MusicalIntervalStep
plain-language theorem explainer
MusicalIntervalStep supplies the carrier type of natural numbers for recording interval steps in the musical realization of universal forcing. It is referenced by musicCost, musicInterpret, and the musicRealization structure that builds a concrete LogicRealization. The declaration is a direct one-line abbreviation to Nat that supplies the arithmetic for pitch-ratio stacking and iteration counts.
Claim. Let $M$ be the type of musical interval steps. Then $M := {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212
background
The MusicRealization module supplies a lightweight musical realization of the universal forcing. The carrier records interval steps whose semantic reading is pitch-ratio stacking; the forced arithmetic is the iteration count of interval composition. MusicalIntervalStep is introduced as the carrier type and is immediately used to define musicCost, which returns 0 on equal steps and 1 otherwise, together with the musicRealization structure that packages Carrier, Cost, compare, and zero into a LogicRealization instance.
proof idea
The declaration is a direct abbreviation MusicalIntervalStep := Nat. No lemmas are applied; the abbreviation simply aliases the natural-number type so that downstream definitions such as musicCost and musicRealization can treat interval steps as ordinary arithmetic objects.
why it matters
MusicalIntervalStep supplies the concrete carrier for musicRealization, which realizes LogicRealization inside the UniversalForcing.MusicRealization module. It thereby provides the arithmetic substrate for interval comparison and feeds the musicCost, musicCost_self, musicCost_symm, musicInterpret, and musicRealization declarations. The construction sits inside the broader forcing chain by furnishing a pitch-ratio model whose iteration counts align with the eight-tick octave structure.
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