Metallic intonation: Difference between revisions

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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 [[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 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.

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.

~~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~~.

== 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.

== 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.

[[Category:Math]]

[[Category:Xenharmonic series]]

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-{{stub}}
+'''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 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.
+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.
-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.
+== 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.
+== 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.
+{{todo|inline=1|expand}}
+[[Category:Math]]
+[[Category:Xenharmonic series]]