Rethinking Dissonance (A Theory of Harmonic Symmetry) 

Here's my 'pet' theory: The notion of 'comparing the overtone with the fundamental' does not take into account several other mathematical and psychoacoustical factors that determine perceived assonance (mathematical consonance between the frequencies). Thus, it cannot fully explain the contradictions mentioned in the video. Here I will explain how digging deeper into number theory and psychoacoustics can give a better foundation for which cultural and personal perceptions of consonance and dissonance can be explained. PART 1: Tartini tones explaining dissonance in context of key center ________________________________________ The psychoacoustic phenomenon of the Tartini-tone (sum and difference tone) is able to explain dissonance taking into account the key centre, and why music across cultures have developed the concept of key centre in one form or another in the first place. When any 2 notes are played, for example, E4 (330hz) and A4 (440hz), there exists (at least) two more psychoacoustically perceived tones. The most prominent being the difference tone, which is the difference between the two frequencies 330hz (E4) and 440hz (A4). That is 110hz (A2). The second most prominent is the sum tone, which is the sum of the two frequencies 330 + 440, which is 770hz (G5 -31.17 cents). However, this tone is rarely ever audible in practice unless the timbre of the tones and conditions are just right. As to explain the dissonance of the above E-A dyad in the context of the key of E, there is no doubt that the function of A is non-tonic, and thus unresolved, However, to explain the dissonance in the context of the key of A, the presence of the psychoacoustic A2 (110hz) makes the dyad feel lacking in that its 'root note' of E4 does not agree with the 'psychoacoustic root note' of A4. In fact, this metric of 'root note agreement' goes further to explain consonances in chords with more than 2 notes. For chords with more than 2 notes, there will be a sum and difference tone created between each of the intervals between each of the note in the chord, however, the combination of the 3 notes will cause an ultimate 'bass' difference tone to occur: A major: A4 (440hz) C#5 (550hz) E5 (660hz) 1. simplify 440:550:660 into 4:5:6 (the simplified ratio represents the nth harmonic starting on any arbitrary note) 2. the 'bass' tone is the fundamental of that arbitrary note. It has the frequency ratio of 1 unit with respect to the 4 units that represents 440hz 3. therefore, the bass tone is 110hz, which is the note A2, which agrees with the root of the A major triad A minor: A4 (440) C5(528) E5(660) 1. simplify 440:528:660 into 10:12:15 (A4 is now the 10th harmonic, C5 is the 12th harmonic, E5 is the 15th harmonic) 2. the 'bass' tone is the fundamental tone. Now we have to find what the 1st harmonic is. 3. If 440hz is the 10th harmonic, 44hz is the fundamental frequency. Therefore, the 'bass' note is F1 (44hz). This does not agree with the root of the A minor triad, therefore explaining why the minor chord is more dissonant than the major chord even though both chords have the exact same interval classes and exact same prime limit (5-limit). This same phenomenon of the tartini tones can also be used to explain how certain chord voicings of the exact same chords will sound more consonant / dissonant than others: by comparing the resultant psychoacoustic tones with the physically sounded tones. It also makes it clear why the augmented and diminished triads are so much more dissonant than the minor triad, or major seventh tetrad. This phenomenon can be made obvious either by the use of distortion (which mimics the non-linearity of the cochlea's sensing of amplitude) or by simply putting your ear close to the sound source. The tartini tones become more prominent as the notes get louder, higher, and purer (like two piccolos playing). Then, one simply needs to listen out for the tartini tones to check if the chord and chord voicing/inversion agrees with the tonality and context in which it is used, and the math above is unnecessary. However, even without doing this, all the above processes are already subconsciously being done by our ears, hence it is how we can even perceive an augmented triad as more dissonant than a minor triad in the first place. PART 2: Deriving the absolute complexity of intervals without any context _______________________________________ Here are two commonly used metrics that objectively define the assonance (mathematical consonance) of an interval between any two notes, without regard for key centre or context: en.xen.wiki/w/Tenney_height en.xen.wiki/w/Benedetti_height I will not go into detail but both metrics rely on the use of the just intonation realisation of intervals and uses the product of the numerator and denominator of the most simplified form of the ratio of the frequencies between those two notes. For example, the perfect fifth is given by the ratio of 3/2 (as in 330hz (E4) & 220hz (A3)), its Benedetti height is 3 x 2 = 6. The major third is given by the ratio of 5/4 (as in 550hz (C#5) & 440hz (A4)), its Benedetti height is 5 x 4 = 20 This method may break down in extensive practice as there are numerous chords and intervals in 12-equal western music that have several different possible just intonation ratios that realise the same chords. PART 3: Deriving perceived complexity of intervals without any context ___________ This is an extremely mathy read, however, it is one of the most precise and practical means of measuring the perceived complexity of the interval between any two notes. Unlike the Benedetti/Tenney height metrics, this method does not depend on having a rational just intonation fraction to represent the interval, and thus it can be used to evaluate the assonance of intervals in any known tuning system. en.xen.wiki/w/Harmonic_Entropy Of course, it is not as simple as a magic formula and there are a few constants to play around with. I will not describe the process as it is too complex to fit it here, however, if anything, simply pay attention to the graphs here: en.xen.wiki/w/Harmonic_Entropy#s.3D17.2C_N.3C10000.2C_sqrt.28n.2Ad.29_weights , which plots the dissonance of the intervals against the interval measured in cents. PART 4: Conclusion Congratulations anyone who has made it this far. There is, in fact, a whole team of nerds, enthusiasts, researchers and professors, who dedicate their lives to the cause of answering the unanswerable truths in the intersection of music, acoustic science, mathematics, history, and biology. Most of us are coincidentally also in the circle of microtonalists, or “Xenharmonists”. Sadly, I don’t think there are many who are good at, or have the time to condense any of this information to fit into a youtube video that appeals to the general music enthusiast, and a youtube comment barely scratching the surface is the best I can do as a hobbyist in this field. If you or any creator out there have made it this far and find this subject appealing enough to you, I’m sure a lot of us would be glad to know that the forefront of music research is being shared with the public. A good place to start would be checking out en.xen.wiki/w/MicrotonalTheory and also, there are numerous communities online that would go on forever about this extreme form of music theory, such as the Xenharmonic Alliance on Facebook, or r/microtonal, amongst many others.

I like Hermann Helmholtz theory, that dissonance arises from lots of overtones in close approximity. So that this closeness leads to a lot of beating due to phase shift in your cochlea, where this beating leads to high energy use of the nerve cellse which is subjectively felt as unpleasentness.

I think that the more frequently an interval is used, the more the ear becomes used to, and that's what we mean by consonance. Less frequently used intervals sound strange and out of place, and that's what makes them dissonant.

I was just planning on working on a composition that explores dissonance, this upload came at the perfect time. Thanks!

This channel is criminally under-subscribed

In physics, the arrow of time is not possible without symmetry breaking. By analogy, perhaps much symmetry in music creates the impression of temporal directionlessness, since a symmetry per se gives no local sense of which part of the structure will break, thus flow, forward.

I really like dissonant sounding things. I think of it as s treat when I hear it applied to popular music, due to how uncommon it is.

liked because i had the same shower thought and you just verbalized it as a side note I find that multiple causation is the most compelling answer to questions like this. Placement in the harmonic series, simplicity of mathematical ratio, and acculturation are all determining factors for how dissonant something sounds, but they work in tandem with each other, moving us one way or the other on the spectrum of dissonance, rather than only one of them being the determining factor to the exclusion of the rest. Like you said, if we were to crop out the rest of the influencing factors that create the emergent aesthetic of dissonance, we would be talking about something other than dissonance.

Thanks. This was great. My teacher have directed me to study in its several depth, symmetry in music. Reason is the 12 tone is itself a symmetrical division of the octave therefore is itself a symmetrical system having within itself an endless (?) amount of subsystems derived that perfectly serve to extend tonal functions or and contrapuntal use or just being representative of any certain identification system.

Thank you so much for your insightful videos man. Really love the content you make!

I love this question. I’m working on a course on reading and writing chord symbols. The target audience is my piano students who want to learn to play from lead sheets. As I work on it I find myself asking your question and several related questions. From the cultural POV, I wonder how much of what we consider ‘needing resolution’ is learned from the musical narratives we spend the most time with. And does that override the laws of physics, Captain? And how do we learn to embrace as consonant what, in another context, is dissonant? Satie taught us to sit with major sevenths. Repetition legitimizes, as Adam Nealy loves to say. When I got to sevenths in my outline I was reminded that one reason major sevenths can sound stable and pretty is because it’s two overlapping perfect fifths; a pair of very stable consonant intervals. But it’s also two major thirds (which, since seeing your video, are perhaps less stable than I’d thought). And those two major thirds are stabilized in their connection to perfect fifths. I’m not sure symmetry is the right word. Bipolar? A Gmaj7 contains the major third from both a G and D major triad. So it’s both I and V. Does hearing major sevenths a la Satie in so many contexts allow us to hear a raw major seventh, G and F#, as softer than any one born pre-Satie could have heard? A dissonance can be tamed by the presence of one or more pitches with which it’s consonant. That’s true of complex chords with multiple extensions and alterations. And it can also create tonic ambiguity. Going further, the narrative expectation of resolution of dissonance can be subverted. Wagner and Ravel come to mind; dissonance ‘resolves’ to new dissonance (as you’ve described elsewhere). How the harmonies are rolled out plays a huge role in steering our perception and expectations. And, of course, jazz is made of this stuff. It’s fascinating to me. Thanks again for an engaging survey of a really interesting topic. Cheers!

To comment on your tangent about unison: on paper, unison is not an interval. However, in practical terms, if you tell the second violins to all play a whole note of C5, regardless of how good they are there will be subtle differences in the notes they’re playing, tenths or hundreds of a hertz. Not enough to sound like different notes harmonizing, but enough to create shifting accentuations and diminishes of particular harmonics. And then there’s the instruments themselves; which string are they playing it on, how old are the strings, the age of the violins, differences in the wood, construction, design, how fast the sweep the bow, all add up to subtle differences in timbre. So I’d say unison is an interval so minute it leaves the realm of harmony and instead expresses a harmonic quality created by the small but numerous differences between two instruments (or different tone generators within one instrument).

Thanks !! Always fascinating and informative !!

In the musical example of I-6/4 - V7 - I at 9:06, the alto should descend to the tonic in the final chord instead of rising to the mediant.

I enjoyed your video. I"m a composer and classical pianist, with a strong interest in psychoacoustics and music psychology. Like many, at one time I went down the numerological harmonic dualism rabbit hole of trying to find a mathematical/acoustic explanation for the 'sadness' of the minor triad! And that's strongly supported by JI and alternate tuning folks, who believe in the idea of number ratios, even though many intervals have multiple number ratios ("Will the real ratio stand up"?). Then I sort of flipped sides, advisedly I believe, by reading research by such as Richard Parncutt and Norman Cook, who have more developed theories about consonance and dissonance of triads, and skepticism that our hearing system has a "number ratio detection device." Your ideas on consonance/dissonance might be stimulated by exposure to their research. In discussing symmetrical chords, you never mentioned the premiere aspect of augmented and diminished chords, which is that they have no clear root. They are very weak in the psychoacoustic concept of root support, aka research by Terhardt and Parncutt. Their theory on the dissonance of symmetric chords is along the lines of gestalt psychology, 'ambiguity', a higher cognitive aspect of the hearing system, where there is not a clear root, and also no clear distinction between the intervals, since they are the same size. It is assumed that this 'dissonance of ambiguity' derives from another concept in psychoacoustics, of pattern recognition, from our hearing system being conditioned by hearing the lower members of the harmonic series during the evolution of our hearing system. (Not the higher members of the harmonic series, but basically positions 1-6, covering the triad intervals P4, M3, and m3.). So a theory from their quarter is that our hearing system is conditioned by hearing chords whose intervals are asymmetric: P4-M3 (first inversion major triad), M3-m3, m3-P4, and their mirror inversions giving the minor triads M3-P4, and m3-M3). This is not harmonic dualism. In the theory of pattern recognition from hearing the harmonic series, it focuses on the chords' shape, which can easily be viewed in either direction...also evidenced in tonal fusion (Stumpt, 1897), there is a consonance from hearing asyymtric intervals, and adjacent intervals which are one semitone apart, as they appear in the harmony series (ie, P8, P5, M3-m3, which are semitone sized 12, 7,5,4 and 3.) The explanation for the dissonance of symmetric chords is more likely along these lines (gestalt ambiguity) than the symmetry breaking you mention. (I will list citations at the end). So the pattern recognition from our exposure to the harmonic series is really only the lower portion of it, such as up till the 6th harmonic. Most music making is dealing with intervals, and not listeners' exploring the harmonic series as microtonalists and spectra lists might, whose upper intervals are basically inaudible. (Also a small point in an otherwise excellent video, when you mention at 7:38 that tritones expand out or in by semitones, in the case of resolution to a minor triad, it is one whole step, and one half step.) What I find most interesting about all of this is that the equally- spaced harmonic series appears in its for shortened form (intervals of semitone size 12-7-5-4-3, or C-C-G-C-e-g) because of the logarithmic perception of frequency, where it is the ratio of frequencies, and not the actual frequencies, which we hear. Thus 2:1 is larger than 3:2. 3:2 is larger than 4:3, and so on, since they are ever-smaller ratios! Even if you're annoyed by this, or don't agree with this, you would certainly enjoy reading these articles, which are crucial research for the topic which you may not have come across: slideplayer.com/slide/6042103/ (Provacative lecture by Richard Parncutt on this issue) www.researchgate.net/publication/251838218_Commentary_on_Cook_Fujisawa%27s_The_Psychophysics_of_Harmony_Perception_Harmony_is_a_Three-Tone_Phenomenon (Debate between Parncutt and Cook on why the augmented triad is felt to be more dissonance than diminished, even though they are both rootless symmetric chords.) They go into interesting follow up on the symmetric chord debate, which is basically: if the intervals of an augmented triad (M3-M3) are more consonant than those of a dim triad (m3-m3), which is it more dissonant to most listeners, even though they are both rootless symmetric chords? The likely conclusion is that because, the dim triad is an essential aspect of functional major minor diatonic harmony; it completely maps to the diatonic scale, and is the upper portion of the ubiquitous V7 chord, and is the vii and ii chord (in minor), whereas the augmented triad is non-diatonic.

Interesting topic. Great points for discussion!

I wonder if anyone has ever done any cross-cultural development studies focussing on children's preferences when it comes to dissonance.

Amazing video! I asked this on another video but could you consider doing a video on Arthur Bliss Frank Bridge or Arnold Bax?

Thanks for that.

8:07 There's also multiple ways to tune the fourth. If you tune V7 according to the harmonic series, the relationship between the 7th of the V chord and the root of the key (normally a perfect fourth) is a 21/16 ratio- very dissonant.

very interesting! right before you mentioned augmented fifths I was about to comment on how dissonant I find minor sixths. I once had a great professor who also believed in the dissonance of symmetric chords/sets/intervals. He would refer to Webern's symphony op. 21, for example.

For me its about harmonic correlation, its intrinsically linked in the sound design/instrument. Some scales work because the timbres have harmonics that match either statically in polyphony or melodically, over time. at the next expected amplitude peak in the waveform. The artist provides that expectation with composition and sound design. It very much depends on the listener too - an engineer tends listens analytically, their punters and sometimes musicians usually don't and have very clear expectations. ?.Consonance is what the mind expects to hear now and next from the harmonics it interprets additively (ear structure). This is why the mind can "get used to" the scales formed by unusual sound design with correlation in harmonic content and scales. ?.Dissonance is unexpected harmonic change statically or over time..

Very interesting

Well as you asked, i hear semitones, tones as dissonances and for some reason augmented 5th is dissonant to me but augmented 2nd or even tritone, isnt. 4ths are good good consonances

Wonderful, in this video I learned that I understand exactly nothing about music theory. But hey, conscious incompetence is still worlds better than unconscious incompetence. It’s only uphill from here!

I'm thinking Korn 1994 self titled album?

Thanks for making videos, why is the lowest note in a chord the most important one? Why do we hear the other notes in relation to the lowest?

Dissonance for me changes as I get used to some sounds in different contexts. Ex. C sharp, D, and D sharp sounds weird in a classical context, but makes more sense in a jazz context.

Cool video! As someone who's very interested in symmetry and worked with many microtonal systems using them, I think there's certainly some merit to the idea that symmetrical structures are "dissonant" - mainly because in symmetrical structures it's harder to orient one's self. Probably good scales for internalizing intervals are delicately balanced between symmetry and asymmetry - too much symmetry, and there's no sense of tonal gravity or hierarchy of voice leading - too much asymmetry and you can't chunk out any recognizable patterns that reduce the act of counting/overthinking numerically. Have you tried 5 equal divisions of the octave and 7 equal divisions of the octave? I think 5-EDO is remarkably concordant, and, even though it's xenharmonic to the 12-EDO pentatonic scale, most ears are generally ready to accept it. Easley Blackwood remarked that the interval that splits the perfect fourth in half is very useful. Sevish and I have talked about stacking intervals and about how stacking the perfect fourth divided in half (semi-fourth or semaphore temperament) seems to be a "gem" in stacking chains of microtonal intervals (the perfect fifth being another). Here's a composition of mine that uses 5-EDO to achieve a concordant sound (at 5:04 to 5:19). soundcloud.com/overtoneshock/fiat-circadia-10-edo

I’m a couple years late, and it’s probably a different wheelhouse from your usual listening, but in the last 10+ years (20 if you’re counting some early examples like Blut aus Nord) there’s been a huge wave of “dissonant black metal,” and “dissonant death metal.” I was curious if you had ever listened to any. After listening to so much of it, I also started finding the dissonance chords to be consonant. You start to feel like “dissonant” really isn’t clear enough, and you have to ask “what is this dissonant in comparison to?” Some good examples include Imperial Triumphant, Plebeian Grandstand, Dodecahedron, and Ad Nauseam. There’s a record label called Total Dissonance Worship where mostly all the bands heavily utilize dissonance in their music. I thought you might get a kick out of that.

For me personally, the only dissonance is the minor 2nd. Not even the minor ninth, because it fits better in certain chords and the spacing gives it a nice sound. But minor seconds are the only ones I consider outright dissonant

Love the videos, beautifully done and very informative! - Perhaps you could solve a music theory disagreement I had recently? - Having posted an article the tritone, and its place in western music concerning tension and release, e.g., harmonically, as in V7 going to I, or melodically, as in Bernstein's Maria, I was criticised for mentioning that the tritone was the only interval that is the same when inverted - #4 being an enharmonic equivalent to flat 5 - six semitones either way. The argument was that the tritone is different depending on temperament, and therefore not the same when inverted. My main point is that harmonic function is not effected by temperament - the gravitational pull of tendency tones might be increased or decreased by temperament, but not harmonic function - is this correct, or am I missing something? - Thanks for a great channel!

I don't think the unison, is an interval either. Great video, classical nerd . Thank you.

👏🏻👏🏻👏🏻

The flat 2 and the blues note definitely feel dissonant, useful notes but super dissonant to me and most people probably

I really like your videos, thank you for continuing to make them. If you want any viewer requests, could you consider doing a video on William Grant Still?

Well, the Prout book from 1900 told me quite well: thirds and sixths are consonant, seconds fourths fifths and sevenths are dissonant!

Ok, here's my humble opinion: Why augmented triad sounds worse than minor/major? Let's find not only intervals, but the whole triads in the harmonic series! The simplest major triad is 4:5:6, nice and simple The simplest minor one is 6:7:9, simple as well The simplest augmented triad i can think of is 12:15:19, which is not as simple as previous ones (There is also 7:9:11, but 9:11 is more of a neutral third) Why perfect fourth was treated as a dissonance? I think that's because of octave equivalence. P5 (2:3) + P8 (1:2) = P12 (1:3) Fifth becomes even more consonant being stretched by an octave. M3 (4:5) + P8 (1:2) = M10 (2:5) Major third as well P4 (3:4) + P8 (1:2) = P11 (3:8) While fourth becomes worse And at the time musicians used meantone or similar tuning systems in which perfect fourth is sharper than in the harmonic series, which made P11 sound even more dissonant.

minor 2nds, tritones, and maj 7th intervals have a dissonant sound to me. at least in 12 TET. I don't deal with microtonality.

Dissonance in Western music: exists Classical Nerd: well yes, but actually, no Balinese gamelan: Hold my overtone series.

Do you know of any composers who work with similar techniques and are also interested in hermetic philosophy? I’ve been looking for some quotational material for a mixed-media piece, and academia strays away from those topics, aside from a single book I found at the music library!!

I have a hard time hearing the I6/4 chord as dissonant. It doesn't really feel completely at rest but it also doesn't feel like it is going anywhere to me, more that is is just a resolution that is more "floaty" if that makes since.

Your pet theory actually makes sense .. that's why in microtonality the "standard" triad contains either a minor or major fifth Consonances that I have observed are : -Major chord with a major fifth -Minor chord with a minor fifth -Minor chord with a major fifth -Minor chord with a minor fifth -Neutral chord with a major fifth -Minor fourth with a minor sixth That is of course not getting into quartal chords and what not but yea ..

Please do Francis Poulenc sometimes for your Great Composers series!🙏😭

If we want to think about this in terms of physics I'd say chords and intervals move in the quantum realm. When observed separately they can be anything, their properties cannot be determined. But when we add context and direction, where it's coming from and where it is going, we can see if, in that instance, it serves as a dissonant or a consonant. Just like quantum particles that give different answers depending on when and where they are observed.

dissonance to me personally: minor 2nd, perfect 4th, tritone, maj. 7th, (min. 9th.)

Funny, I give "your" pet theory as a reason someone might want to let the ninth, eleventh or thirteenth of the scale be equivalent by way of proposing notation for non-octave scales! en.xen.wiki/w/Modal_systematization_of_soid-family_scales

This is relevant to your theory of symmetry causing dissonance. The alternate stacking of major and minor 3rds somehow makes strange extensions like #15 more consonant than a purely diatonic stack. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-HeTygKpi6pI.html

I want to tell my theory as well. I don't know if it is already out there, but my theory goes like that: Let's imagine there is an acustomed ear, who heard all the Western classical music since Renaissance until the present day. For this kind of ear, everything since Machaut until, let's say Thomas Little :), would sound acoustically neutral. I mean that it is no difference between a traditional consonance and a traditional dissonance. But instead, for this kind of person, the consonance would be considered in the context of the correct use of sounds at the right time and the right place. And by contrast, a dissonance would be considered by such person as the incorrect, rather odd use of sounds at the wrong time and the wrong place of the composition. So, in conclusion, a consonance would be the right use of sounds in the right context, and dissonance - the wrong use of sounds in the wrong context.

Too hard for me yet :). Khachaturian should be pronounced like [hachaturj'an]. Like many Armenian surnames it has the last syllable stressed.

Wonderful, in this video I learned that I understand exactly nothing about music theory. But hey, conscious incompetence is still worlds better than unconscious incompetence. It’s only uphill from here!