User:FloraC/Quick reference - Revision history

FloraC: /* Edo sizes */

2026-01-18T10:06:07Z

Edo sizes

← Older revision		Revision as of 10:06, 18 January 2026
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	* Exo: 0–9		* Exo: 0–9
	* Small: 10–38		* Small: 10–38
			** Semitonic: 10–14
			** Subsemitonic: 15–26
			** Dietic: 27–38
	* Medium: 39–79		* Medium: 39–79
	* Large: ~~80–185~~		** Commatic: 39–67
	* Mega: ~~186~~+		** Subcommatic: 68–79
			* Large: 80–190
			** Hemicommatic: 80–132
			** Codettic: 133–137
			** Kleismatic: 138–190
			* Mega: 191+
			** Subkleismatic
			** Hemikleismatic
			** …

FloraC at 10:00, 18 January 2026

2026-01-18T10:00:45Z

← Older revision		Revision as of 10:00, 18 January 2026
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	* Essential interval functions		* Essential interval functions
	* Completes the harmonic series for the first 2 octaves		* Completes the harmonic series for the first 2 octaves
			* Rank is a highly composite number

	; 5-limit (rank-3):		; 5-limit (rank-3):
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	* Categorical characteristics: pivotal and semiambitonal		* Categorical characteristics: pivotal and semiambitonal
	* Completes the harmonic series for the first 3 octaves		* Completes the harmonic series for the first 3 octaves
			* Rank is a highly composite number

	; 11-limit (rank-5):		; 11-limit (rank-5):
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	* Categorical characteristics: ambitonal and semiambitonal		* Categorical characteristics: ambitonal and semiambitonal
	* Completes the harmonic series for the first 4 octaves		* Completes the harmonic series for the first 4 octaves
			* Rank is a highly composite number

	; 23-limit (rank-9):		; 23-limit (rank-9):
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	; 31-limit (rank-11)		; 31-limit (rank-11)
	* Completes the harmonic series for the first 5 octaves		* Completes the harmonic series for the first 5 octaves

	; 37-limit (rank-12)		; 37-limit (rank-12)
			* Rank is a highly composite number

	; 47-limit (rank-15)		; 47-limit (rank-15)
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	* Completes the harmonic series for the first 6 octaves and a fifth		* Completes the harmonic series for the first 6 octaves and a fifth
	* Followed by a record prime gap		* Followed by a record prime gap
			* Rank is a highly composite number

	== Edo sizes ==		== Edo sizes ==

FloraC at 16:53, 10 August 2025

2025-08-10T16:53:21Z

← Older revision		Revision as of 16:53, 10 August 2025
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	== Edo sizes ==		== Edo sizes ==
	* Exo: 0–9		* Exo: 0–9
	* Small: ~~10–37~~		* Small: 10–38
	* Medium: ~~38–79~~		* Medium: 39–79
	* Large: 80–185		* Large: 80–185
	* Mega: 186+		* Mega: 186+

FloraC at 15:18, 10 August 2025

2025-08-10T15:18:34Z

← Older revision		Revision as of 15:18, 10 August 2025
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	== Important prime limits ==		== Important prime limits ==
	; 2-limit (rank-2):		; 2-limit (rank-1):
	* Essential equivalence		* Essential equivalence
	* Completes the harmonic series for the first octave		* Completes the harmonic series for the first octave

FloraC at 15:18, 10 August 2025

2025-08-10T15:18:23Z

← Older revision		Revision as of 15:18, 10 August 2025
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	== Important prime limits ==		== Important prime limits ==
			; 2-limit (rank-2):
			* Essential equivalence
			* Completes the harmonic series for the first octave

	; 3-limit (rank-2):		; 3-limit (rank-2):
	* Essential interval functions		* Essential interval functions
			* Completes the harmonic series for the first 2 octaves

			; 5-limit (rank-3):
			* Completes the harmonic series for the first 2 octaves and a fifth

			; 7-limit (rank-4):
			* Tonality: tonal
			* Categorical characteristics: pivotal and semiambitonal
			* Completes the harmonic series for the first 3 octaves

			; 11-limit (rank-5):
			* Completes the harmonic series for the first 3 octaves and a fifth

	; 13-limit (rank-6):		; 13-limit (rank-6):
	* Essential interval colors		* Essential interval colors
	* Tonality: ~~tonal and~~ microtonal		* Tonality: microtonal
	* Categorical characteristics: ~~pivotal,~~ ambitonal, and semiambitonal		* Categorical characteristics: ambitonal and semiambitonal
	* ~~Mode 8~~		* Completes the harmonic series for the first 4 octaves

	; 23-limit (rank-9):		; 23-limit (rank-9):
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	* Tonality: pseudotonal and pseudomicrotonal		* Tonality: pseudotonal and pseudomicrotonal
	* Categorical characteristics: pseudoambitonal		* Categorical characteristics: pseudoambitonal
	* ~~Mode 12~~		* Completes the harmonic series for the first 4 octaves and a fifth
	* Followed by a record prime gap		* Followed by a record prime gap

	; 31-limit (rank-11)		; 31-limit (rank-11)
	* ~~Mode 16~~		* Completes the harmonic series for the first 5 octaves

	; 37-limit (rank-12)		; 37-limit (rank-12)

	; 47-limit (rank-15)		; 47-limit (rank-15)
	* ~~Mode 24~~		* Completes the harmonic series for the first 5 octaves and a fifth

	; 61-limit (rank-18)		; 61-limit (rank-18)
	* ~~Mode 32~~		* Completes the harmonic series for the first 6 octaves

	; 89-limit (rank-24)		; 89-limit (rank-24)
	* ~~Mode 48~~		* Completes the harmonic series for the first 6 octaves and a fifth
	* Followed by a record prime gap		* Followed by a record prime gap

FloraC at 14:57, 10 August 2025

2025-08-10T14:57:35Z

← Older revision		Revision as of 14:57, 10 August 2025
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	** 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.		** 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.		*** 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.		*** Stacking based a.k.a. 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.		** 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.		*** RTT: this school encompasses stacking based JI and applies approximation for custom structures.
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	* Exo: 0–9		* Exo: 0–9
	* Small: 10–37		* Small: 10–37
	* Medium: ~~38–75~~		* Medium: 38–79
	* Large: ~~76–185~~		* Large: 80–185
	* Mega: 186+		* Mega: 186+

FloraC at 15:18, 28 August 2023

2023-08-28T15:18:10Z

← Older revision		Revision as of 15:18, 28 August 2023
Line 43:		Line 43:
	* Mode 48		* Mode 48
	* Followed by a record prime gap		* Followed by a record prime gap

			== Edo sizes ==
			* Exo: 0–9
			* Small: 10–37
			* Medium: 38–75
			* Large: 76–185
			* Mega: 186+

FloraC: -outdated stuff

2023-08-21T08:33:46Z

-outdated stuff

← Older revision		Revision as of 08:33, 21 August 2023
Line 43:		Line 43:
	* Mode 48		* Mode 48
	* Followed by a record prime gap		* 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'' = ''t''<sub>2</sub>/''t''<sub>1</sub> = ''v''<sub>2</sub>/''v''<sub>1</sub>~~

	~~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 ''t''<sub>2</sub>/''c'' for ''t''<sub>1</sub>,~~

	~~[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''<sub>''n''</sub>}, 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''<sub>''i''</sub>/''c''<sub>''i''</sub> for ''t''<sub>1</sub>,~~

	~~[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 {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>''n''</sub> }}, each prime ''q<sub>i</sub>'' 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.

FloraC at 08:27, 21 August 2023

2023-08-21T08:27:20Z

@@ Line 1: / Line 1: @@
 I call equal temperaments in Tenney-Euclidean tuning "ette".
@@ Line 5: / Line 51: @@
 Some super easy formulae for such a tuning follows.
-== 3-limit TE tuning of ets ==
+=== 3-limit TE tuning of ets ===
 Given a val A, we have Tenney-weighted val V &#61; AW, where W is the Tenney-weighting matrix.
@@ Line 41: / Line 87: @@
 [/math]
-== 3-limit TOP tuning of ets ==
+=== 3-limit TOP tuning of ets ===
 This part is deduced from Paul Erlich's ''Middle Path''.
@@ Line 51: / Line 97: @@
 This ''e'' is also the amount to stretch or compress each prime.
-== General TE tuning of ets ==
+=== General TE tuning of ets ===
 This time we have a sequence c &#61; {''c''<sub>''n''</sub>}, where
@@ Line 67: / Line 113: @@
 [/math]
-== Notes ==
+=== Notes ===
 * For the nullity-1 temperament tempering out {{monzo| ''m''<sub>1</sub> ''m''<sub>2</sub> … ''m''<sub>''n''</sub> }}, each prime ''q<sub>i</sub>'' 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.

FloraC: Created page with "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..."

2022-12-16T11:02:15Z

Created page with "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..."

New page

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'' = ''t''2/''t''1 = ''v''2/''v''1

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 ''t''2/''c'' for ''t''1,

[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 ''t''1,

[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 {{monzo| ''m''1 ''m''2 … ''m''''n'' }}, 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.