the harmonic series for the first 5 octaves


Since
; 37-limit (rank-12)
* Rank is a highly composite number


[math]
; 47-limit (rank-15)
(t_1 - 1)^2 + (t_2 - 1)^2 \\
* Completes the harmonic series for the first 5 octaves and a fifth
&#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
; 61-limit (rank-18)
* Completes the harmonic series for the first 6 octaves


[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]
; 89-limit (rank-24)
* Completes the harmonic series for the first 6 octaves and a fifth
* Followed by a record prime gap
* Rank is a highly composite number


and ''f'' (''x'') &#61; sqrt (''x''/2) is a monotonously increasing function
== Edo sizes ==
 
* Exo: 0–9
''e'' has the same minimum point.
* Small: 10–38
 
** Semitonic: 10–14
Now substitute ''t''<sub>2</sub>/''c'' for ''t''<sub>1</sub>,
** Subsemitonic: 15–26
 
** Dietic: 27–38
[math]
* Medium: 39–79
t_i &#61; \frac {v_i (v_1 + v_2)}{v_1^2 + v_2^2}, i &#61; 1, 2 \\
** Commatic: 39–67
e &#61; \frac { {{!}}v_1 - v_2{{!}} }{\sqrt {2(v_1^2 + v_2^2)} }
** Subcommatic: 68–79
[/math]
* Large: 80–190
 
** Hemicommatic: 80–132
== 3-limit TOP tuning of ets ==
** Codettic: 133–137
This part is deduced from Paul Erlich's ''Middle Path''.
** Kleismatic: 138–190
 
* Mega: 191+
[math]
** Subkleismatic
t_i &#61; \frac {2v_i}{v_1 + v_2}, i &#61; 1, 2 \\
** Hemikleismatic
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.
* The relative interval error space of equal temperaments in TOP tuning seems to be linear.

Latest revision as of 10:06, 18 January 2026

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 a.k.a. 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

2-limit (rank-1)
  • Essential equivalence
  • Completes the harmonic series for the first octave
3-limit (rank-2)
  • Essential interval functions
  • Completes the harmonic series for the first 2 octaves
  • Rank is a highly composite number
5-limit (rank-3)
  • Completes the harmonic series for the first 2 octaves and a fifth
7-limit (rank-4)
  • Tonality: tonal
  • Categorical characteristics: pivotal and semiambitonal
  • Completes the harmonic series for the first 3 octaves
  • Rank is a highly composite number
11-limit (rank-5)
  • Completes the harmonic series for the first 3 octaves and a fifth
13-limit (rank-6)
  • Essential interval colors
  • Tonality: microtonal
  • Categorical characteristics: ambitonal and semiambitonal
  • Completes the harmonic series for the first 4 octaves
  • Rank is a highly composite number
23-limit (rank-9)
  • Limit of classical functional harmony
  • Limit of classical concordance
  • Tonality: pseudotonal and pseudomicrotonal
  • Categorical characteristics: pseudoambitonal
  • Completes the harmonic series for the first 4 octaves and a fifth
  • Followed by a record prime gap
31-limit (rank-11)
  • Completes the harmonic series for the first 5 octaves
37-limit (rank-12)
  • Rank is a highly composite number
47-limit (rank-15)
  • Completes the harmonic series for the first 5 octaves and a fifth
61-limit (rank-18)
  • Completes the harmonic series for the first 6 octaves
89-limit (rank-24)
  • Completes the harmonic series for the first 6 octaves and a fifth
  • Followed by a record prime gap
  • Rank is a highly composite number

Edo sizes

  • Exo: 0–9
  • Small: 10–38
    • Semitonic: 10–14
    • Subsemitonic: 15–26
    • Dietic: 27–38
  • Medium: 39–79
    • Commatic: 39–67
    • Subcommatic: 68–79
  • Large: 80–190
    • Hemicommatic: 80–132
    • Codettic: 133–137
    • Kleismatic: 138–190
  • Mega: 191+
    • Subkleismatic
    • Hemikleismatic
    • …
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