User:FloraC

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Revision as of 14:13, 29 January 2022 by FloraC (talk | contribs) (→Notes)
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Name's Flora Canou (Fumica#5144). Age 22. 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. 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
  • Work out RTT tables for all edos listed in the 13-limit page (relative error < 5.5% and cut off at 494)

Tools

TE Tuning & Temperament Measures Calculator – I made this Python script 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 also tried one on 19et but gave up for multiple reasons.

Q: Why I gave up 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, while in 19et they represent the same ratios better or worse in different keys, and I'm not fond of that. Second, the harmonics of 3, 5, 7, and 13 in 19-et are all flat, so there's not much room to operate. Third, the ambiguity of 4\19 and 15\19 is nice and I want them ambiguous in every key.

I'll pick it up soon.

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 m2 … mn⟩, each prime qi is tuned to
[math]\displaystyle{ \log_2 (q_i) \frac {\sum m_i \log_2 (q_i)}{\sum \vert m_i \vert \log_2 (q_i)} }[/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.
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