Metallic intonation - Revision history

FloraC: Text replacement - "octave" to "octave"

2026-07-14T14:15:20Z

Text replacement - "octave" to "octave"

← Older revision		Revision as of 14:15, 14 July 2026
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	== Harmony ==		== Harmony ==
	If reduced with acoustic phi as the period, the chord formed by the silver and bronze ratios above the root is, coincidentally, a fairly conventional major triad (0¢-402.2¢-692.7¢). This makes it so traditional chord types are easily accessible in metallic intonation systems, but not [[~~2/1\|~~octave]]s, similarly to the [[Carlos Alpha]] tuning.		If reduced with acoustic phi as the period, the chord formed by the silver and bronze ratios above the root is, coincidentally, a fairly conventional major triad (0¢-402.2¢-692.7¢). This makes it so traditional chord types are easily accessible in metallic intonation systems, but not [[octave]]s, similarly to the [[Carlos Alpha]] tuning.

	== Tempered systems ==		== Tempered systems ==

BudjarnLambeth: {{Mathematical interest}}

2025-09-29T10:48:16Z

← Older revision		Revision as of 10:48, 29 September 2025
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			{{Mathematical interest}}
	'''Metallic intonation''' ('''MTI'''){{idiosyncratic}} is a system which uses the irrational [[metallic harmonic series]], based on {{w\|metallic mean}}s, rather than the [[harmonic series]] as the basis for an exact or tempered tuning. It was first described by [[User:CompactStar\|CompactStar]] in 2024. Metallic intonation is related to and can sometimes overlap with [[merciful intonation]]. Except for the [[1/1\|unison]], it consists of only irrational intervals, and is inherently [[nonoctave]]. As the first metallic harmonic, [[acoustic phi]] is a possible candidate to serve as an [[equave]] in the same way as octave. Metallic intonation is suitable for inharmonic [[timbre]]s based on metallic harmonics rather than harmonics.		'''Metallic intonation''' ('''MTI'''){{idiosyncratic}} is a system which uses the irrational [[metallic harmonic series]], based on {{w\|metallic mean}}s, rather than the [[harmonic series]] as the basis for an exact or tempered tuning. It was first described by [[User:CompactStar\|CompactStar]] in 2024. Metallic intonation is related to and can sometimes overlap with [[merciful intonation]]. Except for the [[1/1\|unison]], it consists of only irrational intervals, and is inherently [[nonoctave]]. As the first metallic harmonic, [[acoustic phi]] is a possible candidate to serve as an [[equave]] in the same way as octave. Metallic intonation is suitable for inharmonic [[timbre]]s based on metallic harmonics rather than harmonics.

BudjarnLambeth at 02:00, 15 December 2024

2024-12-15T02:00:33Z

← Older revision		Revision as of 02:00, 15 December 2024
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	{{todo\|inline=1\|expand}}		{{todo\|inline=1\|expand}}
	[[Category:Math]]		[[Category:Math]]
			[[Category:Xenharmonic series]]

BudjarnLambeth: Todo and categories

2024-12-15T02:00:17Z

Todo and categories

← Older revision		Revision as of 02:00, 15 December 2024
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	{{~~Stub~~}}		{{todo\|inline=1\|expand}}
			[[Category:Math]]

CompactStar at 08:01, 22 April 2024

2024-04-22T08:01:24Z

← Older revision		Revision as of 08:01, 22 April 2024
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	== Harmony ==		== Harmony ==
	If reduced with acoustic phi as the period, the chord formed by the silver and bronze ratios above the root is, coincidentally, a fairly conventional major triad (0¢-402.2¢-692.7¢). This makes it so traditional chord types are easily accessible in metallic intonation systems, but not [[2/1\|octave]]s, similarly to the [[Carlos Alpha]] tuning. [[6edφ]] offers a basic equal-tempered approximation of the metallic major triad by steps 0-3-5 (0¢-416.5¢-694.2¢), although with a noticeably sharp third. Systems containing "quasi-equalized" versions of 6edφ, such as [[17edφ]], [[23edφ]], and [[29edφ]] include more accurate approximations.		If reduced with acoustic phi as the period, the chord formed by the silver and bronze ratios above the root is, coincidentally, a fairly conventional major triad (0¢-402.2¢-692.7¢). This makes it so traditional chord types are easily accessible in metallic intonation systems, but not [[2/1\|octave]]s, similarly to the [[Carlos Alpha]] tuning.

			== Tempered systems ==
			[[6edφ]] offers a basic equal-tempered approximation of the metallic major triad by steps 0-3-5 (0¢-416.5¢-694.2¢), although with a noticeably sharp third. Systems containing "quasi-equalized" versions of 6edφ, such as [[17edφ]], [[23edφ]], and [[29edφ]] include more accurate approximations.


	{{Stub}}		{{Stub}}

CompactStar at 07:59, 22 April 2024

2024-04-22T07:59:12Z

← Older revision		Revision as of 07:59, 22 April 2024
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	'''Metallic intonation''' ('''MTI'''){{idiosyncratic}} is a system which uses the irrational [[metallic harmonic series]], based on {{w\|metallic mean}}s, rather than the [[harmonic series]] as the basis for an exact or tempered tuning. It was first described by [[User:CompactStar\|CompactStar]] in 2024. Metallic intonation is related to and can sometimes overlap with [[merciful intonation]]. Except for the [[1/1\|unison]], it consists of only irrational intervals, and is inherently [[nonoctave]]. As the first metallic harmonic, [[acoustic phi]] is a possible candidate to serve as an [[equave]] in the same way as octave. Metallic intonation is suitable for inharmonic ~~timbres which have a spectrum of~~ metallic ~~haromoncs rathr~~ than harmonics.		'''Metallic intonation''' ('''MTI'''){{idiosyncratic}} is a system which uses the irrational [[metallic harmonic series]], based on {{w\|metallic mean}}s, rather than the [[harmonic series]] as the basis for an exact or tempered tuning. It was first described by [[User:CompactStar\|CompactStar]] in 2024. Metallic intonation is related to and can sometimes overlap with [[merciful intonation]]. Except for the [[1/1\|unison]], it consists of only irrational intervals, and is inherently [[nonoctave]]. As the first metallic harmonic, [[acoustic phi]] is a possible candidate to serve as an [[equave]] in the same way as octave. Metallic intonation is suitable for inharmonic [[timbre]]s based on metallic harmonics rather than harmonics.

	In metallic intonation, the metallic means are taken as [[basis element]]s of [[subgroup]]s rather than [[prime]]s, but not all metallic means are necessary because some are analogues of composite integers, in that they can be expressed in terms of other metallic means. For example, the fourth metallic harmonic is a redundant generator because it is the golden ratio (the first metallic harmonic) cubed.		In metallic intonation, the metallic means are taken as [[basis element]]s of [[subgroup]]s rather than [[prime]]s, but not all metallic means are necessary because some are analogues of composite integers, in that they can be expressed in terms of other metallic means. For example, the fourth metallic harmonic is a redundant generator because it is the golden ratio (the first metallic harmonic) cubed.

CompactStar at 07:58, 22 April 2024

2024-04-22T07:58:44Z

← Older revision		Revision as of 07:58, 22 April 2024
Line 1:		Line 1:
	'''Metallic intonation''' ('''MTI'''){{idiosyncratic}} is a system which uses the irrational [[metallic harmonic series]], based on {{w\|metallic mean}}s, rather than the [[harmonic series]] as the basis for an exact or tempered tuning. It was first described by [[User:CompactStar\|CompactStar]] in 2024. Metallic intonation is related to and can sometimes overlap with [[merciful intonation]]. Except for the [[1/1\|unison]], it consists of only irrational intervals, and is inherently [[nonoctave]]. As the first metallic harmonic, [[acoustic phi]] is a possible candidate to serve as an [[equave]] in the same way as octave.		'''Metallic intonation''' ('''MTI'''){{idiosyncratic}} is a system which uses the irrational [[metallic harmonic series]], based on {{w\|metallic mean}}s, rather than the [[harmonic series]] as the basis for an exact or tempered tuning. It was first described by [[User:CompactStar\|CompactStar]] in 2024. Metallic intonation is related to and can sometimes overlap with [[merciful intonation]]. Except for the [[1/1\|unison]], it consists of only irrational intervals, and is inherently [[nonoctave]]. As the first metallic harmonic, [[acoustic phi]] is a possible candidate to serve as an [[equave]] in the same way as octave. Metallic intonation is suitable for inharmonic timbres which have a spectrum of metallic haromoncs rathr than harmonics.
	~~The~~ metallic means are taken as [[basis element]]s of [[subgroup]]s rather than [[prime]]s, but not all metallic means are ~~included~~ because some can be expressed in terms of other metallic means~~, similar to how some integers are composite~~. For example, the fourth metallic harmonic is a redundant generator because it is the golden ratio (the first metallic harmonic) cubed.
			In metallic intonation, the metallic means are taken as [[basis element]]s of [[subgroup]]s rather than [[prime]]s, but not all metallic means are necessary because some are analogues of composite integers, in that they can be expressed in terms of other metallic means. For example, the fourth metallic harmonic is a redundant generator because it is the golden ratio (the first metallic harmonic) cubed.

	== Harmony ==		== Harmony ==

CompactStar at 07:56, 22 April 2024

2024-04-22T07:56:10Z

← Older revision		Revision as of 07:56, 22 April 2024
Line 3:		Line 3:

	== Harmony ==		== Harmony ==
	If reduced with acoustic phi as the period, the chord formed by the silver and bronze ratios above the root is, coincidentally, a fairly conventional major triad (0¢-402.2¢-692.7¢). This makes it so traditional chord types are easily accessible in metallic intonation systems, but not [[2/1\|octave]]s, similarly to the [[Carlos Alpha]] tuning. [[6edφ]] offers a basic equal-tempered approximation of the metallic major triad by steps 0-3-5 (0¢-416.5¢-694.2¢), although with a noticeably sharp third. Systems containing "quasi-equalized" versions of 6edφ, such as [[17edφ]] and [[~~23edφ~~]] include more accurate approximations.		If reduced with acoustic phi as the period, the chord formed by the silver and bronze ratios above the root is, coincidentally, a fairly conventional major triad (0¢-402.2¢-692.7¢). This makes it so traditional chord types are easily accessible in metallic intonation systems, but not [[2/1\|octave]]s, similarly to the [[Carlos Alpha]] tuning. [[6edφ]] offers a basic equal-tempered approximation of the metallic major triad by steps 0-3-5 (0¢-416.5¢-694.2¢), although with a noticeably sharp third. Systems containing "quasi-equalized" versions of 6edφ, such as [[17edφ]], [[23edφ]], and [[29edφ]] include more accurate approximations.


	{{Stub}}		{{Stub}}

CompactStar at 07:55, 22 April 2024

2024-04-22T07:55:55Z

← Older revision		Revision as of 07:55, 22 April 2024
Line 3:		Line 3:

	== Harmony ==		== Harmony ==
	If reduced with acoustic phi as the period, the chord formed by the silver and bronze ratios above the root is, coincidentally, a fairly conventional major triad (0¢-402.2¢-692.7¢). This makes it so traditional chord types are easily accessible in metallic intonation systems, but not [[2/1\|octave]]s, similarly to the [[Carlos Alpha]] tuning. [[6edφ]] offers a basic equal-tempered approximation of the metallic major triad by steps 0-3-5 (0¢-416.5¢-694.2¢), although ~~the sharpness of the major~~ third ~~is more apparent here and it may sound~~ "~~more xenharmonic~~".		If reduced with acoustic phi as the period, the chord formed by the silver and bronze ratios above the root is, coincidentally, a fairly conventional major triad (0¢-402.2¢-692.7¢). This makes it so traditional chord types are easily accessible in metallic intonation systems, but not [[2/1\|octave]]s, similarly to the [[Carlos Alpha]] tuning. [[6edφ]] offers a basic equal-tempered approximation of the metallic major triad by steps 0-3-5 (0¢-416.5¢-694.2¢), although with a noticeably sharp third. Systems containing "quasi-equalized" versions of 6edφ, such as [[17edφ]] and [[23edφ]] include more accurate approximations.
	[[~~Category:Tuning~~]]


	{{Stub}}		{{Stub}}

CompactStar at 03:13, 20 March 2024

2024-03-20T03:13:54Z

← Older revision		Revision as of 03:13, 20 March 2024
Line 1:		Line 1:
	'''Metallic intonation''' ('''MTI'''){{idiosyncratic}} is a system which uses the irrational [[metallic harmonic series]], based on {{w\|metallic mean}}s, rather than the [[harmonic series]] as the basis for an exact or tempered tuning. It is related to and can sometimes overlap with [[merciful intonation]]. Except for the [[1/1\|unison]], it consists of only irrational intervals, and is inherently [[nonoctave]]. As the first metallic harmonic, [[acoustic phi]] is a possible candidate to serve as an [[equave]] in the same way as octave.		'''Metallic intonation''' ('''MTI'''){{idiosyncratic}} is a system which uses the irrational [[metallic harmonic series]], based on {{w\|metallic mean}}s, rather than the [[harmonic series]] as the basis for an exact or tempered tuning. It was first described by [[User:CompactStar\|CompactStar]] in 2024. Metallic intonation is related to and can sometimes overlap with [[merciful intonation]]. Except for the [[1/1\|unison]], it consists of only irrational intervals, and is inherently [[nonoctave]]. As the first metallic harmonic, [[acoustic phi]] is a possible candidate to serve as an [[equave]] in the same way as octave.

	The metallic means are taken as [[basis element]]s of [[subgroup]]s rather than [[prime]]s, but not all metallic means are included because some can be expressed in terms of other metallic means, similar to how some integers are composite. For example, the fourth metallic harmonic is a redundant generator because it is the golden ratio (the first metallic harmonic) cubed.		The metallic means are taken as [[basis element]]s of [[subgroup]]s rather than [[prime]]s, but not all metallic means are included because some can be expressed in terms of other metallic means, similar to how some integers are composite. For example, the fourth metallic harmonic is a redundant generator because it is the golden ratio (the first metallic harmonic) cubed.