User:FloraC/Quick reference

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".

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 = t₂/t₁ = v₂/v₁

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 t₂/c for t₁,

[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

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

This time we have a sequence c = {c_n}, 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_i/c_i for t₁,

[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 [m₁ m₂ … m_n⟩, each prime q_i is tuned to

[math]\displaystyle{ -\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.