the dissonance curve and applet

7/28/2019 The Dissonance Curve and Applet

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The Dissonance Curve and Applet

2 tones of moderate loudness, sounded simultaneously, are said tobe dissonantif the result is unpleasant to listen to, and consonantif

they blend well together, and are pleasing to listen to. The explanationof what causes dissonance (specifically sensory dissonance) is thebeats produced by a harmonic of the first tone, when it is near, butdoes not coincide in frequency with a harmonic of the second.However, if the beats are very slow or extremely fast, the effect is notso unpleasant. There is an interesting discussion of this and more in[Pierce 1983 p.76], including the human ear, and the unpleasantnessof dissonance resulting from 2 different sounds transmitting along thesame nerve fibres. There is also the confounding effect of musicallytrained individuals being sensitive to what intervals they expectto be

consonant or dissonant, based on past experience. This is omittedfrom the mathematical model below.

Let us now consider tones with no harmonics higher than the first, i.e.pure sine waves. Fix tone 1 at frequency f, and denote the frequencyof the other tone by g. As g varies upward from f, there is a region ofroughness or critical bandwidth where there is dissonance; thisdissonance dies down as g continues to increase. The length of thecritical bandwidth depends on the base frequency f, and Pierce givesthe rule of thumb that it extends to about the minor 3rd, that is 6/5 f.The strength of the dissonance sensation is a smooth bump function,reaching a single maximum at about 1/4 of the length of the criticalbandwidth.

What we now have is a weight function for the amount of dissonance

between 2 pure tones. [Sethares 1997] gives an excellent discussiondissonance curves, the experiments of Plomp in the 1960s, and


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provides optimal curve fitted parameters from the data of Plomp andLevelt, which we shall use. [Benson 2002] also provides an informativediscussion.

We can now give a mathematical expression for the dissonance weightfunction as:

d( f, g, Af, Ag ) = AfAg[e-0.84Q-e-1.38Q]

where Q =

( g - f)

--------------

0.021 f + 19

whereAfis the amplitude of f,Ag is the amplitude of g, and g >= f.

Then for standard musical tones with harmonics, we can compute thetotal dissonance by summing the dissonance of each pair ofharmonics. We represent the significant harmonics of tone 1 by thearrayF= ( f1, f2, f3, f4, f5, f6 ) where fi := i * f1and tone 2 by G= ( g1, g2, g3, g4, g5, g6 ) , with corresponding arrays

for the amplitudes of each individual harmonic.

We then loop over each element of array F, and for each elementof F we loop over array G, summing up the result of plugging fi andgj into d(f, g, Af, Ag), with the warning that fis always the minimumterm of {fi, gj}, and gis always the maximum. The following appletdraws the curve as g1 varies over the octave of f1.


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Dissonance Curve Applet

The relative intensity of the first 6 harmonics of a piano wire, struck at1/7 its length, are taken from [Helmholtz 1877 p.79], and they have theapproximate ratios:Plucked: (1.0 0.8 0.6 0.3 0.1 0.03)

Soft hammer: (1.0 1.9 1.1 0.2 0.0 0.05)

Medium hammer: (1.0 2.9 3.6 2.6 1.1 0.2 )

Hard hammer: (1.0 3.2 5.0 5.0 3.2 1.0 )

view the source code

If you save it to disk, change the .txt to .java before you try to javac it.

Discussion of Results for scales

The above applet's diagram tells us how well the frequencies blendwith the base frequency, which we always call C. The reader shouldsystematically experiment with increasing base frequencies and harderhammers ( the harder the hammer, the more prominence of the upperharmonics). A number of interesting phenomena emerge, and we shall

discuss some of them, while wondering which are artifacts of Plomp'smodel, or Sethares' parameters, or the author's own implementation.

Some general observations (by no means all):

1. The greatest consonances occur at both ends of the scale, andat G = 3/2 fin the middle. On either side of these are maxima ofdissonance.

2. At low frequencies, dissonance reigns with only a few shallowpeaks. At high frequencies ( say 2112 Hz ) the curve is in

general much lower. Going too high is probably physicallyunrealistic.

3. Soft hammers damp harmonics and bring out few minima; harderhammers bring out many more, especially at higher frequencies.

4. Minima at F, G, and A persist across the various frequencies andhammers, and they are the deepest. Others come and go.

5. It is probably not meaningful to be concerned about what is theexact lowest point in a wide valley, or to claim a minima locationto match a fraction if the distance is too great (say 5 cents?).This may explain away some theoretically unattractive minima
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like 9/7 and 12/7 for 264Hz, soft hammer, but experimentationreveals many other anomalies!

6. E = 5/4 and E-flat = 6/5 appear sometimes, but are fairly high indissonance. D = 9/8 rarely appears, except at high frequencies,and B = 15/8 never does.

Implications for scales

Let us take as our baseline middle C = 264Hz and the Mediumhammer. Any scale we derive in the primary octave would just betransposed to upper and lower octaves. Recall that the Just IntonationC scale has frequency ratios

C major: (C-D-E-F-G-A-B) = [ 1 9/8 5/4 4/3 3/2 5/3 15/8 ]

C minor: (C-D-Eb-F-G-Ab-Bb) = [ 1 9/8 6/5 4/3 3/2 8/5 9/5 ]

We would like to show that these choices are optimal in somefundamental sense. This will lead us into functional harmony, which isnot well understood by the author, but we'll go as far as we can. Wedefined the C major scale from the notes of the triad chords C, F,G based on harmonics. Here we ask what that means in terms of theamount of dissonance.

The C major triad is certainly very consonant with C, therefore it is inthe scale. The F chord is then a great choice because its gets both theother consonance heavyweights F and A. Why not use A major if A isso good? The problem is the almost maximum dissonance around C# (and noleading tone property to C).

How do we justify the G chord being part of the scale? G is excellentwith C alone, but B and D are terrible, they do not even appear as localminima; 3 octaves up at 2112Hz, we get a deep but wide minima near

D and a bit higher than B-flat ( but never B!). Perhaps this veryweakness can be construed as a strength, though. The notes of the Gchord serve two functions:

G, B and D fill in large interval gaps in (C-E-F-A-C) B has the leading tone property to C ( cf. [Helmholtz 1877 p.285

])

Therefore, we have found that the C major scale to contain the noteswith the following properties:

1. All the tones with maximal consonance against C are included.


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2. The notes form 3 distinct major triads.3. Intervals between notes, while different sizes, do not contain any

gap larger than 204 cents.4. The notes can form at least one dissonant chord G7C = (G-B-D-

F) which "resolves" to C major (or minor).

What we have not established is any reason why the interval sizeshould matter, except for the common sense reasons that theyshouldn't be extrememly small, so as to be readily distinguished. Wehave also not precisely defined the leading tone property, or explainedthe mechanics of how one chord resolves into another. Therefore wecan't say why there shouldn't be other notes also included in the scale.We may vaguely conjecture that the leading tone property issomehow the "frustrated expectation" of the listener to hear the tonic,

and instead hearing something close.

Idea: It would be interesting to see if one could construct amathematical function to indicate what chord ( if any) a given chordshould resolve to, perhaps a "dissonance metric" to give the shortestpath to a satisfactorily consonant chord. Also, by plotting thedissonance value of chords, could one study classical music in adynamical systems approach as trajectories winding around ondissonance surfaces, and interacting with minima, like a particle in apotential energy well?

We may reasonably postulate the following as another musical axiom,and justify it by the lack of general acceptance of 20th century atonalmusic:

Resolution Axiom: A critically important function of harmony in musicis the "tension" of dissonant harmony resolving into the "relaxation" ofconsonance, usually the tonic chord.

For the minor scales, we again have the dissonance minima at C, F, Gand A; we form the minor triads for C,F and G ( Am is already in Cmajor ). With the Fm we lose the strong consonance of A with C andinstead have Ab. Historically, people had trouble accepting a minorchord ending a composition, but it sounds perfectly acceptable today.This fact could play havoc with creating the above mentioned chordresolution function.

Developing a Theory of Chords from Harmonic Principles...


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[There is asummary of the resultsI found at the end of this page. If toomany things discussed here are unfamiliar, you might want to readthisintroduction first]. Suggestions for improvement are most welcome!

Let us say a consonant chordis a set of notes characterized by

1. some notes share one or more low order harmonics2. there are no strongly dissonant clashes between the other pairs

of low order harmonics.

We say low orderbecause high harmonics can be very dissonantagainst the fundamental frequency, but ordinarily those harmonics areso weak as to not be noticed.

We can form 4 types of consonant chords:

1. A root tone and other tones matching harmonics of the rootby octave equivalence ( Like C major, shown below).

2. All tones sharing a common harmonic (Like C minor).3. Notes matching harmonics notby octave equivalence (like F = c-

f-a).4. Matching some harmonics by octave equivalence and some

not. (This also characterizes a minor triad!)

There seem to be two types of chords in practice: those which arebased on harmonic principles (e.g. C major and C minor) and thosewhich are based on chosing a pattern of notes on the keyboard (e.g.C-Augmented or C-Diminished-6). Here is alist of C chords, drawn incircle of 5ths diagrams and a catalog of some of their properties. In thepresent discussion, a companion to the one just linked to, the goal is tostudy what chords are possible from simple acoustical principles withharmonics.

The figure that I conceived of below is useful for graphically showing

the relationships between the harmonics; I have fancifully called ittheHarmonic Tower:

The vertical axis is marked by the harmonics of a note of a fixedfrequency, call it c. The horizontal axis is marked by a listing of allfractions between 1 and 2 in terms of increasing denominator (luckilythey can quickly stop being listed because they become irrelevant,either by only matching too high a harmonic of c, or by being toodissonant against the 1st harmonic of c, the fundamental). It makessense to list the fractions along the horizontal axis like this becausethey correspond to a string vibrating as a whole (1), as halves (3/2), as
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a segment of one third its length (4/3),and a two thirds segment (5/3),and so forth. This is the way that harmonic bodies vibrate!

The green strip, of width about a minor 3rd, roughly marks the so-calledcritical bandwidth . (If you follow the link, you will see the graphof a bump function representing dissonance). Imagine sliding the stripup and down to compare the harmonics of 2 notes, lining its bottomedge up with one of the harmonic horizontal tic marks in the column ofthe note that you are interested in. If the harmonic marking in another

column lies inside the strip (as opposed to on the edge), then the twonotes will produce unpleasant beats when sounded together and thusbe dissonant.

Remember that consonant chords are characterized bymatching some harmonics. As we shall see, some chords match onsome harmonics very nicely, but almostmatch on others, which meansthese near misses lie in the critial bandwidth and the pleasing quality ofthe matches is nullified by the unpleasantness of the clashes; thus thechord cannot be called consonant.
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Look at the notes of C major (c-g-e) and notice how they matchthe 3rd and 5th harmonics of c by octave equivalence (Yellowboxes on diagram).

Look at the notes of C minor (c-g-eb) all having a match on thesame line as the 6th harmonic of c.

Observe the triad C-sus (c-g-f), and how it is poised to be awonderfully strong consonant chord (the harmonic matching isso low) but this is offset by its higher dissonance (due to theclash of the 1st harmonics of g and f).

Let us try to identify all the combinations and see what chords we get:We require that the note c always be present, thus we always have theharmonic 1. We extended the list of harmonics all the way up to the9th, so that we could get bb = 9/5 and d= 9/8. Note that we are

missing b= 15/8 , because it is just too implausible to go that high andmatch the 15th harmonic of c with the 8th harmonic of b (highharmonics are typically inaudible).

Indexes Matching Harmonics Note Names Chord Name Fractions Dissonance

(0,1) (1, 3* ) c - g 5th 1 3/2 0.1595

(0,2) (1, 4 ) c - f 4th 1 4/3 0.3669

(0,3) (1, 5 ) c - a 6th 1 5/3 0.2348

(0,4) (1, 5* ) c - e major 3rd 1 5/4 0.4796

(0,5) (1, 7* ) c - 7/4 1 7/4 0.3956

(0,6) (1, 6 ) c - e minor 3rd 1 6/5 0.4798

(0,7) (1, 7 ) c - 7/5 1 7/5 0.5524

(0,8) (1, 8 ) c - a minor 6th 1 8/5 0.5212

(0,9) (1, 9 ) c - b 1 9/5 0.4037

(0,10) (1, 7 ) c - 7/6 1 7/6 0.6648

(0,11) (1, 8 ) c - 8/7 1 8/7 0.6813

(0,12) (1, 9 ) c - 9/7 1 9/7 0.6249

(0,13) (1, 9* ) c - d 1 9/8 0.7065

Duplicates are omitted (for example (1,3) and (1,6) are both c-g).Asterisks symbolize harmonics that are matched by an octaveequivalent note and are thus more strongly matched. In the theoryespoused by Terhardt(see References )when 2 or more notes aresounded together there are two (sometimes competing) components ofharmony:

The amount of sensory dissonance from beating harmonics
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The amount of identification of a root tone, via the auditoryphenomena of virtual pitch

If the harmonics match by octave equivalence, then root identificationis enhanced. To calculate dissonance in the last column of each table,we used the Helmholtz-Plomp model with parameters from Sethares'book(see References ). Getting dissonance values of chords to comeout so that they agree with the experience of musicians is a trickybusiness, so what we have here is just one possibility. The calculateddissonance value is very sensitive to how you weight the harmonics.We only summed the dissonance of only the first 6 harmonics (withequal weights), even though we matched harmonics in chords up tothe 9th. Otherwise nice chords like C major came out way toodissonant! Also we measured the dissonance of a chord as the

dissonance of the 2 worst notes, not summing all possible dissonancecombinations. The source code to calculate these tables is availableafter the last table.

IndexesMatching

HarmonicsNote

NamesChordName

Fractions Dissonance

(0,1,2) (1, 3*, 4) c - f - g C sus 1 4/3 3/2 0.5637

(0,1,3) (1, 3*, 5) c - g - a 1 3/2 5/3 0.5535

(0,1,4) (1, 3*, 5*) c - e - g C major 1 5/4 3/2 0.4796

(0,1,5) (1, 3*, 7*) c - g - 7/4 1 3/2 7/4 0.5020

(0,1,6) (1, 3*, 6) c - e - g C minor 1 6/5 3/2 0.4798

(0,1,7) (1, 3*, 7) c - 7/5 - g 1 7/5 3/2 0.8344

(0,1,8) (1, 3*, 8) c - g - ab 1 3/2 8/5 0.8535

(0,1,9) (1, 3*, 9) c - g - b 1 3/2 9/5 0.4037

(0,1,10) (1, 3*, 7) c - 7/6 - g 1 7/6 3/2 0.6648

(0,1,11) (1, 3*, 8) c - 8/7 - g 1 8/7 3/2 0.6813

(0,1,12) (1, 3*, 9) c - 9/7 - g 1 9/7 3/2 0.6249

(0,1,13) (1, 3*, 9*) c - d - g 1 9/8 3/2 0.7065

(0,2,3) (1, 4, 5) c - f - a F major 1 4/3 5/3 0.3904

(0,2,4) (1, 4, 5*) c - e - f 1 5/4 4/3 0.9091

(0,2,5) (1, 4, 7*) c - f - 7/4 1 4/3 7/4 0.4629

(0,2,6) (1, 4, 6) c - e - f 1 6/5 4/3 0.6535

(0,2,7) (1, 4, 7) c - f - 7/5 1 4/3 7/5 1.0148

(0,2,8) (1, 4, 8) c - f - ab F minor 1 4/3 8/5 0.5212

(0,2,9) (1, 4, 9) c - f - b 1 4/3 9/5 0.4037

(0,2,10) (1, 4, 7) c - 7/6 - f 1 7/6 4/3 0.6648

(0,2,11) (1, 4, 8) c - 8/7 - f 1 8/7 4/3 0.6813
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(0,2,12) (1, 4, 9) c - 9/7 - f 1 9/7 4/3 1.0657

(0,2,13) (1, 4, 9*) c - d - f 1 9/8 4/3 0.7065

(0,3,4) (1, 5, 5*) c - e - a A minor 1 5/4 5/3 0.4796

(0,3,5) (1, 5, 7*) c - a - 7/4 1 5/3 7/4 0.9767

(0,3,6) (1, 5, 6) c - e - a A dim 1 6/5 5/3 0.4898

(0,3,7) (1, 5, 7) c - 7/5 - a 1 7/5 5/3 0.5524

(0,3,8) (1, 5, 8) c - a - a 1 8/5 5/3 1.0406

(0,3,9) (1, 5, 9) c - a - b 1 5/3 9/5 0.7025

(0,3,10) (1, 5, 7) c - 7/6 - a 1 7/6 5/3 0.6648

(0,3,11) (1, 5, 8) c - 8/7 - a 1 8/7 5/3 0.6813

(0,3,12) (1, 5, 9) c - 9/7 - a 1 9/7 5/3 0.6249

(0,3,13) (1, 5, 9*) c - d - a 1 9/8 5/3 0.7065

(0,4,5) (1, 5*, 7*) c - e - 7/4 1 5/4 7/4 0.4855

(0,4,6) (1, 5*, 6) c - e - e 1 6/5 5/4 1.0635

(0,4,7) (1, 5*, 7) c - e - 7/5 1 5/4 7/5 0.6054

(0,4,8) (1, 5*, 8) c - e - ab 1 5/4 8/5 0.5643

(0,4,9) (1, 5*, 9) c - e - bb 1 5/4 9/5 0.5082

(0,4,10) (1, 5*, 7) c - 7/6 - e 1 7/6 5/4 0.8964

(0,4,11) (1, 5*, 8) c - 8/7 - e 1 8/7 5/4 0.7583

(0,4,12) (1, 5*, 9) c - e - 9/7 1 5/4 9/7 1.0359

(0,4,13) (1, 5*, 9*) c - d - e 1 9/8 5/4 0.7065

(0,5,6) (1, 7*, 6) c - e - 7/4 1 6/5 7/4 0.4997

(0,5,7) (1, 7*, 7) c - 7/5 - 7/4 1 7/5 7/4 0.5524

(0,5,8) (1, 7*, 8) c - ab - 7/4 1 8/5 7/4 0.6163

(0,5,9) (1, 7*, 9) c - 7/4 - b 1 7/4 9/5 1.0571

(0,5,10) (1, 7*, 7) c - 7/6 - 7/4 1 7/6 7/4 0.6648

(0,5,11) (1, 7*, 8) c - 8/7 - 7/4 1 8/7 7/4 0.6813

(0,5,12) (1, 7*, 9) c - 9/7 - 7/4 1 9/7 7/4 0.6249

(0,5,13) (1, 7*, 9*) c - d - 7/4 1 9/8 7/4 0.7065(0,6,7) (1, 6, 7) c - e - 7/5 1 6/5 7/5 0.5845

(0,6,8) (1, 6, 8) c - eb - ab Ab major 1 6/5 8/5 0.5212

(0,6,9) (1, 6, 9) c - eb - bb 1 6/5 9/5 0.4798

(0,6,10) (1, 6, 7) c - 7/6 - e 1 7/6 6/5 1.0305

(0,6,11) (1, 6, 8) c - 8/7 - e 1 8/7 6/5 1.0401

(0,6,12) (1, 6, 9) c - eb - 9/7 1 6/5 9/7 0.8866

(0,6,13) (1, 6, 9*) c - d - e 1 9/8 6/5 0.9421

(0,7,8) (1, 7, 8) c - 7/5 - a 1 7/5 8/5 0.5524(0,7,9) (1, 7, 9) c - 7/5 - b 1 7/5 9/5 0.5524


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(0,7,10) (1, 7, 7) c - 7/6 - 7/5 1 7/6 7/5 0.6648

(0,7,11) (1, 7, 8) c - 8/7 - 7/5 1 8/7 7/5 0.6813

(0,7,12) (1, 7, 9) c - 9/7 - 7/5 1 9/7 7/5 0.7362

(0,7,13) (1, 7, 9*) c - d - 7/5 1 9/8 7/5 0.7065

(0,8,9) (1, 8, 9) c - a - b 1 8/5 9/5 0.5212

(0,8,10) (1, 8, 7) c - 7/6 - ab 1 7/6 8/5 0.6648

(0,8,11) (1, 8, 8) c - 8/7 - a 1 8/7 8/5 0.6813

(0,8,12) (1, 8, 9) c - 9/7 - a 1 9/7 8/5 0.6249

(0,8,13) (1, 8, 9*) c - d - ab 1 9/8 8/5 0.7065

(0,9,10) (1, 9, 7) c - 7/6 - b 1 7/6 9/5 0.6648

(0,9,11) (1, 9, 8) c - 8/7 - b 1 8/7 9/5 0.6813

(0,9,12) (1, 9, 9) c - 9/7 - b 1 9/7 9/5 0.6249

(0,9,13) (1, 9, 9*) c - d - bb 1 9/8 9/5 0.7065

(0,10,11) (1, 7, 8) c - 8/7 - 7/6 1 8/7 7/6 0.9257

(0,10,12) (1, 7, 9) c - 7/6 - 9/7 1 7/6 9/7 0.7055

(0,10,13) (1, 7, 9*) c - d - 7/6 1 9/8 7/6 1.0679

(0,11,12) (1, 8, 9) c - 8/7 - 9/7 1 8/7 9/7 0.6813

(0,11,13) (1, 8, 9*) c - d - 8/7 1 9/8 8/7 0.8079

(0,12,13) (1, 9, 9*) c - d - 9/7 1 9/8 9/7 0.7065

n = 78

Some observations and summary of results and open questions:

1. We certainly get some expected chords C and Cm, F and Fm,and Am.

2. We notably don't get G, because we don't get the note b (itsharmonic relationship to c is just too distant).

3. Curiously, A-dim = c-eb-a shows up and with quite a lowdissonance value.

4. Is C-aug = c-e-ab? If so, it is in the list, and we have constructedthe augmented and diminished chords from harmonic principles!But is C-aug = c-e-g# really?

5. We also get Ab major, which is a little suprising.6. Evidently, good consonant chords can be made using 7/4 such

as c-f-7/4 and c-eb-7/4. This may have implications for what I callthepropagation problem below.

7. The chord F major = c-f-a is made with no octave equivalencematching...to c. It certainly does match f's harmonics by octaveequivalence. Is that always the case for triads made by all non-

O.E. matching? In other words, is what I called "case 3" at thebeginning of this page actually not a case at all?


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8. I am probabaly not worrying enough about inversions like c-e-g,e-g-c, g-c-e not having the same dissonance

9. The dissonance values are still not "right", since for example Fminor = c-f-ab has a higher dissonance than c-f-bb on the linebelow it.

10. The "Propagation Problem": Western music is based onthe major chord. What other chords could form scales in a similarmanner and possibly allow similiar harmonic functions, likecadences? This would first mean solving what I callthepropagation problem: How to start from the C major triad andgenerate the C major scale and the other 11 standard keys insome kind of a way that is mathematical and optimal. I attempt todo thishere, but it is not complete.

Theorem If you create a scale, say the C major scale, by major chordslinked at the 3rd harmonic, like F - C - G, then you automatically havecreated minor chords as well. Likewise, 3 linked minor chords createmajor chords.

4-Chords

Here is a listing of the lowest dissonance 4 note chords:

IndexesMatching

HarmonicsNote Names

Chord

NameFractions Dissonance

(0,1,2,3) (1, 3*, 4, 5) c - f - g - a F61 - 4/3 - 3/2 -

5/30.5637

(0,1,2,5) (1, 3*, 4, 7*) c - f - g - 7/41 - 4/3 - 3/2 -

7/40.5637

(0,1,2,9) (1, 3*, 4, 9) c - f - g - bb C7 sus1 - 4/3 - 3/2 -

9/50.5637

(0,1,3,4) (1, 3*, 5, 5*) c - e - g - a C61 - 5/4 - 3/2 -

5/30.5535

(0,1,3,6) (1, 3*, 5, 6) c - eb - g - a Cm61 - 6/5 - 3/2 -

5/30.5535

(0,1,4,5) (1, 3*, 5*, 7*) c - e - g - 7/4 Tetrad1 - 5/4 - 3/2 -

7/40.5020

(0,1,4,9) (1, 3*, 5*, 9) c - e - g - bb C7F1 - 5/4 - 3/2 -

9/50.5082

(0,1,5,6) (1, 3*, 7*, 6) c - eb - g - 7/41 - 6/5 - 3/2 -

7/40.5020

(0,1,6,9) (1, 3*, 6, 9) c - eb - g - bb Cm71 - 6/5 - 3/2 -

9/50.4798

(0,2,8,9) (1, 4, 8, 9) c - f - ab - bb 1 - 4/3 - 8/5 -9/5

0.5212
http://jjensen.org/musicTheory.htmlhttp://jjensen.org/musicTheory.htmlhttp://jjensen.org/musicTheory.htmlhttp://jjensen.org/musicTheory.html


13/13

(0,3,6,7) (1, 5, 6, 7) c - eb - 7/5 - a1 - 6/5 - 7/5 -

5/30.5845

(0,4,8,9) (1, 5*, 8, 9) c - e - ab - bb1 - 5/4 - 8/5 -

9/50.5643

(0,5,6,7) (1, 7*, 6, 7) c - e - 7/5 -7/4

1 - 6/5 - 7/5 -7/4

0.5845

(0,6,7,8) (1, 6, 7, 8)c - e - 7/5 -

ab

1 - 6/5 - 7/5 -

8/50.5845

(0,6,7,9) (1, 6, 7, 9)c - e - 7/5 -

bb

1 - 6/5 - 7/5 -

9/50.5845

(0,6,8,9) (1, 6, 8, 9) c - eb - ab - bb1 - 6/5 - 8/5 -

9/50.5212

(0,7,8,9) (1, 7, 8, 9)c - 7/5 - a -

bb

1 - 7/5 - 8/5 -

9/50.5524

1. The table for the 4 note combinations is so big (286 rows) that itis on its own page:4 note chords table.

2. One chord which is notably missing is C-maj-7 = c-e-g-band thisis again because we don't generate the note b with our schemeof low order harmonics.
http://jjensen.org/4-chords.htmlhttp://jjensen.org/4-chords.htmlhttp://jjensen.org/4-chords.htmlhttp://jjensen.org/4-chords.html

the dissonance curve and applet

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