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Daniel Shawcross Wilkerson Begun 23 September 2006; first published version 19 February 2012; second version 9 July 2014.Abstract and IntroductionMost music theory books are like medieval medical textbooks: theycontain unjustified superstition, non-reasoning, and funny symbolsglorified by Latin phrases. How does music, in particular harmony,actually work, presented as a real, scientifictheory of music?The core to our approach is to consider not only thePhysical phenomena of nature but also theComputational phenomena of any machine that must listen toand make sense of sound, such as the human brain; by adding thiscomputational approach to that of the physical, heretofore unexplainedphenomena of harmony simply fall out, that is, are suddenly derived ina straightforward manner. In particular we derive from firstprinciples of Physics and Computation the following three fundamentalphenomena of music:the Major Scale,the Standard Chord Dictionary, andthe difference in feeling between the Major and Minor Triads.While the Major Scale has been independently derived before byothers in a similar manner as we do here [Helmholtz1863, p. 300], [Birkhoff1933, p. 92], I believe the derivation ofthe Standard Chord Dictionary as well as the difference in feelingbetween the Major and Minor Triads to be an original contribution toscience and art. We think our observations should convertstraightforwardly into an algorithm for classifying the basic aspectsof tonal music in a manner similar to the way a human would.We also examine the theory of the heretofore agreed-upon authorityon this subject, 19th-century German Physicist Hermann Helmholtz[Helmholtz1863], and show that his theory,while making correct observations, and while qualifying as scientific,fails to actually explain the three observed phenomena listed above;Helmholtz isn't really wrong, he just fails to be really right. Thefailure of the theory of Helmholtz as an explanation for harmony ismade starkly clear by a simple observation that has somehow heretoforegone unacknowledged: his theory says that notes are in "concord"because the sound playing them together is "less worse" than theplaying together of notes that are in less concord; but this theory isonly subtractive, that is, additional notes can onlypenalize, some merely less than others, and so the most harmonioussound should be a single note by itself(!) and harmony would not existas a phenomenon of music at all. Yet harmony does exist, anddoes so because people experience harmony as adding somethingbeyond the playing of a single note; this additive nature of harmonyis easily explained by our theory. We also consider the more recentand more computational theory of Terhardt [Terhardt1974-PCH] (and others) and show that,while his approach (and, it seems, that of others following in histhread) also attempts a computational explanation and derives someobservations that seem to resemble some of those of the initial partof our analysis, we seem to go further.I intend this article to be satisfying to scientists as an originalcontribution to science (as a set of testable conjectures that explainobserved phenomena), yet I also intend it to be approachable bymusicians and other curious members of the general public who may havelong wondered at the curious properties of tonal music and beenfrustrated by the lack of satisfying, readable exposition on thesubject. Therefore I have written in a deliberately plain andconversational style, avoiding unnecessarily formal language; BenjaminFranklin and Richard Feynman often wrote in a plain and conversationalstyle, so if you don't like it, to quote Richard Feynman, "Don't bugme man!"Table of Contents1 The Problem of Music1.1 Modern "Music Theory" Reads Like a Medieval Medical Textbook1.2 What is a Satisfactory, Scientific Theory?1.3 Music "Theory" is Not a Scientific Theory of Anything1.4 Can we Make a Satisfactory Theory of Music?1.5 Physical Science: Harmonics Everywhere1.5.1 Timbre: Systematic Distortions from the Ideal Harmonic Series1.6 Computational Science: as Fundamental as Physical Science1.6.1 Algorithms are Universal2 Living in a Computational Cartoon2.1 Searching for Harmonics2.1.1 Virtual Pitch: Hearing the Harmonic Series Even When it is Not There2.1.2 Using Greatest Common Divisor as the Missing Fundamental2.1.3 Even Animals Seem to Compute the Ideal Harmonic Series2.2 Artifacts of Optimization2.2.1 Relative Pitch: Differences Between Sounds2.2.2 Octaves: Sounds Normalized to a Factor of Two2.3 Harmony: Sweetness is the Ideal2.3.1 Recreating an Ideal Harmonic Series using Instruments having Systematically-Distorted Timbre2.3.2 Harmony Induces Two Kinds of Intervals: Horizontal Within the Note and Vertical Across the Notes2.3.3 Vertical Intervals Have Pure Ratios2.3.4 Vertical Intervals Have Balanced Amplitudes2.3.5 Vertical Intervals Are All The Same Ratio2.3.6 Harmony is Sweeter Than Sweet2.4 Interestingness: Just Enough Complexity2.4.1 The Simplicity of Theme2.4.2 The Complexity of Ambiguity2.5 Recognition: Feature Vectors2.5.1 Soft Computing2.5.2 False Recognition2.5.3 Cubism: Partial Recognition Due to Redundant, Over-Determined Feature Vectors3 Harmonic Music Explained3.1 The Major Triad3.2 The Major Scale3.2.1 Interlocking Triads3.2.2 Using Logarithms to Visualize Distances Between Tones/Notes3.2.3 The Keyboard Revealed3.3 Scales and Keys3.3.1 Changing Key: Playing Other Groups of Triads3.3.2 Key Changes Break Harmony3.3.3 Just versus Equal Tuning3.4 The Minor3.4.1 The Minor Triad3.4.2 The Minor as Auditory Cubism3.4.3 Minor Scales3.5 Chords3.5.1 The Standard Chord Dictionary3.5.2 How to Turn Sweetness into Mud: Over-Using Octaves3.5.3 Chords from the Harmonic Series3.5.4 Chords Inducing Ambiguity3.5.5 Chords Using the Minor Triad3.5.6 Chords Preserving Intervals but not Harmonics4 Miscellaneous Objections4.1 But what about the Circle of Fifths!4.1.1 Fifths make a Circle4.1.2 The Circle of Fifths is Just a Combinatorial Coincidence4.1.3 The Circle of Fifths Allows for Cool Chord Transitions4.1.4 The Symmetries of the Circle of Fifths are a Terrible Red Herring4.2 But Other Cultures Have Different Musical Scales!4.2.1 A Culture May Simply not be Fully Exploiting All of the Universal Harmonic Features4.2.2 But The Nasca People Of Peru Use A Linear, Not A Logarithmic, Scale!4.3 But You Can Make a Piece of Music Based Entirely on That Utterly Un-Harmonic Interval, the Augmented Fourth!4.4 But I've Been a Musician All My Life / Studied Music In College and I've Never Heard Any of This Before!5 Helmholtz Fails to Explain Harmony5.0.1 Overview of the Theory of Helmholtz5.0.2 The Theory of Helmholtz Implies that Harmony Should Not Exist as a Phenomenon of Music At All5.0.3 Helmholtz did not have Access to Computer Science5.1 Helmholtz's Theory Relies Only On Interfering Overtones, But Harmony Is Something More5.2 Helmholtz's Theory Does Not Imply Virtual Pitch5.3 Helmholtz's Theory is that Pleasure is Only the Absence of Pain5.3.1 Harmony is Rapture5.4 Helmholtz's Theory Fails to Fully Explain the Qualitative Difference Between the Major and Minor Triads5.5 Helmholtz Isn't Really Wrong, He Just Fails To Be Really Right6 Other Modern Theories, such as Terhardt and 'Fusion or pattern matching' Theory6.1 Terhardt Recognizes that the Brain is Listening For Something6.2 Terhardt Does Not Explain Suspended and Minor Chords7 Future Work: Towards A Unifying Theory of Music7.1 Melody as Arpeggio7.1.1 Scale As Theme: Melodic Association From Harmonic Association7.1.2 Streaming: Multiple Similar Phenomenon Occurring Consecutively Are Explained By The Brain As One Thing Moving7.1.3 Melody can Easily Create Interesting Ambiguities7.2 The Role of Narrative Generally7.3 Embodiment and Emotion7.4 A Proposal For A Unifying Physical and Computational Theory of Music8 Acknowledgements9 References1 The Problem of MusicPeople push different keys on a piano; some combinations andpatterns sound good; others do not. How does that work? Looking at apiano, it is laid out in the following pattern (w=white, b=black)
Hmm, the white and black keys mostly just alternate, yetthese alternating regions last for 5 and then 7 keys and then that 5/7region-pair repeats, and where these regions meet there are twoadjacent white keys. There seems to be a pattern, but it is quite anodd one.The piano keyboard seems really weird andad-hoc.Doesn't it seem that something as simple as sound should have a simpledevice for producing it?Further, this weirdness is not specific just to the piano: the keylayout reflects the Major Scale [maj] which is thebasis of all Western music. Is that black-white pattern somehowfundamental to sound and music itself? Or are they really just acultural coincidence, combinations of sounds that we have heard overand over since infancy and been trained to associate with differentemotions? Is something fundamental to the ear and to sound itselfthat is going on here or not?1.1 Modern "Music Theory" Reads Like a Medieval Medical TextbookThese questions have bothered me literally for decades (startingwhen I was about ten, looking at our piano keyboard and asking"what?!"; I basically wrote the above Section 1 "The Problem of Music"at that time). Consulting "music theory" never helped me either, asReading a music theory book is like reading a medievalmedical textbook: such books are full of unjustified superstition,non-reasoning, and funny symbols glorified by Latin phrases.For example, here is the first page from a famous book on JazzTheory, "Jazz Improvisation 1: Tonal and Rhythmic Principles" by JohnMehegan [Mehegan1959]. Recall, this isthe first page of Lesson 1 of Section 1 of Book 1, the veryfirst thing the student reads!"Each of the twelve scales is a frame forming the harmonicsystem."What is a "scale"? Where do they come from? For what purpose arethere or how does it emerge that there are twelve exactly?What is a "harmonic system" and what does it mean to say a scale"frames" it?"Diatonic harmony moves in two directions: Horizontal andVertical."Really?! They both look pretty diagonal to me. Oh, but it'sDiatonic! That sounds Latin so I guess these people are smart."By combining these two movements... we derive thescale-tone seventh chords in the key of C."What is a "chord"? What is a "key"? WHAT THE HECK ARE THEYTALKING ABOUT!You can't start a science textbook likethat. You have to start with simple observations humans can make. Youhave to build up complex structures from simple ones. You have tomotivate your distinctions.Even if you say "A chord is 3 or more notes played together" that'salso almost the definition of a "key" as well; for what purpose do wehave this distinction? You could say "well the notes of a key areplayed together but not at the same time," but that also is true of anarpeggio-ed chord; again what's the distinction? Even if you say "a Cmajor chord is C-E-G" there is no motivation as to how it is thatC-E-G sound good together and other combinations of notes do not.This "music theory" reminds me a bit of Richard Feynman'sdescription of a science textbook he reviewed for the Californiaschool board as told in '"Surely You're Joking, Mr. Feynman!":Adventures of a Curious Character', [Feynman1985,p. 270-271], (emphasis in the original):For example, there was a book that started out with four pictures:first there was a wind-up toy; then there was an automobile then therewas a boy riding a bicycle; then there was something else. Andunderneath each picture it said, "What makes it go?"I thought, "I know what it is: They're going to talk aboutmechanics, how the springs work inside the toy; about chemistry, howthe engine of the automobile works; and biology, about how the muscleswork."It was the kind of thing my father would have talked about: "Whatmakes it go? Everything goes because the sun is shining." And thenwe would have fun discussing it:"No, the toy goes because the spring is wound up," I would say."How did the spring get wound up?" he would ask."I wound it up.""And how did you get moving?""From eating.""And food grows only because the sun is shining. So it's because thesun is shining that all these things are moving." That would get theconcept across that motion is simply the transformation of thesun's power.I turned the page. The answer was, for the wind-up toy, "Energymakes it go." And for the boy on the bicycle, "Energy makes it go."For everything, "Energy makes it go."Now that doesn't mean anything. Suppose it's "Wakalixes."That's the general principle: "Wakalixes makes it go." There's noknowledge coming in. The child doesn't learn anything; it's just aword!1.2 What is a Satisfactory, Scientific Theory?Further, a scientific theory of something is expected to have acertain "explanatory power". But what is "explanatory power"? Is itjust whatever we like? Consider the old explanations of disease; hereis one: evil spirits inhabit you [dem].Well, did anyone ever see these spirits? Were theexperiences of these spirits universal across human kind? Where theresome general rules of how the spirits behaved? How many there were?What would appease them?Another theory was Humorism [hum]: that there werefour different fluids in the body: blood, black bile, yellow bile, andphlegm; when they got out of balance, you had a disease. Ok, this isbetter than arbitrary spirits, but did anyone measure the relativelevels of these fluids? Could someone predict sickness by observingthese fluids get out of balance? Could you make someone better by,say, draining blood from them? "Treatment" based on this theory seemto have been long practiced, but did anyone measure to see ifdraining blood really made people better versus a controlgroup that did not have their blood drained?Now we have an new theory called modern medicine. It is much morecomplex, but let's take a subset of it: there are little creaturescalled bacteria that live everywhere. Certain kinds can live in yourbody and the results of their activity, such as their excretions, getyour body out of normal working order, and thus you become sick. Ifyou give chemicals to a person that are more toxic to the bacteriathan the person, does the person get better? Yes [Mobley-antibiotics] ! Even when compared to a control group?Yes! Can we see these little bacteria in a microscope? Yes!Ok, this is much more satisfactory as a scientific theory.Now, let us step back and consider what makes us more satisfiedwith this theory. What is going on such that it is a better theory? For one thing, the theory is mechanical: we have somemechanism, consistent with our understanding of inanimate matter today(physics and chemistry) such that the operation of the mechanismcorresponds with what we observe (Scientific Method) [sci]. Further, this mechanism is deterministic andprecise: there isn't much arbitrariness in the mechanism: wecan compute rather well how sick someone will get and how much toxinwe have to give them to kill the bacteria and not the person. This mechanism is universal: there is no appeal tobeliefs or cultural norms: people throughout the world get sick in thesame way and the medicines work on them, with but small differencesthat can be further explained by another mechanism called genetics. This mechanical explanation is simple andminimal (Occam's razor) [occ]. We cansee the parts working. Lastly, the mechanism is factored -- made up ofindependent parts -- and the complexity of the observed phenomena isemergent -- arising naturally from the operation of theparts. That is, these parts of the explanation of disease all operateindependently: (1) how the body works such that the bacterialexcretions disrupt it, (2) how bacteria works such that the toxinkills it, (3) how the toxicity to the human depends on the size of thehuman, etc.Physicist Richard Feynman gave a series of lectures where heattempted to encapsulate the basic nature of how science is done andthe kind of results it produces; these were published as "TheCharacter of Physical Law" [Feynman1965]. Here is a brilliant paragraph on howto know when you have finally found the truth. [Feynman1965, p. 171] (underlining added, not inthe original):One of the most important things in this 'guess -- computeconsequences -- compare with experiment' business is to know when youare right. It is possible to know when you are right way ahead ofchecking all the consequences. You can recognizetruth by its beauty and simplicity. It is always easy when youhave made a guess, and done two or three little calculations to makesure that it is not obviously wrong, to know that it is right. Whenyou get it right, it is obvious that it is right -- at least if youhave any experience -- because usually what happens is that more comes out than goes in. Your guess is, infact, that something is very simple. If you cannot see immediatelythat it is wrong, and it is simpler than it was before, then it isright. The inexperienced, and crackpots, and people like that, makeguesses that are simple, but you can immediately see that they arewrong, so that does not count. Others, the inexperienced students,make guesses that are very complicated and it sort of looks as if it isall right, but I know it is not true because the truth always turns outto be simpler than you thought.Using Computer Science terminology, I summarize Feynman's point asfollows.The more factored a theoryand the more emergent the observed phenomenafrom the theory, the more satisfying the theory.The Ptolemaic [ptol] modelof the solar system puts the earth at the center. This explanationreally does explain the movements, especially when epicycles[epi] are added,but it is rather complex and ad hoc: how does it emerge that we needepicycles? The Copernican [cop]system is also another explanation of the solar system that puts thesun at the center. This second explanation only requires Newton's lawsof motion plus gravity. The consequences of Newton's laws arecomplex and even hard to simulate, even on a modern computer, but thelaws themselves are quite simple and independent andmechanical and factored and observable etc. Even further, thenotation used in this theory easily reflects the underlyingunderstanding in the theory: it allows for easy calculationswhen making predictions of the theory. All in all, the Copernicansystem is quite a quite satisfying explanation, or theory, of themotions of planets in the solar system because, not only does itexplain the observed phenomena, it is factored into simpleparts and the observed phenomena are emergent from theinteractions of those parts. Consequently, we use the Copernicansystem today (