Relationship between Bohlen–Pierce and octave-ful temperaments - Revision history

Lériendil: "octarod" is a really old name and this isn't referring to the 11-limit extension

2025-10-09T13:56:27Z

"octarod" is a really old name and this isn't referring to the 11-limit extension

← Older revision		Revision as of 13:56, 9 October 2025
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	The rank-2 temperament under discussion here is the 3.5.7 temperament known as [[BPS]] that tempers out only 245/243 (not any other commas such as 3125/3087). Its mapping matrix is {{val\|1 1 2}}, {{val\|0 2 -1}} and the two generators are ~1902 cents (the period, [[3/1]]) and ~440 cents (which represents a sharpened 9/7, two of which make a 5/3 because 245/243 vanishes.) In the world of tritave equivalency, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the Bohlen–Pierce "Lambda" scale.		The rank-2 temperament under discussion here is the 3.5.7 temperament known as [[BPS]] that tempers out only 245/243 (not any other commas such as 3125/3087). Its mapping matrix is {{val\|1 1 2}}, {{val\|0 2 -1}} and the two generators are ~1902 cents (the period, [[3/1]]) and ~440 cents (which represents a sharpened 9/7, two of which make a 5/3 because 245/243 vanishes.) In the world of tritave equivalency, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the Bohlen–Pierce "Lambda" scale.

	If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[~~octarod~~]]".		If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[sensamagic]]".

	What if we want a rank-2 system? In that case we must find a suitable interval in the system to represent 2/1. In theory any of the many 7-limit temperaments tempering out 245/243 (the [[sensamagic family]]) could be considered a relative of Bohlen–Pierce. But most of them divide either 3/1 or the ~440 cent generator into some number of equal parts, and so don't have the same generators/lattice as rank-2 BP itself. (In math terms the "index" is greater than 1.) For example, [[magic]] divides 3/1 into 5 equal parts. [[Superpyth]], on the other hand, leaves 3/1 unsplit but makes 9/7 no longer the generator—instead it has complexity 6.		What if we want a rank-2 system? In that case we must find a suitable interval in the system to represent 2/1. In theory any of the many 7-limit temperaments tempering out 245/243 (the [[sensamagic family]]) could be considered a relative of Bohlen–Pierce. But most of them divide either 3/1 or the ~440 cent generator into some number of equal parts, and so don't have the same generators/lattice as rank-2 BP itself. (In math terms the "index" is greater than 1.) For example, [[magic]] divides 3/1 into 5 equal parts. [[Superpyth]], on the other hand, leaves 3/1 unsplit but makes 9/7 no longer the generator—instead it has complexity 6.

ArrowHead294 at 13:07, 10 March 2025

2025-03-10T13:07:57Z

← Older revision		Revision as of 13:07, 10 March 2025
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	<references group="note" />		<references group="note" />

	[[Category:~~Bohlen-Pierce~~]]		[[Category:Bohlen–Pierce]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

ArrowHead294: /* Relationship of rank-2 Bohlen–Pierce–Stearns temperament to octave-ful temperaments */

2025-01-10T14:22:23Z

Relationship of rank-2 Bohlen–Pierce–Stearns temperament to octave-ful temperaments

← Older revision		Revision as of 14:22, 10 January 2025
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	== Relationship of rank-2 Bohlen–Pierce–Stearns temperament to octave-ful temperaments ==		== Relationship of rank-2 Bohlen–Pierce–Stearns temperament to octave-ful temperaments ==
	The rank-2 temperament under discussion here is the 3.5.7 temperament known as [[BPS]] that tempers out only 245/243 (not any other commas such as 3125/3087). Its mapping matrix is {{val\|1 1 2}}, {{val\|0 2 -1}} and the two generators are ~1902 cents (the period, [[3/1]]) and ~440 cents (which represents a sharpened 9/7, two of which make a 5/3 because 245/243 vanishes.) In ~~3/1-equivalence~~ world, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the BP "Lambda" scale.		The rank-2 temperament under discussion here is the 3.5.7 temperament known as [[BPS]] that tempers out only 245/243 (not any other commas such as 3125/3087). Its mapping matrix is {{val\|1 1 2}}, {{val\|0 2 -1}} and the two generators are ~1902 cents (the period, [[3/1]]) and ~440 cents (which represents a sharpened 9/7, two of which make a 5/3 because 245/243 vanishes.) In the world of tritave equivalency, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the Bohlen–Pierce "Lambda" scale.

	If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[octarod]]".		If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[octarod]]".

	What if we want a rank-2 system? In that case we must find a suitable interval in the system to represent 2/1. In theory any of the many 7-limit temperaments tempering out 245/243 (the [[sensamagic family]]) could be considered a relative of Bohlen–Pierce. But most of them divide either 3/1 or the ~440 cent generator into some number of equal parts, and so don't have the same generators/lattice as rank-2 BP itself. (In math terms the "index" is greater than 1.) For example, [[magic]] divides 3/1 into 5 equal parts. [[Superpyth]], on the other hand, leaves 3/1 unsplit but makes 9/7 no longer the ~~generator - instead~~ it has complexity 6.		What if we want a rank-2 system? In that case we must find a suitable interval in the system to represent 2/1. In theory any of the many 7-limit temperaments tempering out 245/243 (the [[sensamagic family]]) could be considered a relative of Bohlen–Pierce. But most of them divide either 3/1 or the ~440 cent generator into some number of equal parts, and so don't have the same generators/lattice as rank-2 BP itself. (In math terms the "index" is greater than 1.) For example, [[magic]] divides 3/1 into 5 equal parts. [[Superpyth]], on the other hand, leaves 3/1 unsplit but makes 9/7 no longer the generator—instead it has complexity 6.

	To find a rank-2 temperament more closely analogous to Lambda temperament but containing the prime 2, 2/1 must be mapped to some already existing note of this temperament's genchain rather than a new note obtained by splitting up the 3/1 period or ~440 cent generator. There are two obvious candidates for this mapping of 2/1: +7 generators (which corresponds to 125/63 in the 3.5.7 subgroup) and -6 generators (which corresponds to 49/25 in the 3.5.7 subgroup). In 13ed3 these notes are "enharmonically equivalent" to each other as 13 = 7 + 6; this corresponds to tempering of the additional comma 3125/3087.		To find a rank-2 temperament more closely analogous to Lambda temperament but containing the prime 2, 2/1 must be mapped to some already existing note of this temperament's genchain rather than a new note obtained by splitting up the 3/1 period or ~440 cent generator. There are two obvious candidates for this mapping of 2/1: +7 generators (which corresponds to 125/63 in the 3.5.7 subgroup) and −6 generators (which corresponds to 49/25 in the 3.5.7 subgroup). In 13ed3 these notes are "enharmonically equivalent" to each other as {{nowrap\|13 {{=}} 7 + 6}}; this corresponds to tempering of the additional comma 3125/3087.

	=== Sensi ===		=== Sensi ===
	Consider the first possibility, that 2/1 is mapped to +7 generators in the BP lambda scale, therefore tempering out [[126/125]]. Seven generators up from C is H#<ref group="note">This page uses the standard where the nominals are assigned starting from C to the LssLsLsLs mode of the scale. See [[4L 5s (3/1-equivalent)#Notation]] for a further discussion.</ref>, so this means H# represents 2/1 above C. In [[13edt]], H# is only 1170 cents, so in order for this to accurately represent 2/1 the generator needs to be widened by a few cents, from ~440 cents up to ~443 cents.		Consider the first possibility, that 2/1 is mapped to +7 generators in the BP lambda scale, therefore tempering out [[126/125]]. Seven generators up from C is H♯<ref group="note">This page uses the standard where the nominals are assigned starting from C to the LssLsLsLs mode of the scale. See [[4L 5s (3/1-equivalent)#Notation]] for a further discussion.</ref>, so this means H# represents 2/1 above C. In [[13edt]], H♯ is only 1170 cents, so in order for this to accurately represent 2/1 the generator needs to be widened by a few cents, from ~440 cents up to ~443 cents.

	The result of doing this is [[sensi]] temperament.		The result of doing this is [[sensi]] temperament.

	=== Hedgehog ===		=== Hedgehog ===
	Now consider what happens if 2/1 is mapped to -6 generators instead, therefore tempering out [[50/49]]. Six generators down from C (in the Lambda scale notation) is Jb, so now Jb represents 2/1 above C. In this case the generator must be narrowed from ~440 to ~436 cents in order for Jb to end up at a reasonable 2/1.		Now consider what happens if 2/1 is mapped to −6 generators instead, therefore tempering out [[50/49]]. Six generators down from C (in the Lambda scale notation) is J♭, so now J♭ represents 2/1 above C. In this case the generator must be narrowed from ~440 to ~436 cents in order for J♭ to end up at a reasonable 2/1.

	Furthermore, since 2/1 is being mapped to (2, -6) (up two ~1902 cent periods and down six ~436 cent generators), and both 2 and -6 are even numbers, that means 2/1 splits into two equal parts in this system which each represents [[7/5]]~[[10/7]]. (This is in contrast to sensi in which 2/1 is intact and doesn't split into any equal parts.)		Furthermore, since 2/1 is being mapped to (2, −6) (up two ~1902 cent periods and down six ~436 cent generators), and both 2 and −6 are even numbers, that means 2/1 splits into two equal parts in this system which each represents [[7/5]]~[[10/7]]. (This is in contrast to sensi in which 2/1 is intact and doesn't split into any equal parts.)

	The result of doing this is [[hedgehog]] temperament.		The result of doing this is [[hedgehog]] temperament.

ArrowHead294: ArrowHead294 moved page Relationship between Bohlen-Pierce and octave-ful temperaments to Relationship between Bohlen–Pierce and octave-ful temperaments: Per wikipedia:MOS:ENDASH

2024-11-20T16:14:27Z

ArrowHead294 moved page Relationship between Bohlen-Pierce and octave-ful temperaments to Relationship between Bohlen–Pierce and octave-ful temperaments: Per wikipedia:MOS:ENDASH

← Older revision	Revision as of 16:14, 20 November 2024
(No difference)

ArrowHead294 at 16:13, 20 November 2024

2024-11-20T16:13:59Z

← Older revision		Revision as of 16:13, 20 November 2024
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	Just as conventional music theory is associated with multiple temperaments (the [[rank]]-1 temperament [[12edo\|12EDO]] and the rank-2 temperament [[meantone]]), the [[Bohlen–Pierce]] system is based on at least two different temperaments. The first is [[13ed3\|13EDT]] (13ED3), the rank-1 temperament where 3/1 is divided into 13 equal parts. Many explanations of Bohlen–Pierce simply state that this equal division '''is''' Bohlen–Pierce. But just as a description of conventional music theory wouldn't be complete if you just said it's 12 equal divisions of 2/1 and left it at that (ignoring diatonic scales, key signatures, and so on), the Bohlen–Pierce system is also strongly associated with the rank-2 temperament (of the 3.5.7 [[just intonation subgroup\|subgroup]]) tempering out [[245/243]]. This temperament has been referred to by various names including "Lambda temperament" or "BP Diatonic temperament", although the standard name by which it is referred to on this wiki is [[~~Bohlen–Pierce-Stearns~~]].		Just as conventional music theory is associated with multiple temperaments (the [[rank]]-1 temperament [[12edo\|12EDO]] and the rank-2 temperament [[meantone]]), the [[Bohlen–Pierce]] system is based on at least two different temperaments. The first is [[13ed3\|13EDT]] (13ED3), the rank-1 temperament where 3/1 is divided into 13 equal parts. Many explanations of Bohlen–Pierce simply state that this equal division '''is''' Bohlen–Pierce. But just as a description of conventional music theory wouldn't be complete if you just said it's 12 equal divisions of 2/1 and left it at that (ignoring diatonic scales, key signatures, and so on), the Bohlen–Pierce system is also strongly associated with the rank-2 temperament (of the 3.5.7 [[just intonation subgroup\|subgroup]]) tempering out [[245/243]]. This temperament has been referred to by various names including "Lambda temperament" or "BP Diatonic temperament", although the standard name by which it is referred to on this wiki is [[Bohlen–Pierce–Stearns]].

	== Relationship of 13EDT to octave-ful temperaments ==		== Relationship of 13EDT to octave-ful temperaments ==

ArrowHead294 at 16:13, 20 November 2024

2024-11-20T16:13:49Z

← Older revision		Revision as of 16:13, 20 November 2024
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	Just as conventional music theory is associated with multiple temperaments (the [[rank]]-1 temperament [[12edo\|12EDO]] and the rank-2 temperament [[meantone]]), the [[~~Bohlen-Pierce~~]] system is based on at least two different temperaments. The first is [[13ed3\|13EDT]] (13ED3), the rank-1 temperament where 3/1 is divided into 13 equal parts. Many explanations of ~~Bohlen-Pierce~~ simply state that this equal division '''is''' ~~Bohlen-Pierce~~. But just as a description of conventional music theory wouldn't be complete if you just said it's 12 equal divisions of 2/1 and left it at that (ignoring diatonic scales, key signatures, and so on), the ~~Bohlen-Pierce~~ system is also strongly associated with the rank-2 temperament (of the 3.5.7 [[just intonation subgroup\|subgroup]]) tempering out [[245/243]]. This temperament has been referred to by various names including "Lambda temperament" or "BP Diatonic temperament", although the standard name by which it is referred to on this wiki is [[~~Bohlen-Pierce~~-Stearns]].		Just as conventional music theory is associated with multiple temperaments (the [[rank]]-1 temperament [[12edo\|12EDO]] and the rank-2 temperament [[meantone]]), the [[Bohlen–Pierce]] system is based on at least two different temperaments. The first is [[13ed3\|13EDT]] (13ED3), the rank-1 temperament where 3/1 is divided into 13 equal parts. Many explanations of Bohlen–Pierce simply state that this equal division '''is''' Bohlen–Pierce. But just as a description of conventional music theory wouldn't be complete if you just said it's 12 equal divisions of 2/1 and left it at that (ignoring diatonic scales, key signatures, and so on), the Bohlen–Pierce system is also strongly associated with the rank-2 temperament (of the 3.5.7 [[just intonation subgroup\|subgroup]]) tempering out [[245/243]]. This temperament has been referred to by various names including "Lambda temperament" or "BP Diatonic temperament", although the standard name by which it is referred to on this wiki is [[Bohlen–Pierce-Stearns]].

	== Relationship of 13EDT to octave-ful temperaments ==		== Relationship of 13EDT to octave-ful temperaments ==
	Equal-tempered ~~Bohlen-Pierce~~ or 13EDT is a rank-1 temperament whose mapping is {{val\|13 19 23}}. It tempers out infinitely many commas including 245/243 and 3125/3087. What happens if we add the prime 2 back in to this system? If we add it as an independent dimension, we get a rank-2 temperament whose mapping (with the unusual basis ordering 3.5.7.2) is {{val\|13 19 23 8}}, {{val\|0 0 0 1}}. So in addition to the ~146 cent step of the ~~Bohlen-Pierce~~ equal temperament, there is another generator of about 28 cents which is used only for intervals with the prime 2 (in other words, ratios that contain even numbers). This temperament has a name, [[bohpier]] (named after ~~Bohlen-Pierce~~ because of this relationship).		Equal-tempered Bohlen–Pierce or 13EDT is a rank-1 temperament whose mapping is {{val\|13 19 23}}. It tempers out infinitely many commas including 245/243 and 3125/3087. What happens if we add the prime 2 back in to this system? If we add it as an independent dimension, we get a rank-2 temperament whose mapping (with the unusual basis ordering 3.5.7.2) is {{val\|13 19 23 8}}, {{val\|0 0 0 1}}. So in addition to the ~146 cent step of the Bohlen–Pierce equal temperament, there is another generator of about 28 cents which is used only for intervals with the prime 2 (in other words, ratios that contain even numbers). This temperament has a name, [[bohpier]] (named after Bohlen–Pierce because of this relationship).

	If we take that same abstract temperament, bohpier, and re-express it with the standard 7-limit basis ordering 2.3.5.7, we get the mapping {{val\|1 0 0 0}}, {{val\|0 13 19 23}}. The generators when viewed through this lens are ~1200 cents and ~146 cents, and the unique thing you can immediately see from the mapping matrix is that 3, 5, and 7 are all multiples of the ~146 cent generator, with no octaves required for any of them. This makes sense because if you remove octaves from bohpier and use only the generator chain, it becomes equal-tempered ~~Bohlen-Pierce~~ again.		If we take that same abstract temperament, bohpier, and re-express it with the standard 7-limit basis ordering 2.3.5.7, we get the mapping {{val\|1 0 0 0}}, {{val\|0 13 19 23}}. The generators when viewed through this lens are ~1200 cents and ~146 cents, and the unique thing you can immediately see from the mapping matrix is that 3, 5, and 7 are all multiples of the ~146 cent generator, with no octaves required for any of them. This makes sense because if you remove octaves from bohpier and use only the generator chain, it becomes equal-tempered Bohlen–Pierce again.

	Bohpier has a near-equal MOS of 8 notes, and [[41edo\|41EDO]] is a particularly good equal temperament that [[support]]s it. Therefore 41EDO secretly contains a version of ~~Bohlen-Pierce~~ you can get by taking every fifth note. (Other EDOs are [[33edo\|33EDO]] and [[49edo\|49EDO]] but they are not so good.)		Bohpier has a near-equal MOS of 8 notes, and [[41edo\|41EDO]] is a particularly good equal temperament that [[support]]s it. Therefore 41EDO secretly contains a version of Bohlen–Pierce you can get by taking every fifth note. (Other EDOs are [[33edo\|33EDO]] and [[49edo\|49EDO]] but they are not so good.)

	~~== Relationship of rank-2 Bohlen-Pierce-Stearns temperament to octave-ful temperaments ==~~

			== Relationship of rank-2 Bohlen–Pierce–Stearns temperament to octave-ful temperaments ==
	The rank-2 temperament under discussion here is the 3.5.7 temperament known as [[BPS]] that tempers out only 245/243 (not any other commas such as 3125/3087). Its mapping matrix is {{val\|1 1 2}}, {{val\|0 2 -1}} and the two generators are ~1902 cents (the period, [[3/1]]) and ~440 cents (which represents a sharpened 9/7, two of which make a 5/3 because 245/243 vanishes.) In 3/1-equivalence world, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the BP "Lambda" scale.		The rank-2 temperament under discussion here is the 3.5.7 temperament known as [[BPS]] that tempers out only 245/243 (not any other commas such as 3125/3087). Its mapping matrix is {{val\|1 1 2}}, {{val\|0 2 -1}} and the two generators are ~1902 cents (the period, [[3/1]]) and ~440 cents (which represents a sharpened 9/7, two of which make a 5/3 because 245/243 vanishes.) In 3/1-equivalence world, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the BP "Lambda" scale.

	If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[octarod]]".		If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[octarod]]".

	What if we want a rank-2 system? In that case we must find a suitable interval in the system to represent 2/1. In theory any of the many 7-limit temperaments tempering out 245/243 (the [[sensamagic family]]) could be considered a relative of ~~Bohlen-Pierce~~. But most of them divide either 3/1 or the ~440 cent generator into some number of equal parts, and so don't have the same generators/lattice as rank-2 BP itself. (In math terms the "index" is greater than 1.) For example, [[magic]] divides 3/1 into 5 equal parts. [[Superpyth]], on the other hand, leaves 3/1 unsplit but makes 9/7 no longer the generator - instead it has complexity 6.		What if we want a rank-2 system? In that case we must find a suitable interval in the system to represent 2/1. In theory any of the many 7-limit temperaments tempering out 245/243 (the [[sensamagic family]]) could be considered a relative of Bohlen–Pierce. But most of them divide either 3/1 or the ~440 cent generator into some number of equal parts, and so don't have the same generators/lattice as rank-2 BP itself. (In math terms the "index" is greater than 1.) For example, [[magic]] divides 3/1 into 5 equal parts. [[Superpyth]], on the other hand, leaves 3/1 unsplit but makes 9/7 no longer the generator - instead it has complexity 6.

	To find a rank-2 temperament more closely analogous to Lambda temperament but containing the prime 2, 2/1 must be mapped to some already existing note of this temperament's genchain rather than a new note obtained by splitting up the 3/1 period or ~440 cent generator. There are two obvious candidates for this mapping of 2/1: +7 generators (which corresponds to 125/63 in the 3.5.7 subgroup) and -6 generators (which corresponds to 49/25 in the 3.5.7 subgroup). In 13ed3 these notes are "enharmonically equivalent" to each other as 13 = 7 + 6; this corresponds to tempering of the additional comma 3125/3087.		To find a rank-2 temperament more closely analogous to Lambda temperament but containing the prime 2, 2/1 must be mapped to some already existing note of this temperament's genchain rather than a new note obtained by splitting up the 3/1 period or ~440 cent generator. There are two obvious candidates for this mapping of 2/1: +7 generators (which corresponds to 125/63 in the 3.5.7 subgroup) and -6 generators (which corresponds to 49/25 in the 3.5.7 subgroup). In 13ed3 these notes are "enharmonically equivalent" to each other as 13 = 7 + 6; this corresponds to tempering of the additional comma 3125/3087.

	=== Sensi ===		=== Sensi ===
	Consider the first possibility, that 2/1 is mapped to +7 generators in the BP lambda scale, therefore tempering out [[126/125]]. Seven generators up from C is H#<ref>This page uses the standard where the nominals are assigned starting from C to the LssLsLsLs mode of the scale. See [[4L 5s (3/1-equivalent)#Notation]] for a further discussion.</ref>, so this means H# represents 2/1 above C. In [[13edt]], H# is only 1170 cents, so in order for this to accurately represent 2/1 the generator needs to be widened by a few cents, from ~440 cents up to ~443 cents.		Consider the first possibility, that 2/1 is mapped to +7 generators in the BP lambda scale, therefore tempering out [[126/125]]. Seven generators up from C is H#<ref group="note">This page uses the standard where the nominals are assigned starting from C to the LssLsLsLs mode of the scale. See [[4L 5s (3/1-equivalent)#Notation]] for a further discussion.</ref>, so this means H# represents 2/1 above C. In [[13edt]], H# is only 1170 cents, so in order for this to accurately represent 2/1 the generator needs to be widened by a few cents, from ~440 cents up to ~443 cents.

	The result of doing this is [[sensi]] temperament.		The result of doing this is [[sensi]] temperament.
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	Note that although the period and generator of hedgehog are usually given as (~600 cents, ~164 cents), an equally valid choice of generator is (~600 cents, ~436 cents), because 436 is the half-octave complement of 164. When 3/1 is taken as the period, the ~436 cent generator must be used (because the ~164 cent one doesn't generate all the notes).		Note that although the period and generator of hedgehog are usually given as (~600 cents, ~164 cents), an equally valid choice of generator is (~600 cents, ~436 cents), because 436 is the half-octave complement of 164. When 3/1 is taken as the period, the ~436 cent generator must be used (because the ~164 cent one doesn't generate all the notes).

	==Notes==		== Notes ==
	<references />		<references group="note" />

	[[Category:Bohlen-Pierce]]		[[Category:Bohlen-Pierce]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Lériendil: /* Relationship of rank-2 Bohlen-Pierce-Stearns temperament to octave-ful temperaments */

2024-08-28T12:00:12Z

Relationship of rank-2 Bohlen-Pierce-Stearns temperament to octave-ful temperaments

← Older revision		Revision as of 12:00, 28 August 2024
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	== Relationship of rank-2 Bohlen-Pierce-Stearns temperament to octave-ful temperaments ==		== Relationship of rank-2 Bohlen-Pierce-Stearns temperament to octave-ful temperaments ==

	The rank-2 temperament under discussion here is the 3.5.7 temperament known as [[BPS]] that tempers out only 245/243 (not any other commas such as 3125/3087). Its mapping matrix is {{val\|1 1 2}}, {{val\|0 2 -1}} and the two generators are ~1902 cents (the period, [[3/1]]) and ~440 cents (which represents a sharpened 9/7, two of which make a 5/3 because of 245/243 vanishes.) In 3/1-equivalence world, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the BP "Lambda" scale.		The rank-2 temperament under discussion here is the 3.5.7 temperament known as [[BPS]] that tempers out only 245/243 (not any other commas such as 3125/3087). Its mapping matrix is {{val\|1 1 2}}, {{val\|0 2 -1}} and the two generators are ~1902 cents (the period, [[3/1]]) and ~440 cents (which represents a sharpened 9/7, two of which make a 5/3 because 245/243 vanishes.) In 3/1-equivalence world, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the BP "Lambda" scale.

	If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[octarod]]".		If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[octarod]]".

Lériendil at 11:26, 28 August 2024

2024-08-28T11:26:40Z

← Older revision		Revision as of 11:26, 28 August 2024
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	Just as conventional music theory is associated with multiple temperaments (the [[rank]]-1 temperament [[12edo\|12EDO]] and the rank-2 temperament [[meantone]]), the [[Bohlen-Pierce]] system is based on at least two different temperaments. The first is [[13ed3\|13EDT]] (13ED3), the rank-1 temperament where 3/1 is divided into 13 equal parts. Many explanations of Bohlen-Pierce simply state that this equal division '''is''' Bohlen-Pierce. But just as a description of conventional music theory wouldn't be complete if you just said it's 12 equal divisions of 2/1 and left it at that (ignoring diatonic scales, key signatures, and so on), the Bohlen-Pierce system is also strongly associated with the rank-2 temperament (of the 3.5.7 [[just intonation subgroup\|subgroup]]) tempering out [[245/243]]. This temperament has been referred to by various names including "Lambda temperament" or "BP Diatonic temperament".		Just as conventional music theory is associated with multiple temperaments (the [[rank]]-1 temperament [[12edo\|12EDO]] and the rank-2 temperament [[meantone]]), the [[Bohlen-Pierce]] system is based on at least two different temperaments. The first is [[13ed3\|13EDT]] (13ED3), the rank-1 temperament where 3/1 is divided into 13 equal parts. Many explanations of Bohlen-Pierce simply state that this equal division '''is''' Bohlen-Pierce. But just as a description of conventional music theory wouldn't be complete if you just said it's 12 equal divisions of 2/1 and left it at that (ignoring diatonic scales, key signatures, and so on), the Bohlen-Pierce system is also strongly associated with the rank-2 temperament (of the 3.5.7 [[just intonation subgroup\|subgroup]]) tempering out [[245/243]]. This temperament has been referred to by various names including "Lambda temperament" or "BP Diatonic temperament", although the standard name by which it is referred to on this wiki is [[Bohlen-Pierce-Stearns]].

	== Relationship of 13EDT to octave-ful temperaments ==		== Relationship of 13EDT to octave-ful temperaments ==

Lériendil: /* Hedgehog */

2024-08-28T11:24:45Z

Hedgehog

← Older revision		Revision as of 11:24, 28 August 2024
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	Note that although the period and generator of hedgehog are usually given as (~600 cents, ~164 cents), an equally valid choice of generator is (~600 cents, ~436 cents), because 436 is the half-octave complement of 164. When 3/1 is taken as the period, the ~436 cent generator must be used (because the ~164 cent one doesn't generate all the notes).		Note that although the period and generator of hedgehog are usually given as (~600 cents, ~164 cents), an equally valid choice of generator is (~600 cents, ~436 cents), because 436 is the half-octave complement of 164. When 3/1 is taken as the period, the ~436 cent generator must be used (because the ~164 cent one doesn't generate all the notes).

	~~The Moon, however, is not so big.<ref>R. Smith, "Size of the Moon", ''Scientific American'', 46 (April 1978): 44–46.</ref>~~

	==Notes==		==Notes==

Lériendil: /* Relationship of rank-2 Bohlen-Pierce-Stearns temperament to octave-ful temperaments */

2024-08-28T11:24:34Z

Relationship of rank-2 Bohlen-Pierce-Stearns temperament to octave-ful temperaments

← Older revision		Revision as of 11:24, 28 August 2024
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	=== Sensi ===		=== Sensi ===
	Consider the first possibility, that 2/1 is mapped to +7 generators in the BP lambda scale, therefore tempering out [[126/125]]. Seven generators up from C is H#, so this means H# represents 2/1 above C. In [[13edt]], H# is only 1170 cents, so in order for this to accurately represent 2/1 the generator needs to be widened by a few cents, from ~440 cents up to ~443 cents.		Consider the first possibility, that 2/1 is mapped to +7 generators in the BP lambda scale, therefore tempering out [[126/125]]. Seven generators up from C is H#<ref>This page uses the standard where the nominals are assigned starting from C to the LssLsLsLs mode of the scale. See [[4L 5s (3/1-equivalent)#Notation]] for a further discussion.</ref>, so this means H# represents 2/1 above C. In [[13edt]], H# is only 1170 cents, so in order for this to accurately represent 2/1 the generator needs to be widened by a few cents, from ~440 cents up to ~443 cents.

	The result of doing this is [[sensi]] temperament.		The result of doing this is [[sensi]] temperament.
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	Note that although the period and generator of hedgehog are usually given as (~600 cents, ~164 cents), an equally valid choice of generator is (~600 cents, ~436 cents), because 436 is the half-octave complement of 164. When 3/1 is taken as the period, the ~436 cent generator must be used (because the ~164 cent one doesn't generate all the notes).		Note that although the period and generator of hedgehog are usually given as (~600 cents, ~164 cents), an equally valid choice of generator is (~600 cents, ~436 cents), because 436 is the half-octave complement of 164. When 3/1 is taken as the period, the ~436 cent generator must be used (because the ~164 cent one doesn't generate all the notes).

			The Moon, however, is not so big.<ref>R. Smith, "Size of the Moon", ''Scientific American'', 46 (April 1978): 44–46.</ref>

			==Notes==
			<references />

	[[Category:Bohlen-Pierce]]		[[Category:Bohlen-Pierce]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]