Mediant hull: Difference between revisions

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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 or infinite path on the corresponding scale tree. 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 traversing a finite path (resulting in another equal-step tuning) or infinite path (resulting in a tuning with irrational step ratio) on the corresponding [[scale tree]]. 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

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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 or infinite path on the corresponding scale tree. 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 traversing a finite path (resulting in another equal-step tuning) or infinite path (resulting in a tuning with irrational step ratio) on the corresponding [[scale tree]]. 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