User:FloraC

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Name's Flora Canou (Fumica#5144). Age 23. I speak English & Chinese Mandarin. I currently work on mostly microtonal theories especially RTT.

I contributed to the n-EDO Retuner plugin for MuseScore and made a fork with key signatures re-ordered into fifths for my own use.

I explored and documented the sensamagic dominant chord and the hemimage bleeding chord, based on my understanding of septimal voice leading.

I explored the canou family of temperaments, and a few others in User:FloraC/Temperament proposal.

Long term projects:

  • Cleanup for all temperament pages
  • Rework scale trees for mos pages

Music

Microtonal/xenharmonic releases

12et releases

  • Avant l'Avenir (2020–2022)
  • Favonius (2018–2020)
  • Eruption in Silence (2016–2018)

Tools

Temperament Evaluator – I made this set of Python scripts to compute TE tunings, badnesses, optimal patent vals, etc.

Writings

Well temperaments

I developed well temperaments on 12et and 17et which can be seen here.

I've also been trying to develop one on 19et but no satisfactory result as of now.

Q: What are the difficulties in developing a 19wt?

A: First, unlike 12- and 17et with ambiguous major and minor thirds, 19et's thirds are close enough to 5-limit JI that interpreting them otherwise is like a force. In 12- and 17et, those intervals can represent different ratios in different keys, whereas in 19et, they represent the same ratios better or worse in different keys. The effect isn't satisfactory. Second, the harmonics of 3, 5, 7, and 13 in 19et are all flat, so there's not much room to operate. Third, the ambiguity of 4\19 and 15\19 is an important characteritics, and those should be ambiguous in every key.

Q: What are the solutions?

A: For 19et to have any room to operate, octave stretch must be employed. For 4\19 and 15\19 not deviating too much, hemitwelfth is used as a generator.

Q: It's possible to make octave stretched well temperaments?

A: Yes it's possible. Just one more argument than pure-octave. Issue is I haven't got a satisfactory result.

Quick reference

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

Detail

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

Detail

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

Detail

This time we have a sequence c = {cn}, 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 ti/ci 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 m2mn, each prime qi 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.