User:Moremajorthanmajor/5L 2s (8/3-equivalent) - Revision history

ArrowHead294 at 16:52, 3 March 2025

2025-03-03T16:52:08Z

← Older revision		Revision as of 16:52, 3 March 2025
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	*Chroma-negative generator: 679.218 cents (2\5) to 727.734 cents (3\7)		*Chroma-negative generator: 679.218 cents (2\5) to 727.734 cents (3\7)

	{{Scale ~~tree\|~~5L 2s<8/3>}}		{{MOS tuning spectrum\|Scale Signature=5L 2s<8/3>}}

	Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include 17/12 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 above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include 17/12 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.

FloraC: Uncategorize

2024-08-11T15:39:44Z

Uncategorize

← Older revision		Revision as of 15:39, 11 August 2024
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	==Approaches to Functional Harmony==		==Approaches to Functional Harmony==
	{{see also\| Diatonic functional harmony}}		{{see also\| Diatonic functional harmony}}

	~~[[Category:Diatonic\| ]] <!-- main article -->~~
	~~[[Category:7-tone scales]]~~

Moremajorthanmajor: /* Scale tree */

2023-07-07T23:39:07Z

Scale tree

← Older revision		Revision as of 23:39, 7 July 2023
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	If 4\7 (four degrees of 7ED8/3) is at one extreme and 3\5 (three degrees of 5ED8/3) 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 12ED8/3.		If 4\7 (four degrees of 7ED8/3) is at one extreme and 3\5 (three degrees of 5ED8/3) 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 12ED8/3.

	If we carry this freshman-summing out a little further, new, larger [[ED8/3]]<nowiki/>s pop up in our continuum.		If we carry this freshman-summing out a little further, new, larger [[Ed8/3\|ED8/3]]<nowiki/>s pop up in our continuum.

Moremajorthanmajor: /* Scale tree */

2023-07-07T23:38:35Z

Scale tree

← Older revision		Revision as of 23:38, 7 July 2023
Line 104:		Line 104:
	If 4\7 (four degrees of 7ED8/3) is at one extreme and 3\5 (three degrees of 5ED8/3) 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 12ED8/3.		If 4\7 (four degrees of 7ED8/3) is at one extreme and 3\5 (three degrees of 5ED8/3) 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 12ED8/3.

	If we carry this freshman-summing out a little further, new, larger [[~~EDXI~~]]<nowiki/>s pop up in our continuum.		If we carry this freshman-summing out a little further, new, larger [[ED8/3]]<nowiki/>s pop up in our continuum.

Moremajorthanmajor at 23:38, 20 June 2023

2023-06-20T23:38:23Z

← Older revision		Revision as of 23:38, 20 June 2023
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	\|Tuning=5L 2s<8/3>}}		\|Tuning=5L 2s<8/3>}}

	~~One way of distinguishing the '''17/12 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 [[17ed8/3]]<nowiki/>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<8/3>}}Among the most well-known variants of this '''17/12 diatonic''' MOS proper are [[17ed8/3]]<nowiki/>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 at 01:00, 11 June 2023

2023-06-11T01:00:48Z

← Older revision		Revision as of 01:00, 11 June 2023
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	{{Infobox MOS		{{Infobox MOS
	\| ~~Name =~~		\|Tuning=5L 2s<8/3>}}
	~~\| Equave~~ = 8/3
	~~\| Periods = 1~~
	~~\| nLargeSteps = 5~~
	~~\| nSmallSteps = 2~~
	~~\| Equalized = 4~~
	~~\| Collapsed = 3~~
	~~\| Pattern = LLLsLLs~~
	~~\| Neutral = 3L 4s~~
	}}

	One way of distinguishing the '''17/12 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 [[17ed8/3]]<nowiki/>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 '''17/12 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 [[17ed8/3]]<nowiki/>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.

Moremajorthanmajor at 23:54, 3 June 2023

2023-06-03T23:54:29Z

← Older revision		Revision as of 23:54, 3 June 2023
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	}}		}}

	One way of distinguishing the '''17/12 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 [[~~17edXI\|17EDXI~~]]<nowiki/>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 '''17/12 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 [[17ed8/3]]<nowiki/>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''==
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	*L L L s L L s		*L L L s L L s

	Insert 2 for L and 1 for s and you'll get the [[~~12edXI\|12EDXI~~]] diatonic.		Insert 2 for L and 1 for s and you'll get the [[12ed8/3]] diatonic.

	*2 2 2 1 2 2 1		*2 2 2 1 2 2 1

	When L=3, s=1, you have ~~17EDXI~~ of standard practice: 3 3 3 1 3 3 1		When L=3, s=1, you have 17ED8/3 of standard practice: 3 3 3 1 3 3 1

	When L=3, s=2, you have [[~~19edXI\|19EDXI~~]]: 3 3 3 2 3 3 2		When L=3, s=2, you have [[19ed8/3]]: 3 3 3 2 3 3 2

	When L=4, s=1, you have [[~~22edXI\|22EDXI~~]]: 4 4 4 1 4 4 1		When L=4, s=1, you have [[22ed8/3]]: 4 4 4 1 4 4 1

	When L=4, s=3, you have [[~~26edXI\|26EDXI~~]]: 4 4 4 3 4 4 3		When L=4, s=3, you have [[26ed8/3]]: 4 4 4 3 4 4 3

	When L=5, s=1, you have [[~~27edXI\|27EDXI~~]]: 5 5 5 1 5 5 1		When L=5, s=1, you have [[27ed8/3]]: 5 5 5 1 5 5 1

	When L=5, s=2, you have [[~~29edXI\|29EDXI~~]]: 5 5 5 2 5 5 2		When L=5, s=2, you have [[29ed8/3]]: 5 5 5 2 5 5 2

	When L=5, s=3, you have [[~~31EDXI~~]]: 5 5 5 3 5 5 3		When L=5, s=3, you have [[31ed8/3]]: 5 5 5 3 5 5 3

	When L=5, s=4, you have [[~~33edXI\|33EDXI~~]]: 5 5 5 4 5 5 4		When L=5, s=4, you have [[33ed8/3]]: 5 5 5 4 5 5 4

	So you have scales where L and s are nearly equal, which approach [[~~7edXI\|7EDXI~~]]:		So you have scales where L and s are nearly equal, which approach [[7ed8/3]]:

	*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 [[~~5edXI\|5EDXI~~]]:		And you have scales where s becomes so small it approaches zero, which would give us [[5ed8/3]]:

	*1 1 1 0 1 1 0 = 1 1 1 1 1		*1 1 1 0 1 1 0 = 1 1 1 1 1
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	==Tuning ranges==		==Tuning ranges==
	===Parasoft to ultrasoft===		===Parasoft to ultrasoft===
	"17/12 Flattone" systems, such as [[~~26edo\|26EDXI~~]].		"17/12 Flattone" systems, such as [[26ed8/3]].

	===Hyposoft===		===Hyposoft===
	"17/12 Meantone" (more properly "septimal meantone") systems, such as [[~~31edo\|31EDXI~~]].		"17/12 Meantone" (more properly "septimal meantone") systems, such as [[31ed8/3]].

	===Hypohard===		===Hypohard===
	The near-just part of the region is of interest mainly for those interested in 17/12 [[Pythagorean tuning]] and large, accurate EDO systems based on close-to-Pythagorean fifths, such as [[~~41edXI~~\|~~41EDXI~~]] and [[~~53edXI~~\|~~53EDXI~~]]. This class of tunings is called 17/12 [[schisma\|schismic]] temperament; these tunings can approximate 5<sup>17/12</sup>-limit harmonies very accurately by [[tempering out]] 17/12 of a small comma called the [[schisma]]. (Technically, ~~12EDXI~~ tempers out the 17/12 schisma and thus is a 17/12 schismic tuning, but it is nowhere near as accurate as 17/12 schismic tunings can be.)		The near-just part of the region is of interest mainly for those interested in 17/12 [[Pythagorean tuning]] and large, accurate EDO systems based on close-to-Pythagorean fifths, such as [[41ed8/3\|41ED8/3]] and [[53ed8/3\|53ED8/3]]. This class of tunings is called 17/12 [[schisma\|schismic]] temperament; these tunings can approximate 5<sup>17/12</sup>-limit harmonies very accurately by [[tempering out]] 17/12 of a small comma called the [[schisma]]. (Technically, 12ED8/3 tempers out the 17/12 schisma and thus is a 17/12 schismic tuning, but it is nowhere near as accurate as 17/12 schismic tunings can be.)

	The sharp-of-just part of this range includes so-called "17/12 [[neogothic]]" or "17/12 parapyth" systems, which tune the 17/12 diatonic major third slightly sharply of [[11/7]] (around [[128/81]]) and the diatonic minor third slightly flatly of [[13/11]] (around [[32/27]]). Good 17/12 neogothic ~~EDXIs~~ include [[~~29edo\|29EDXI~~]] and [[~~46edXI\|46EDXI~~]]. [[~~17edXI\|17EDXI~~]] is often considered the sharper end of the neogothic spectrum; its major third at 800 cents is considerably more discordant than in flatter 17/12 neogothic tunings.		The sharp-of-just part of this range includes so-called "17/12 [[neogothic]]" or "17/12 parapyth" systems, which tune the 17/12 diatonic major third slightly sharply of [[11/7]] (around [[128/81]]) and the diatonic minor third slightly flatly of [[14/11]] (around [[81/64]]). Good 17/12 neogothic ED8/3s include [[29ed8/3]] and [[46ed8/3]]. [[17ed8/3]] is often considered the sharper end of the neogothic spectrum; its major third at 800 cents is considerably more discordant than in flatter 17/12 neogothic tunings.

	===Parahard to ultrahard===		===Parahard to ultrahard===
	"17/12 Archy" systems such as [[~~17edo\|17EDXI~~]], [[~~22edo\|22EDXI~~]], and [[~~27edo\|27EDXI~~]].		"17/12 Archy" systems such as [[17ed8/3]], [[22ed8/3]], and [[27ed8/3]].

	==Modes==		==Modes==
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	==Scale tree==		==Scale tree==
	If 4\7 (four degrees of ~~7EDXI~~) is at one extreme and 3\5 (three degrees of ~~5EDXI~~) 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 ~~12EDXI~~.		If 4\7 (four degrees of 7ED8/3) is at one extreme and 3\5 (three degrees of 5ED8/3) 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 12ED8/3.

	If we carry this freshman-summing out a little further, new, larger [[EDXI]]<nowiki/>s pop up in our continuum.		If we carry this freshman-summing out a little further, new, larger [[EDXI]]<nowiki/>s pop up in our continuum.

Moremajorthanmajor at 01:45, 29 May 2023

2023-05-29T01:45:39Z

Show changes

Fredg999: Fredg999 moved page 5L 2s (8/3-equivalent) to User:Moremajorthanmajor/5L 2s (8/3-equivalent) over redirect: Page not ready for the main namespace

2023-05-20T05:22:14Z

Fredg999 moved page 5L 2s (8/3-equivalent) to User:Moremajorthanmajor/5L 2s (8/3-equivalent) over redirect: Page not ready for the main namespace

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

Moremajorthanmajor: Moremajorthanmajor moved page User:Moremajorthanmajor/5L 2s (8/3-equivalent) to 5L 2s (8/3-equivalent) over redirect

2023-05-19T02:57:45Z

Moremajorthanmajor moved page User:Moremajorthanmajor/5L 2s (8/3-equivalent) to 5L 2s (8/3-equivalent) over redirect

← Older revision	Revision as of 02:57, 19 May 2023
(No difference)

← Older revision		Revision as of 01:00, 11 June 2023
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	{{Infobox MOS		{{Infobox MOS
	\| ~~Name =~~		\|Tuning=5L 2s<8/3>}}
	~~\| Equave~~ = 8/3
	~~\| Periods = 1~~
	~~\| nLargeSteps = 5~~
	~~\| nSmallSteps = 2~~
	~~\| Equalized = 4~~
	~~\| Collapsed = 3~~
	~~\| Pattern = LLLsLLs~~
	~~\| Neutral = 3L 4s~~
	}}

	One way of distinguishing the '''17/12 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 [[17ed8/3]]<nowiki/>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 '''17/12 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 [[17ed8/3]]<nowiki/>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.