Mediant hull - Revision history

Arseniiv: added a link to mediant

2023-02-09T11:30:44Z

added a link to mediant

← Older revision		Revision as of 11:30, 9 February 2023
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	This works similarly for rank-3 and higher scales: for example, <code>diasem(7\|\|26\|\|31)</code> specifies the mediant hull of the 7edo (L:m:s = 1:1:0), [[26edo]] (L:m:s = 4:2:1) and [[31edo]] (L:m:s = 5:2:1) tunings for [[diasem]].		This works similarly for rank-3 and higher scales: for example, <code>diasem(7\|\|26\|\|31)</code> specifies the mediant hull of the 7edo (L:m:s = 1:1:0), [[26edo]] (L:m:s = 4:2:1) and [[31edo]] (L:m:s = 5:2:1) tunings for [[diasem]].

			== See also ==
			* [[Mediant]]

	[[Category:Terms]]		[[Category:Terms]]

Inthar at 06:16, 20 September 2021

2021-09-20T06:16:00Z

← Older revision		Revision as of 06:16, 20 September 2021
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	The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by ~~traversing~~ a finite (resulting in another equal-step tuning) or infinite (resulting in a tuning with irrational step ratio) Farey sum of the starting values. We write the mediant hull of tunings by putting double bars <code>\|\|</code> between the tunings. For example, <code>5L 2s(7\|\|12)</code>, the mediant hull of 7edo [[5L 2s\|diatonic]] and 12edo diatonic, is [[TAMNAMS\|soft-of-basic]] diatonic. (Note: Double bars are used for formatting on Discord, so you should use <code>``</code> or escape the double bars.)		The '''mediant hull''' or '''Farey hull''' of a set of [[equal-step tuning]]s for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by taking a finite (resulting in another equal-step tuning) or infinite (resulting in a tuning with irrational step ratio) Farey sum of the starting values. We write the mediant hull of tunings by putting double bars <code>\|\|</code> between the tunings. For example, <code>5L 2s(7\|\|12)</code>, the mediant hull of 7edo [[5L 2s\|diatonic]] and 12edo diatonic, is [[TAMNAMS\|soft-of-basic]] diatonic. (Note: Double bars are used for formatting on Discord, so you should use <code>``</code> or escape the double bars.)

	Formally, given a 2-step scale structure ''S'' with equave ''E'' and a set of equal-step tunings ''T<sub>1</sub>, ..., T<sub>r</sub>'', given by generators ''m<sub>1</sub>\n<sub>1</sub>edE, ..., m<sub>r</sub>\n<sub>r</sub>edE'' for S, the ''mediant hull'' or ''Farey hull'' of ''T<sub>1</sub>, ..., T<sub>r</sub>'' is		Formally, given a 2-step scale structure ''S'' with equave ''E'' and a set of equal-step tunings ''T<sub>1</sub>, ..., T<sub>r</sub>'', given by generators ''m<sub>1</sub>\n<sub>1</sub>edE, ..., m<sub>r</sub>\n<sub>r</sub>edE'' for S, the ''mediant hull'' or ''Farey hull'' of ''T<sub>1</sub>, ..., T<sub>r</sub>'' is

Inthar: /* Properties */

2021-09-19T19:42:59Z

Properties

← Older revision		Revision as of 19:42, 19 September 2021
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	The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 2) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.		The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 2) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.

	In particular, the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), is the Farey hull of the degenerate tunings X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 and X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1. Each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).		In particular, the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), the Farey hull of the degenerate tunings X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 and X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1, is an (''r'' − 1)-simplex. Each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).

	== Notation ==		== Notation ==

Inthar: /* Properties */

2021-09-19T19:42:28Z

Properties

← Older revision		Revision as of 19:42, 19 September 2021
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	The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 2) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.		The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 2) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.

	In particular, the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), is the Farey hull of the degenerate tunings X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 and X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1. ~~each~~ hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).		In particular, the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), is the Farey hull of the degenerate tunings X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 and X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1. Each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).

	== Notation ==		== Notation ==

Inthar: /* Properties */

2021-09-19T19:42:19Z

Properties

← Older revision		Revision as of 19:42, 19 September 2021
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	The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 2) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.		The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 2) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.

	In particular, ~~for~~ the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).		In particular, the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), is the Farey hull of the degenerate tunings X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 and X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1. each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).

	== Notation ==		== Notation ==

Inthar at 19:35, 19 September 2021

2021-09-19T19:35:38Z

← Older revision		Revision as of 19:35, 19 September 2021
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	Formally, given a 2-step scale structure ''S'' with equave ''E'' and a set of equal-step tunings ''T<sub>1</sub>, ..., T<sub>r</sub>'', given by generators ''m<sub>1</sub>\n<sub>1</sub>edE, ..., m<sub>r</sub>\n<sub>r</sub>edE'' for S, the ''mediant hull'' or ''Farey hull'' of ''T<sub>1</sub>, ..., T<sub>r</sub>'' is		Formally, given a 2-step scale structure ''S'' with equave ''E'' and a set of equal-step tunings ''T<sub>1</sub>, ..., T<sub>r</sub>'', given by generators ''m<sub>1</sub>\n<sub>1</sub>edE, ..., m<sub>r</sub>\n<sub>r</sub>edE'' for S, the ''mediant hull'' or ''Farey hull'' of ''T<sub>1</sub>, ..., T<sub>r</sub>'' is
	<math>T_1 \mid\mid T_2 \mid\mid \cdots \mid\mid T_r = \operatorname{cl}\bigg\{\dfrac{a_1 m_1 + \cdots + a_r m_r}{(a_1 n_1 + \cdots + a_r n_r)~~edE~~} : (a_1, ..., a_r) \in \mathbb{Z}^r_{\geq 0} \setminus (0, 0, ..., 0) \bigg\}</math>		<math>T_1 \mid\mid T_2 \mid\mid \cdots \mid\mid T_r = \operatorname{cl}\bigg\{\dfrac{a_1 m_1 + \cdots + a_r m_r}{(a_1 n_1 + \cdots + a_r n_r)\mathrm{ed}E} : (a_1, ..., a_r) \in \mathbb{Z}^r_{\geq 0} \setminus (0, 0, ..., 0) \bigg\}</math>

	where <math>\operatorname{cl}</math> denotes the topological closure (specifying the generator is enough, given the period, since the period doesn't change upon taking the mediant). The generalization to higher-rank scale structures (requiring more than one non-period generator) is obvious.		where <math>\operatorname{cl}</math> denotes the topological closure (specifying the generator is enough, given the period, since the period doesn't change upon taking the mediant). The generalization to higher-rank scale structures (requiring more than one non-period generator) is obvious.

Inthar: /* Properties */

2021-09-17T04:22:51Z

Properties

← Older revision		Revision as of 04:22, 17 September 2021
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	Todo; rank 3 example		Todo; rank 3 example

	The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 1) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.		The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 2) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.

	In particular, for the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).		In particular, for the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).

Inthar: /* Properties */

2021-09-17T04:21:56Z

Properties

← Older revision		Revision as of 04:21, 17 September 2021
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	== Properties ==		== Properties ==
			Todo; rank 3 example

	The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 1) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.		The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 1) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.

Inthar: /* Properties */

2021-09-17T04:19:07Z

Properties

← Older revision		Revision as of 04:19, 17 September 2021
Line 9:		Line 9:
	The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 1) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.		The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 1) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.

	In particular, for the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0, (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).		In particular, for the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0 (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).

	== Notation ==		== Notation ==

Inthar at 04:18, 17 September 2021

2021-09-17T04:18:46Z

← Older revision		Revision as of 04:18, 17 September 2021
Line 9:		Line 9:
	The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 1) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.		The Farey hull of ''r'' non-collinear tunings of a rank ''r'' scale is an (''r'' − 1)-dimensional simplex with vertices the ''r'' starting values. Each face (of dimension ''r'' − 1) of this simplex is the Farey hull of a choice of ''r'' − 1 of the starting tunings.

	In particular, for the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub>, corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0.		In particular, for the set of all possible tunings of a scale structure (with step sizes X<sub>1</sub> ≥ X<sub>2</sub> ≥ ... ≥ X<sub>r</sub>), each hyperface corresponds to the degenerate version of the scale (with ''r'' − 1 step sizes) with X<sub>i</sub> = X<sub>i+1</sub> (resp. X<sub>r</sub> = 0), corresponding to omitting the vertex (degenerate tuning) with step ratio X<sub>1</sub>:... :X<sub>i</sub>:X<sub>i+1</sub>:...:X<sub>r</sub> = 1:...:1:0:...:0, (resp. X<sub>1</sub>:...:X<sub>r</sub> = 1:...:1).

	== Notation ==		== Notation ==