Cubes of Sound, Tonal Space: A Geometric Representation of Consonancesand other Tone Intervals

Hårklau, H. Cubes of Sound, Tonal Space: A Geometric Representation of Consonances and other Tone Intervals 2010 (revised 2012) Internet 6 p 08362 Herda, A. Soul and Kosmos. Menelaos and the Shield of Euphorbos in Didyma Berlin/Athens/Tübingen Internet 13 pThe Center for Hellenic Studies | 3100 Whitehaven Street, NW. 2008 08363 Horky, P. Aristotle on mathematical mathematical pythagoreanism in the 4th century 3 - 36 In: Plato and Pythagoreanism ch 1 2013 08364 Izdebska, A. The Pythagorean Metaphysics of Numbers in the Works of the Ikhwān al-̣Safāʼ and al-Shahrastāni .......p 361 - 374 in Renger and Stavru (eds.), Pythagorean Knowledge from the Ancient to the Modern World: Askesis, Religion, Science, Wiesbaden: Harrassowitz Verlag 2016, 467-518.

Cubes"of"Sound,"Tonal"Space:"A"Geometric"Representation"of" Consonances"and"other"Tone"Intervals" Halvard"Hårklau"2010"(revised"2012)" " " " " Abstract( A" while" ago," I" needed" to" compare" an" intruiging" natural" phenomenon" with" musical" harmonies," and" different" systems" of" tuning" were" tested." As" it" turned" out," both" the" traditional" Pythagorean" tuning" and"the"just"tuning"of"the"Renaissance"seemed"to"be"relevant."Hence,"there"was"a"need"to"construct" a"system"that"compiced"both"these"“musical"paradigms”."For"the"sake"of"a"comprehencive"overview," a"geometric"representation"was"worked"out,"and"surpicing"and"beautiful"features"developed"as"the" system"crystallized,"litterally"into"a"square"lattice." While"it"is"customary"to"represent"tone"ratios"as"direct"functions"of"frequencies,"with"higher"number" representing"higher"pitch,"in"this"essay"the"ratios"relate"to"string"lenghts,"as"in"a"monocord"(which"are" resiprocally" proportional" to" the" frequencies);" hence," unless" otherwise" stated," " lower" numbers" represent"higher"pitches;"i.e."1:2"is"one"octave"higher"than"the"fundamental,"while"2:1"is"one"octave" lower." " " Introduction( [yet"to"be"written]" "

" ( Figure( 1:( “The" Platonic" Lambda”," or" an" interpretation"of"the"Tetractys"using"the"series"of" tone"intervals"from"Plato´s"Timaios"(odd"series"1V 3V9V27"and"even"series"1V2V4V8)"and"filling"in"the" gaps."Each"step"downwards"towards"the"left"can" be" concevided" as" performing" an" operator" of" doubling" (x2," musical" octaves)," while" each" step" downwards" towards" the" right" performs" the" operator" of" tripling" (x3," musical" duodecims" or" 12ths)." A" horizontal" step" towards" the" right" performs"the"operator"x3:2,"musical"fifths." " " " Figure(2:(A"musical"fifth"(3:2)"can"be"defined"as"a" harmonious" combination" of" thirds:" Either" a" major" (5:4)" or" minor" (6:5)" third" in" sequence" or" vice" versa." If" we" let" the" numbers" correspond" to" relative" string" lengths," larger" numbers" correspond" to" lower" pitch;" hence," this" example" illustrates" a" “downward" tuning”" from" the" fundamental," as" was" the" custom" in" ancient" Greece" " " " Figure( 3:" Performing" the" inverse" operations" in" the" opposite" directions" gives" symmetric" upper" and" lower" sets" of" triads," and" we" obtain" a" hexagonal"figure"with"point"corresponding"to"six" different"tone"intervals"from"the"fundamental"in" the"centre."" " "

The planes defined by the thirds and the fifths are separated by fourths. Each of these planes can be considere as composed of an infinete system of Major and Minor Triads.

F F* Cc Cr 7 C-major red A-major red C-minor blue A-minor blue

" ( Figure(10:"´The"Pythagorean"Plane."The"horizontal"row"with"bluish"background"illustrates"a"series"of" twelve"fifths,"which"is"a"way"of"defining"a"chromatic"Pythagorean"scale."The"diagonals"with"greenish" background"illustrate"a"diatonic"scale"(from"C"to"B,"corresponding"to"modern"CVmajor)."This"together" with" series" marked" yellow" comprises" a" complete" enharmonic" Pythagorean" scale" of" 17" notes." The" purple"rectangle"is"the"unit,"illustrated"in"the"previous"figure"as"a"central"plane"of"a"unit"cube."Note" that" one" diagonal" step" upwards" to" the" left" is" a" whole" tone" step" upwards," while" a" diagonal" step" downwards"to"the"left"is"an"octave"upwards."A"fourth"upwards"is"one"step"downwards,"while"a"fifth" upwards"is"one"step"to"the"left."We"leave"it"as"an"exercise"to"the"reader"to"calculate"the"exact"tone" values,"with"middle"C"as"1"as"reference,"and"to"fill"in"the"remaining"tones"in"the"scheme." " "