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.