Mediant hull: Difference between revisions

Line 1:

The '''mediant hull''' or '''Farey hull''' of a set of [[~~edo~~]] ~~tunings~~ for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by taking a finite number of mediants of the starting edo tunings, or as a limit of a sequence of finite mediants. 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 [[diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic.

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 number of mediants of the starting edo tunings, or as a limit of a sequence of finite mediants. 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 [[diatonic]] and 12edo diatonic, is [[TAMNAMS|soft-of-basic]] diatonic.

Formally, given a 2-step scale structure ''S'' and a set of edo tunings ''T1, ..., Tr'', given by generators ''m1/n1, ..., mr/nr'' for S, the ''mediant hull'' or ''Farey hull'' of ''T1, ..., Tr'' is

Revision as of 16:57, 14 September 2021 view source Inthar (talk \| contribs) autopatrolled, emailconfirmed 15,004 edits mNo edit summary ← Older edit		Revision as of 16:57, 14 September 2021 view source Inthar (talk \| contribs) autopatrolled, emailconfirmed 15,004 edits No edit summary Newer edit →
Line 1:		Line 1:
	The '''mediant hull''' or '''Farey hull''' of a set of [[~~edo~~]] ~~tunings~~ for a given scale structure is the tuning range defined as the set of all tunings that can be obtained by taking a finite number of mediants of the starting edo tunings, or as a limit of a sequence of finite mediants. 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 [[diatonic]] and 12edo diatonic, is [[TAMNAMS\|soft-of-basic]] diatonic.		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 number of mediants of the starting edo tunings, or as a limit of a sequence of finite mediants. 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 [[diatonic]] and 12edo diatonic, is [[TAMNAMS\|soft-of-basic]] diatonic.

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