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 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 [[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. | |||
== 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¢). | 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. | |||
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[[Category:Math]] | |||
[[Category:Xenharmonic series]] | |||
Latest revision as of 02:00, 15 December 2024
Metallic intonation (MTI)[idiosyncratic term] is a system which uses the irrational metallic harmonic series, based on metallic means, rather than the harmonic series as the basis for an exact or tempered tuning. It was first described by CompactStar in 2024. Metallic intonation is related to and can sometimes overlap with merciful intonation. Except for the 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 based on metallic harmonics rather than harmonics.
In metallic intonation, the metallic means are taken as basis elements of subgroups rather than primes, 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
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 octaves, 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.
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