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** 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.  
** 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.  
*** 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.  
*** 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.  
** 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.  
*** RTT: this school encompasses stacking based JI and applies approximation for custom structures.  
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== Important prime limits ==
== Important prime limits ==
; 2-limit (rank-1):
* Essential equivalence
* Completes the harmonic series for the first octave
; 3-limit (rank-2):  
; 3-limit (rank-2):  
* Essential interval functions
* 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):  
; 13-limit (rank-6):  
* Essential interval colors
* Essential interval colors
* Tonality: tonal and microtonal
* Tonality: microtonal
* Categorical characteristics: pivotal, ambitonal, and semiambitonal
* Categorical characteristics: ambitonal and semiambitonal
* Mode 8
* Completes the harmonic series for the first 4 octaves
* Rank is a highly composite number


; 23-limit (rank-9):  
; 23-limit (rank-9):  
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* Tonality: pseudotonal and pseudomicrotonal
* Tonality: pseudotonal and pseudomicrotonal
* Categorical characteristics: pseudoambitonal
* Categorical characteristics: pseudoambitonal
* Mode 12
* Completes the harmonic series for the first 4 octaves and a fifth
* Followed by a record prime gap
* Followed by a record prime gap


; 31-limit (rank-11)
; 31-limit (rank-11)
* Mode 16
* Completes the harmonic series for the first 5 octaves


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


; 47-limit (rank-15)
; 47-limit (rank-15)
* Mode 24
* Completes the harmonic series for the first 5 octaves and a fifth


; 61-limit (rank-18)
; 61-limit (rank-18)
* Mode 32
* Completes the harmonic series for the first 6 octaves


; 89-limit (rank-24)
; 89-limit (rank-24)
* Mode 48
* Completes the harmonic series for the first 6 octaves and a fifth
* Followed by a record prime gap
* Followed by a record prime gap
* Rank is a highly composite number


== Tuning equal temperaments ==
== Edo sizes ==
I call equal temperaments in Tenney-Euclidean tuning "ette".
* Exo: 0–9
 
* Small: 10–38
3-limit TE tuning, which is my preferred tuning for most ets, is "ette3".
** Semitonic: 10–14
 
** Subsemitonic: 15–26
Some super easy formulae for such a tuning follows.
** Dietic: 27–38
 
* Medium: 39–79
=== 3-limit TE tuning of ets ===
** Commatic: 39–67
Given a val A, we have Tenney-weighted val V = AW, where W is the Tenney-weighting matrix.
** Subcommatic: 68–79
 
* Large: 80–190
If T is the Tenney-weighted tuning map, then for any et, for obvious reasons,
** Hemicommatic: 80–132
 
** Codettic: 133–137
[math]t_2/v_2 = t_1/v_1[/math]
** Kleismatic: 138–190
 
* Mega: 191+
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>
** Subkleismatic
 
** Hemikleismatic
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.
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