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| | | #REDIRECT [[Second]] |
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| A '''neutral second''', as a concrete interval region, is typically near 150 cents in size, distinct from [[Semitone|semitones]] of roughly 100 cents and [[Major second|major seconds]] of roughly 200 cents. A rough tuning range of the neutral second is 130 to 170 cents, and they characteristically are half of some kind of [[minor third]]. Neutral seconds are not found in diatonic, but they do exist in a diatonic functional context as "neutral" diatonic intervals.
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| They are one of the most distinctive-sounding yet versatile xenharmonic intervals, which makes them highly valuable.
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| ==Function==
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| The large step of armotonic is always some sort of neutral second, which functions similar to both a whole tone and semitone.
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| A step of 13ed3 is a middle neutral second. It is notable for having an unusually good approximation of LCJI for a system generated by neutral seconds.
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| Supraminor/submajor scales, such as 2.3.17/7 blackdye, have a large number of neutral seconds.
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| '''Porcupine neutral seconds''' are larger neutral seconds most commonly between 2\15 and 3\22 which, consistent with Porcupine temperament, split a sharp 6/5 in half and a flat 4/3 into thirds. This functionally makes them a very flat minor whole tone (~10/9), and are thus an easy way to make otherwise uninteresting progressions sound xenharmonic.
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| In larger edos, it's possible to have a similar tuning of intervals, but without Porcupine tempering. In the 100b val, the tempered 10/9 is a slightly larger 168¢ in exchange for making the tempered 81/80 48¢, a much more usable aberrisma than Porcupine typically offers.
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| == In just intonation ==
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| === By prime-limit ===
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| The [[Pythagorean tuning|3-limit]] does not have a simple neutral second, so we start with the 5-limit:
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| * The 5-limit '''acute minor second''' or '''large limma''' is a ratio of [[27/25]], and is about 133{{c}}.
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| * The 7-limit neutral second is a ratio of [[35/32]], and is about 155{{c}}.
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| ** There is also a 7-limit '''swetismic neutral second''', which is a ratio of [[49/45]], and is about 147{{c}}.
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| * The 11-limit neutral/submajor seconds are the ratios of [[12/11]] and [[11/10]], which are about 151{{c}} and 165{{c}}, respectively; 11/10 in particular can also be analyzed as a [[major second]]. Despite that, it is also here for completeness.
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| * The 13-limit neutral/supraminor seconds are the ratios of [[14/13]] and [[13/12]], which are about 128{{c}} and 139{{c}}, respectively; 14/13 in particular can also be analyzed as a [[semitone]]. Despite that, it is also here for completeness.
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| == In mos scales ==
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| Intervals between 120 and 171{{c}} generate the following [[MOS|mos]] scales:
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| These tables start from the last monolarge mos generated by the interval range.
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| Scales with more than 12 notes are not included.
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| {| class="wikitable"
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| !Range
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| ! colspan="2" |Mos
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| |-
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| |120–133{{c}}
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| |[[1L 8s]]
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| |[[9L 1s]]
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| |-
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| |133–150{{c}}
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| |[[1L 7s]]
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| |[[8L 1s]]
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| |-
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| |150–171{{c}}
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| |[[1L 6s]]
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| |[[7L 1s]]
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| |}
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| ==Categorization==
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| ===Proposal: Ground's Neutral Second Categorization System===
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| As neutral intervals, neutral seconds can be named based on interval splitting.
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| {| class="wikitable"
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| |+ Names Based on Interval Splitting (with Nearby Edo Intervals)
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| |-
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| ! ¢ !! Definition !! Name (accepted names are bold) !! Edo !! ¢ !! Error ¢
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| | 119.443 || '''15/14''' || '''Septimal Major Semitone''' || 1\10 || 120.000 || 0.557
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| | 121.243 || 7/6 / <sup>4</sup>√(7/5) || Quadranseptimal Supraminor Second || 1\10 || 120.000 || -1.243
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| | 124.511 || <sup>4</sup>√(4/3) || Quadranpyth Supraminor Second || 3\29 || 124.138 || -0.373
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| | 128.298 || '''14/13''' || '''Tridecimal Supraminor Second''' || 3\28 || 128.571 || 0.273
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| | 133.435 || √(7/6) || Semiseptal Neutral Second || 1\9 || 133.333 || -0.102
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| | 138.573 || '''13/12''' || '''Tridecimal Neutral Second''' || 3\26 || 138.462 || -0.111
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| | 140.391 || <sup>5</sup>√(3/2) || Quintanpyth Neutral Second || 2\17 || 141.176 || 0.785
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| | 145.628 || <sup>4</sup>√(7/5) || Quadranseptimal Neutral Second || 4\33 || 145.455 || -0.174
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| | 150.637 || '''12/11''' || '''Undecimal Neutral Second''' || 1\8 || 150.000 || -0.637
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| | 157.821 || √(6/5) || Semipental Neutral Second || 5\38 || 157.895 || 0.074
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| | 165.004 || '''11/10''' || '''Undecimal Submajor Second''' || 4\29 || 165.517 || 0.513
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| | 166.015 || <sup>3</sup>√(4/3) || Trienpyth Submajor Second || 4\29 || 165.517 || -0.498
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| | 170.013 || 6/5 / <sup>4</sup>√(7/5) || Quadranseptimal Submajor Second || 1\7 || 171.429 || 1.415
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| | 175.489 || <sup>4</sup>√(3/2) || Quadranpyth Major Second || 6\41 || 175.610 || 0.121
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| |}
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| {{Interval regions}}
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