User:Moremajorthanmajor/5L 2s (5/3-equivalent) - Revision history

ArrowHead294 at 16:47, 3 March 2025

2025-03-03T16:47:54Z

← Older revision		Revision as of 16:47, 3 March 2025
Line 93:		Line 93:
	\|3\5\|\|		\|3\5\|\|
	\|}If we carry this freshman-summing out a little further, new, larger ED5/3s pop up in our continuum.		\|}If we carry this freshman-summing out a little further, new, larger ED5/3s pop up in our continuum.
	{{Scale ~~tree\|~~5L 2s<5/3>}}Tunings above 7\12 on this chart are called "positive tunings" (as they greaten the size of the fifth) and include 3/4 meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.		{{MOS tuning spectrum\|Scale Signature=5L 2s<5/3>}}Tunings above 7\12 on this chart are called "positive tunings" (as they greaten the size of the fifth) and include 3/4 meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

	Tunings below 7\12 on this chart are called "negative tunings" and they include 3/4 Pythagorean tuning itself (well approximated by 31\53) as well as 3/4 superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 10\17 through 13\22, the thirds fall closer to 5-limit than 7-limit intervals: 6:5 as opposed to 7:6.		Tunings below 7\12 on this chart are called "negative tunings" and they include 3/4 Pythagorean tuning itself (well approximated by 31\53) as well as 3/4 superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 10\17 through 13\22, the thirds fall closer to 5-limit than 7-limit intervals: 6:5 as opposed to 7:6.

Moremajorthanmajor: /* Hypohard */

2023-07-24T04:21:15Z

Hypohard

← Older revision		Revision as of 04:21, 24 July 2023
Line 37:		Line 37:
	The near-just part of the region is of interest mainly for those interested in “3/4” [[Pythagorean tuning]] and large, accurate eds systems based on close-to-Pythagorean fifths, such as 41ED5/3 and 53ED5/3. This class of tunings is called [[schisma\|trischismic]] temperament; these tunings can approximate 5<sup>3/4</sup>-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12ED5/3 tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)<!--(see [[5L 2s/Temperaments#Schismic]])-->.		The near-just part of the region is of interest mainly for those interested in “3/4” [[Pythagorean tuning]] and large, accurate eds systems based on close-to-Pythagorean fifths, such as 41ED5/3 and 53ED5/3. This class of tunings is called [[schisma\|trischismic]] temperament; these tunings can approximate 5<sup>3/4</sup>-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12ED5/3 tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)<!--(see [[5L 2s/Temperaments#Schismic]])-->.

	The sharp-of-just part of this range includes so-called “3/4 [[neogothic]]" or "3/4 parapyth" systems, which tune the diatonic major third slightly flatly of [[6/5]] and the diatonic minor third slightly sharply of [[12/11]]. Good 3/4 neogothic EDSs include 29ED5/3 and 46ED5/3. 17ED5/3 is often considered the sharper end of the 3/4 neogothic spectrum; its major third at ~~313~~ cents (~~417~~ śata) is considerably more concordant than in flatter neogothic tunings.		The sharp-of-just part of this range includes so-called “3/4 [[neogothic]]" or "3/4 parapyth" systems, which tune the diatonic major third slightly flatly of [[6/5]] and the diatonic minor third slightly sharply of [[12/11]]. Good 3/4 neogothic EDSs include 29ED5/3 and 46ED5/3. 17ED5/3 is often considered the sharper end of the 3/4 neogothic spectrum; its major third at 312 cents (416 śata) is considerably more concordant than in flatter neogothic tunings.
	===Parahard to ultrahard===		===Parahard to ultrahard===
	"3/4 Archy" systems such as 17ED5/3, 22ED5/3, and 27ED5/3.		"3/4 Archy" systems such as 17ED5/3, 22ED5/3, and 27ED5/3.

Moremajorthanmajor: /* Hypohard */

2023-07-24T04:19:43Z

Hypohard

← Older revision		Revision as of 04:19, 24 July 2023
Line 37:		Line 37:
	The near-just part of the region is of interest mainly for those interested in “3/4” [[Pythagorean tuning]] and large, accurate eds systems based on close-to-Pythagorean fifths, such as 41ED5/3 and 53ED5/3. This class of tunings is called [[schisma\|trischismic]] temperament; these tunings can approximate 5<sup>3/4</sup>-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12ED5/3 tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)<!--(see [[5L 2s/Temperaments#Schismic]])-->.		The near-just part of the region is of interest mainly for those interested in “3/4” [[Pythagorean tuning]] and large, accurate eds systems based on close-to-Pythagorean fifths, such as 41ED5/3 and 53ED5/3. This class of tunings is called [[schisma\|trischismic]] temperament; these tunings can approximate 5<sup>3/4</sup>-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12ED5/3 tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)<!--(see [[5L 2s/Temperaments#Schismic]])-->.

	The sharp-of-just part of this range includes so-called “3/4 [[neogothic]]" or "3/4 parapyth" systems, which tune the diatonic major third slightly flatly of [[6/5]] and the diatonic minor third slightly sharply of [[12/11]]. Good 3/4 neogothic EDSs include ~~29EDQ~~ and 46ED5/3. 17ED5/3 is often considered the sharper end of the 3/4 neogothic spectrum; its major third at 313 cents (417 śata) is considerably more concordant than in flatter neogothic tunings.		The sharp-of-just part of this range includes so-called “3/4 [[neogothic]]" or "3/4 parapyth" systems, which tune the diatonic major third slightly flatly of [[6/5]] and the diatonic minor third slightly sharply of [[12/11]]. Good 3/4 neogothic EDSs include 29ED5/3 and 46ED5/3. 17ED5/3 is often considered the sharper end of the 3/4 neogothic spectrum; its major third at 313 cents (417 śata) is considerably more concordant than in flatter neogothic tunings.
	===Parahard to ultrahard===		===Parahard to ultrahard===
	"3/4 Archy" systems such as 17ED5/3, 22ED5/3, and 27ED5/3.		"3/4 Archy" systems such as 17ED5/3, 22ED5/3, and 27ED5/3.

Moremajorthanmajor at 23:33, 20 June 2023

2023-06-20T23:33:35Z

← Older revision		Revision as of 23:33, 20 June 2023
Line 1:		Line 1:
	{{Infobox MOS		{{Infobox MOS
	\|Tuning=5L 2s<5/3>}}		\|Tuning=5L 2s<5/3>}}
	~~One way~~ of ~~distinguishing the~~ '''3/4''' '''diatonic''' ~~scale is by considering it a [[MOS scale\|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. Among the most well-known variants of this~~ MOS proper are 12ED5/3's diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.		{{MOS intro\|Scale Signature=5L 2s<5/3>}}Among the most well-known variants of this '''3/4''' '''diatonic''' MOS proper are 12ED5/3's diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.
	==On the term ''diatonic''==		==On the term ''diatonic''==
	In [[TAMNAMS]] (which is the convention on all pages on scale patterns on the wiki), [[diatonic]] exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering\|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized'' (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.		In [[TAMNAMS]] (which is the convention on all pages on scale patterns on the wiki), [[diatonic]] exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering\|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized'' (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.

Moremajorthanmajor at 00:59, 11 June 2023

2023-06-11T00:59:20Z

← Older revision		Revision as of 00:59, 11 June 2023
Line 1:		Line 1:
	{{Infobox MOS		{{Infobox MOS
	\| ~~Name =~~		\|Tuning=5L 2s<5/3>}}
	~~\| Periods = 1~~
	~~\| nLargeSteps = 5~~
	~~\| nSmallSteps = 2~~
	~~\| Equalized = 4~~
	~~\| Collapsed = 3~~
	~~\| Pattern = LLLsLLs~~
	~~\| Neutral = 3L 4s~~
	~~\| Equave~~=5/3}}
	One way of distinguishing the '''3/4''' '''diatonic''' scale is by considering it a [[MOS scale\|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. Among the most well-known variants of this MOS proper are 12ED5/3's diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.		One way of distinguishing the '''3/4''' '''diatonic''' scale is by considering it a [[MOS scale\|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. Among the most well-known variants of this MOS proper are 12ED5/3's diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.
	==On the term ''diatonic''==		==On the term ''diatonic''==

Moremajorthanmajor: /* Scale tree */

2023-06-04T00:03:42Z

Scale tree

← Older revision		Revision as of 00:03, 4 June 2023
Line 100:		Line 100:
	\|-		\|-
	\|3\5\|\|		\|3\5\|\|
	\|}If we carry this freshman-summing out a little further, new, larger ~~EDSs~~ pop up in our continuum.		\|}If we carry this freshman-summing out a little further, new, larger ED5/3s pop up in our continuum.
	{{Scale tree\|5L 2s<5/3>}}Tunings above 7\12 on this chart are called "positive tunings" (as they greaten the size of the fifth) and include 3/4 meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.		{{Scale tree\|5L 2s<5/3>}}Tunings above 7\12 on this chart are called "positive tunings" (as they greaten the size of the fifth) and include 3/4 meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Moremajorthanmajor: /* Scale tree */

2023-06-04T00:01:31Z

Scale tree

← Older revision		Revision as of 00:01, 4 June 2023
Line 92:		Line 92:
	*[[3_4Cotoneum7]]		*[[3_4Cotoneum7]]
	==Scale tree==		==Scale tree==
	If 4\7 (four degrees of ~~7EDS~~) is at one extreme and 3\5 (three degrees of ~~5EDS~~) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of ~~12EDO~~:		If 4\7 (four degrees of 7ED5/3) is at one extreme and 3\5 (three degrees of 5ED5/3) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12ED5/3:
	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
Line 108:		Line 108:

	5L 2s contains the pentatonic MOS [[2L_3s (major sixth equivalent) \|2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L_5s (major sixth equivalent) \|7L 5s]] or [[5L_7s (major sixth equivalent) \|5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (675¢, 700$).		5L 2s contains the pentatonic MOS [[2L_3s (major sixth equivalent) \|2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L_5s (major sixth equivalent) \|7L 5s]] or [[5L_7s (major sixth equivalent) \|5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (675¢, 700$).

	==Related Scales==		==Related Scales==
	{{main\| 5L 2s (major sixth equivalent) MODMOSes }} ''and [[5L 2s (major sixth equivalent) Muddles]]''		{{main\| 5L 2s (major sixth equivalent) MODMOSes }} ''and [[5L 2s (major sixth equivalent) Muddles]]''

Moremajorthanmajor at 23:57, 3 June 2023

2023-06-03T23:57:49Z

← Older revision		Revision as of 23:57, 3 June 2023
Line 9:		Line 9:
	\| Neutral = 3L 4s		\| Neutral = 3L 4s
	\| Equave=5/3}}		\| Equave=5/3}}
	One way of distinguishing the '''3/4''' '''diatonic''' scale is by considering it a [[MOS scale\|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. Among the most well-known variants of this MOS proper are ~~12EDS~~'s diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.		One way of distinguishing the '''3/4''' '''diatonic''' scale is by considering it a [[MOS scale\|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. Among the most well-known variants of this MOS proper are 12ED5/3's diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.
	==On the term ''diatonic''==		==On the term ''diatonic''==
	In [[TAMNAMS]] (which is the convention on all pages on scale patterns on the wiki), [[diatonic]] exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering\|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized'' (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.		In [[TAMNAMS]] (which is the convention on all pages on scale patterns on the wiki), [[diatonic]] exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering\|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized'' (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.
Line 15:		Line 15:
	The 5L 2s MOS scale has this generalized form.		The 5L 2s MOS scale has this generalized form.
	*L L s L L L s		*L L s L L L s
	Insert 2 for L and 1 for s and you'll get the ~~12edo~~ diatonic of standard practice.		Insert 2 for L and 1 for s and you'll get the 12ed5/3 diatonic of standard practice.
	*2 2 1 2 2 2 1		*2 2 1 2 2 2 1
	When L=3, s=1, you have ~~17EDS~~: 3 3 1 3 3 3 1		When L=3, s=1, you have 17ED5/3: 3 3 1 3 3 3 1

	When L=3, s=2, you have ~~19EDS~~: 3 3 2 3 3 3 2		When L=3, s=2, you have 19ED5/3: 3 3 2 3 3 3 2

	When L=4, s=1, you have ~~22EDS~~: 4 4 1 4 4 4 1		When L=4, s=1, you have 22ED5/3: 4 4 1 4 4 4 1

	When L=4, s=3, you have ~~26EDS~~: 4 4 3 4 4 4 3		When L=4, s=3, you have 26ED5/3: 4 4 3 4 4 4 3

	When L=5, s=1, you have ~~27EDS~~: 5 5 1 5 5 5 1		When L=5, s=1, you have 27ED5/3: 5 5 1 5 5 5 1

	When L=5, s=2, you have ~~29EDS~~: 5 5 2 5 5 5 2		When L=5, s=2, you have 29ED5/3: 5 5 2 5 5 5 2

	When L=5, s=3, you have ~~31EDS~~: 5 5 3 5 5 5 3		When L=5, s=3, you have 31ED5/3: 5 5 3 5 5 5 3

	When L=5, s=4, you have ~~33EDS~~: 5 5 4 5 5 5 4		When L=5, s=4, you have 33ED5/3: 5 5 4 5 5 5 4

	So you have scales where L and s are nearly equal, which approach ~~7EDS~~:		So you have scales where L and s are nearly equal, which approach 7ED5/3:
	*1 1 1 1 1 1 1		*1 1 1 1 1 1 1
	And you have scales where s becomes so small it approaches zero, which would give us ~~5EDS~~:		And you have scales where s becomes so small it approaches zero, which would give us 5ED5/3:
	*1 1 0 1 1 1 0 = 1 1 1 1 1		*1 1 0 1 1 1 0 = 1 1 1 1 1
	==Tuning ranges==		==Tuning ranges==
	===Parasoft to ultrasoft===		===Parasoft to ultrasoft===
	"3/4 Flattone" systems, such as ~~26EDS~~.		"3/4 Flattone" systems, such as 26ED5/3.
	===Hyposoft===		===Hyposoft===
	"3/4 Meantone" (more properly "3/4 septimal meantone") systems, such as ~~31EDS~~.		"3/4 Meantone" (more properly "3/4 septimal meantone") systems, such as 31ED5/3.
	===Hypohard===		===Hypohard===
	The near-just part of the region is of interest mainly for those interested in “3/4” [[Pythagorean tuning]] and large, accurate eds systems based on close-to-Pythagorean fifths, such as ~~41EDS~~ and ~~53EDS~~. This class of tunings is called [[schisma\|trischismic]] temperament; these tunings can approximate 5<sup>3/4</sup>-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, ~~12EDS~~ tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)<!--(see [[5L 2s/Temperaments#Schismic]])-->.		The near-just part of the region is of interest mainly for those interested in “3/4” [[Pythagorean tuning]] and large, accurate eds systems based on close-to-Pythagorean fifths, such as 41ED5/3 and 53ED5/3. This class of tunings is called [[schisma\|trischismic]] temperament; these tunings can approximate 5<sup>3/4</sup>-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12ED5/3 tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)<!--(see [[5L 2s/Temperaments#Schismic]])-->.

	The sharp-of-just part of this range includes so-called “3/4 [[neogothic]]" or "3/4 parapyth" systems, which tune the diatonic major third slightly flatly of [[6/5]] and the diatonic minor third slightly sharply of [[12/11]]. Good 3/4 neogothic EDSs include ~~29EDS~~ and ~~46EDS~~. ~~17EDS~~ is often considered the sharper end of the 3/4 neogothic spectrum; its major third at 313 cents (417 śata) is considerably more concordant than in flatter neogothic tunings.		The sharp-of-just part of this range includes so-called “3/4 [[neogothic]]" or "3/4 parapyth" systems, which tune the diatonic major third slightly flatly of [[6/5]] and the diatonic minor third slightly sharply of [[12/11]]. Good 3/4 neogothic EDSs include 29EDQ and 46ED5/3. 17ED5/3 is often considered the sharper end of the 3/4 neogothic spectrum; its major third at 313 cents (417 śata) is considerably more concordant than in flatter neogothic tunings.
	===Parahard to ultrahard===		===Parahard to ultrahard===
	"3/4 Archy" systems such as ~~17EDS~~, ~~22EDS~~, and ~~27EDS~~.		"3/4 Archy" systems such as 17ED5/3, 22ED5/3, and 27ED5/3.
	==Modes==		==Modes==
	Diatonic modes have standard names from classical music theory:		Diatonic modes have standard names from classical music theory:

Moremajorthanmajor at 02:12, 29 May 2023

2023-05-29T02:12:41Z

Show changes

Fredg999: Fredg999 moved page 5L 2s (5/3-equivalent) to User:Moremajorthanmajor/5L 2s (5/3-equivalent) over redirect: Page not ready for the main namespace

2023-05-20T05:22:39Z

Fredg999 moved page 5L 2s (5/3-equivalent) to User:Moremajorthanmajor/5L 2s (5/3-equivalent) over redirect: Page not ready for the main namespace

← Older revision	Revision as of 05:22, 20 May 2023
(No difference)