User:FloraC

Name's Flora Canou (Fumica#5144).

English & Chinese Mandarin;

Mostly microtonal theory currently.

I contributed to the n-EDO Retuner plugin for MuseScore and made a fork which has key signatures reordered 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.

Tools

TE Tuning & Temperament Measures Calculator – I made this Python script to compute TE tunings, badnesses, optimal patent vals, etc.

Writings

• Critique on Functional Just System

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.

• Flora's 12-note well temperament
• Flora's 17-note well temperament

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.

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

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 any temperament tempering out [m₁ m₂ … m_n⟩, each prime p_i is tuned to log₂ (p_i)(Σ_{i = 1}ⁿ m_i log₂ (p_i))/(Σ_{i = 1}ⁿ |m_i| log₂ (p_i)).
• For ets, TOP tuning and TE tuning are close but not identical.

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• Analysis on the 13-limit just intonation space: episode i
• Analysis on the 13-limit just intonation space: episode ii
• Critique on D&D's terminology
• Critique on Functional Just System
• Flora's 12-note well temperament
• Flora's 17-note well temperament
• Fokker analysis of rank-3 scales
• Fumica's edo impressions
• Fundamental principles to musical sense
• Hard problems of harmony and psychoacoustically supported optimization
• Infobox Interval
• Mathematically defined ET tier list
• On the canon of music
• Proposed standard ear-training waveform
• Quick reference
• Reasonable commas
• Sandbox
• Sequence of equal temperaments by error/Supplement
• Style guide
• Table of 540edo intervals
• Temperament data
• Temperament name proposal
• There is not a third side of the river
• common.css