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In reading my new book 'Complete Classical Music Guide', I understand the following:

If two notes are separated by a consonant the sound is pleasing to the ears. If they are of a dissonant interval they clash with each other. Therefore they have to be resolved by a suitable consonance.

However, I do not understand (as it does not explain):

Intervals considered dissonant have changed since the Middle Ages

Is this because:

  • we have later accepted that these “sounds” are not correct, as they were originally listed because ...

or

  • it is wholly opinion at the forefront of why intervals were once considered dissonant?

I would be most grateful if someone could explain why dissonant intervals have changed as so?

Changed in what way?

complete classical music guide book dissonant intervals


EDIT 30/12

This subject has particularly fascinated me. I have done some further reading.

I read the following in a PDF article, available here (A History of 'Consonance' and 'Dissonance' by James Tenney)

Musically speaking, there are not really dissonant intervals, but only dissonant notes, Which note is dissonant in an interval physically (acoustically) dissonant. This depends on the clang to which that interval has to be referred By thus distinguishing dissonant.. .notes in place of the old system of intervals and chords, a much clearer view of chords is obtained.

Whilst an 'Interval' is the difference between two pitches not necessarily a harmony per-se, when these notes are played together they dramatically tell the story of the piece, so I find the above quotation remarkably odd; "there are not really dissonant intervals".

This is more interesting from The Harvard Dictionary of Music by Don Randel, available here:

During the Middle Ages, theorists generally believed that polyphonic textures were primarily consonant in nature and those imperfect consonances played a subordinate role to perfect consonances. They recognised that phrases begin and end on perfect consonance and that the final perfect consonance is approached by step in contrary motion from the nearest imperfect consonance.


Are we suggesting then here that this is all perceived by many many factors like temperament, genre, range, and the actual interval itself? Almost to a point (as my original post suggests) that opinion is of the highest principle here?

This is what is confusing: An augmented second and a minor third are identical. But in traditional understanding, the first one is considered a dissonance, and the other consonance.

The big question then is can it be argued then that an octave could be dissonant?

  • I've read that the perfect fourth was considered dissonant in the Middle Ages but consonant today. – Dekkadeci Dec 29 '17 at 0:18
  • @Dekkadeci are you sure it wasn't the other way around? – leftaroundabout Dec 29 '17 at 0:47
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    As far as I know, major and minor thirds were not considered consonant in Europe until the 1500s. In fact, this was partially the reason for the development of meantone temperament and its variants - to preserve the quality of the thirds while ensuring the perfect fourth and fifth were not altered by too many cents. – syntonicC Dec 29 '17 at 15:47
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    @leftaroundabout, I might be mixing it up with the times I've read that perfect fourths were considered dissonant in counterpoint (such as ars-nova.com/Theory%20Q&A/Q85.html and en.wikibooks.org/wiki/Counterpoint/First_Species). Hold on, en.wikipedia.org/wiki/Perfect_fourth indicates this, according to The New Grove Dictionary of Music and Musicians, second edition: "In the 15th century the fourth came to be regarded as dissonant on its own, and was first classed as a dissonance by Johannes Tinctoris in his Terminorum musicae diffinitorium (1473)." – Dekkadeci Dec 30 '17 at 6:54
  • @syntonicC you don't need any temperament for choral music, it's perfectly possible and natural to sing 5-limit (or even 7- or higher) just intonation. This seemingly just didn't occur to middle-age composers. – leftaroundabout Dec 30 '17 at 10:53
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+100

OK, so first a clarification of the contexts:

The quoted materials are mostly concerned with Western European music. The music of the Middle Ages almost certainly refers to the liturgical music of that time and place, so we're mostly talking about Gregorian Chant, organum, motets, etc. When talking about changes in the later eras, I suspect they're mostly talking about changes that had entirely solidified by the time of Classical composers like Mozart and Beethoven. If we push further into the twentieth and twenty-first century music of that lineage we see further changes in interval usage, as we would if we push into other genres such as jazz, rock, pop and hip-hop.

Next, an executive summary:

I suspect that the first book is mostly referring to a shift from the perfect fourth being mostly treated as a consonance in the Middle Ages to being treated as a dissonance in some contexts in the Classical era. Concomitant with this shift is a shift from a tendency to treat what we now call imperfect consonances (major and minor thirds and sixths) as dissonances in the Middle Ages to the consonances they generally are in the Classical era. We could also speak of a much later shift from Classical to Jazz: in the former the seventh is virtually always a dissonance while it's often a consonance in the latter.

To go into more detail, I think it's extremely useful to shore up the sometimes loose definitions of "consonant" and "dissonant." In particular, I think it would be helpful to use the terms "concordant" and "discordant" to refer to objective measures of how simply the frequencies of two notes relate to each other. This is shown somewhat in the picture you posted that includes an illustration of the highly concordant interval of the octave—wherein the periods of the two waveforms are highly correlated—and the fairly discordant interval of a second. The waveforms of the second take far longer before they get back to their starting relationship (not shown in the image, since the image would have to be too long for the page width) than the octave, which shows the waveforms getting back to their starting relationship twice in just a brief span. Although there might be some ambiguities and debate about a highly-specific ranking of every possible interval, it's fair to say generally that perfect unisons and octaves are extremely concordant; perfect fourths and fifths and quite concordant; major and minor thirds and sixths are moderately concordant; minor sevenths and major seconds are fairly discordant; major sevenths and minor seconds are highly discordant. This is purely based on how the pitches interact with each other in terms of their periodic waveforms.

If we let those terms stand for a more "scientific" understanding of intervallic relationships, we can then use "consonant" and "dissonant" to refer to the way that composers (in the broadest sense of that term) actually use the intervals within the context of their piece and genre. A dissonant interval is one that tends to be treated as unstable: an interval that needs to resolve to a more consonant interval. A consonant interval is one that is treated as stable; it's an interval that can be resolved to. Because this is entirely about style and context, this is the aspect of interval definitions that has shifted the most over the years. Unfortunately, there's one more complication that has to be discussed before talking about that.

For the most part, introductory texts treat pitch and tuning as far clearer than they actually are. They tend to assume equal temperament, which is fine for beginners, but the question of historical shifts in interval usage cannot be separated from the issue of temperament. There are lots of questions on this site that will give you the details, but suffice to say that the earliest tunings used during the Middle Ages for fixed-pitch instruments (instruments like fretted strings or keyboards where you have to set the exact tuning of the pitches in advance and can't easily modify the tuning during performance) was Pythagorean. This is a system where all the notes are tuned according to perfect fifths. If you do this—essentially tune your piano by going around the circle of fifths and getting every fifth truly perfect—then your major thirds will be quite discordant compared to our modern equally-tempered major thirds and will be very discordant compared to a pure major third. Again, the specifics of these differences are complicated, but you can find out more by looking up "just intonation", "temperament" and "equal temperament" among other topics. The upshot as far as this question is concerned is that it isn't surprising that composers of liturgical music in the Middle Ages treated the major third as a dissonance that needed to resolve to purer intervals such as the fourth and fifth. In part this is because it is less common to treat an interval as discordant as the Pythagorean major third as a point of resolution.

It's vastly oversimplified, but very generally speaking, different attitudes start to emerge (over the course of centuries) during the Renaissance. The sweetness of pure thirds (which were easy to explore in a capella singing for example) became more popular, and people started tuning fixed-pitch instruments so that they had pure (or purer) thirds as well. Composers started using thirds (and sixths) as primary consonances and so needed to adjust the tuning of fixed-pitch instruments to make those thirds more concordant as well. As the third became a primary consonance, an ambiguous status for the perfect fourth emerges. Although the perfect fourth was just as concordant as it was before (more concordant than a major third!) it was only a half step away from being a third. This led to a natural "gravity" as—in the context of the music that was being written—the top note of those fourths tended to "want" to resolve down a half step and become a consonant third. The objective concordance of the perfect fourth didn't change, but its contextual meaning did. When the bottom note of the fourth is the lowest note of an entire harmony, the fourth became a clear and strong dissonance.

The fourth didn't become more discordant, it just became more dissonant.

And, just to reiterate, this is only in certain contexts even within music of the Classical era. When the fourth is between two upper notes within a larger harmony, it's perfectly consonant. The fourth is only treated like a dissonance when the bottom note of the fourth is also the lowest note of the entire harmony. (Look up the cadential 6/4 chord for the quintessential example of this, although also see the conversation in the comments to this answer between me and @leftaroundabout, which adds some interesting subtleties to this issue). The distinction between concordance/discordance on the one hand and consonance/dissonance on the other also helps to clarify your question about the augmented second versus the minor third. First off, in many non-equal tempered scenarios, those two intervals are not the same. The augmented second will be wider and far more discordant than the pure minor third. The books you're quoting, however, are talking about equally-tempered instruments like the modern piano, and on those it is true that the augmented second and minor third interval are the same objective relationship. As a result, it would be entirely correct to say that they are equally concordant (or discordant), but because "consonance" and "dissonance" are contextually defined, it isn't ridiculous to say that an augmented second is dissonant while a minor third is consonant. That tends to be their usage in music of the Classical era: a composer would almost never spell out the interval as an augmented second unless they intended it to be performed as an instability that needs to resolve. The spelling of notes in the score is not simply an instruction about what buttons to push, it can also indicate the rhetorical intent of the composer.

So, to sum up, the fourth was initially treated as a consonance equal to the perfect fifth in the Middle Ages of Europe and thirds tended to be dissonances that resolve to them. As the status of thirds began to change, the fourth became ambiguous and is often a dissonance that needs to resolve to a third. Historically, there was some shift in the concordance of thirds due to changing standards of tuning, but the primary shift is a purely contextual one toward consonance. More recently, we hear something similar happen to the treatment of the seventh in jazz music, where it's often a perfectly consonant interval despite its relative discordance.

EDIT TO ADD: I just saw where you asked for sources. One problem is that this kind of question falls into an unfortunate hole in the literature. On the one hand, it's too basic for discussion in academic journals; on the other hand, the historical evolution of something so specific is seen as too tangential for an introductory text. I think most textbooks of both music theory and music history at least mention it, sort of like the book that prompted your question. I know that both the Clendinning/Marvin and Kostka/Payne music theory texts at least devote a paragraph to the topic. I don't have my copy on hand, but I think there are a few paragraphs in the Burkholder/Grout/Palisca textbook on this as well—however I recall that is just discussed across the course of the text as it moves from early music into the Baroque, not in one clear place.

Perhaps the best place to look for authoritative sources is in Grove Music (now called Oxford Music). The article on the Fourth in there is pretty brief, but includes this paragraph:

The 4th has a unique position in Western music because it has been regarded as a Perfect interval (like the unison, 5th and octave) and a dissonance at the same time. In ancient Greek music the basis of melody was the Tetrachord, a set of four pitches encompassed by a 4th. The earliest forms of medieval parallel Organum favoured it as the interval between the vox organalis and vox principalis. With the further development of polyphonic music in the 12th and 13th centuries, the 5th replaced the 4th as the most important Consonance after the octave and the unison. By the 15th century the 4th appeared as a consonance only between the upper parts of a vertical sonority, for example in 6-3 chords of the fauxbourdon style and at 8-5-1 cadences (e.g. D–A–D); composers of the later 15th century, including Du Fay, sometimes deliberately avoided the 4th in three-part writing (see Non-quartal harmony), and Tinctoris deemed it a dissonance in his Terminorum musicae diffinitorium (c1473).

You'll notice that the concord-discord/consonance-dissonance distinction I talk about is far from universal, and the terms are used differently in most texts on the topic. Ultimately, this is the kind of information I learned during a semester-long graduate course on The History of Theory. As is common in such courses, there was no textbook, we engaged with primary historical sources. We began with ancient theorists like Aristoxenus and, later, Gaudentius, then moved to later writers like Boethius et al. For the most part, these writers don't talk much about the shift in thinking, instead, one has to intuit and observe the shift between them. Anyway, I hope this at least helps situate your understanding a bit.

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    Excellent discussion. I'd remark that a ₄⁶ chord is in fact more discordant than a root-position major chord, if a wide voicing is used (perhaps with the bass note octave-doubled, as common in orchestral settings): 3:8:10 vs 2:5:6. In a close voicing, as typical on guitar, it is not more discordant, and fifth-in-bass chords are indeed commonly used by guitarists, not usually perceived as dissonant. – leftaroundabout Jan 1 '18 at 15:23
  • @leftaroundabout This is an interesting frame that I haven't considered before. It does seem like a wider spacing certainly emphasizes the instability more, although I'm not sure why 6:8:10 would be that much closer in stability to 2:5:6 than 3:8:10. Definitely true that this issue is more-or-less irrelevant to proper guitar style, but I've always taken the shift away from 6/4 as dissonant to be more of a genre issue than an instrumental one. When baroque and classical composers write for lute/guitar do they still use 6/4 chords primarily for unstable functions like passing and cadential? – Pat Muchmore Jan 2 '18 at 15:18
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    Well, you don't hear 6:8:10 but 3:4:5 in a close-voiced ₄⁶ chord, and I daresay that is quite concordant indeed. — Sure, when classical composers use ₄⁶ it's usually e.g. predominant rather than tonic because genre convention. Then again, in classical guitar, the voicings tend to be wider than in folk/rock styles, and classical training emphasises the need to make the bass note stand out clearly below the rest. – leftaroundabout Jan 2 '18 at 15:31
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My view (perhaps not quite mainstream-agreed) is that nothing has changed in what intervals were dissonant, between the middle ages and today. Rather they simply didn't use some of the intervals that we use as consonances today, vice versa.

Middle-age harmony was strictly Pythagorean in nature, meaning all intervals were constructed by multiplying or dividing frequencies with the numbers 3 and 2. For instance, you get a perfect fifth when starting out with some frequency (e.g. A3: 220 Hz), multiplying by 3 (660 Hz) and dividing by 2 (330 Hz), which as you'll notice is almost exactly the frequency of an E4 on a modern piano (329.6 Hz). An octave simply has the double frequency, and so on.

From this construction, you get as the most natural intervals: octave, fifth, fourth, major ninth, major second...

OTOH, to construct, say, something like a C♯ from that A3, you'd need to first go to B3 (twice up a fifth, i.e. multiply with 3×3 = 9 and divide by 2×2 = 4, then down another octave; compound factor: ⁹⁄₈) and then up another full step, which makes the total frequency ratio ⁸¹⁄₆₄. It's pretty far-fetched to call that a consonant interval, and indeed if you search the internet for “Pythagorean major third” you'll find it sounds pretty jarring. This interval is not really used in post-1700 music at all, not so much because it sounds dissonant as because the instruments can't play it.

Because in the Renaissance, musicians (re-) discovered another note quite close to that “C♯” of 278.4 Hz, one that can't be constructed from Pythagorean tuning. Namely, if you just multiply by ⁵⁄₄, you end up at 275 Hz, which gives a very sweet, harmonious-sounding interval. Ever since, instruments were built with that consonant major third in mind, and that interval is a mainstay of Western music. But it's not the interval that folks in the middle ages, rightly, considered dissonant, though the frequencies are quite similar.

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    While it's true that medieval treatises on tuning favored Pythagorean interavals, it's an open question how musicians tuned in practice. I seriously doubt that medieval singers aimed for Pythagorean thirds in the many pieces, especially in English and Alpine music, that were primarily based on thirds. But I guess there's no way of knowing for sure. – Scott Wallace Jan 1 '18 at 20:36
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Here is an excerpt from a review of Beethoven's Op. 111 Sonata, written in 1823, soon after it was published:

Beethoven ... is suffering under a privation that to a musician is intolerable - he is almost totally bereft of the sense of hearing ... The Sonata, op. 111 consists of two movements ... In [the first movement] are visible some of those dissonances the harshness of which may have escaped the observation of the composer.

Here's a performance of the Op. 111. Perhaps you will decide whether or not you agree with the critic.

The point, then, is that dissonance is a purely subjective concept, at least as far as we currently know.

This is borne out by the fact that the concept of what is and isn't dissonant has been under constant revision over the past few hundred years. It's interesting to realize that in the 1400s the tritone (B to F, for example) was referred to as diabolus in musica, and its use was strictly forbidden in sacred music. Presumably because its "obvious" dissonance was something that couldn't come from God. Over time, we have found our way to the idea that since dissonance is a subjective concept, we can define musical structure any way we like, without regard to whether music written using that structure "sounds dissonant" or not.

However, I got into a very interesting discussion on the subject some years ago, talking about the idea of resonance. If the human body resonates to sound -- and the laws of physics suggest that it does, as does any other physical object -- then it follows that certain frequencies could create resonances that are objectively beneficial to health and certain other frequencies could create resonances that are objectively less so. For example, it would seem likely that there are frequencies that could set up vibrations in tumors that would cause them to break apart, and others that could damage internal organs.

If it came about that some intervals had more beneficial effects on the health of the body than others, then that might be considered an objective standard of consonance vs. dissonance.

So, perhaps 200 years from now, we will look back on the discussions of "subjective dissonance" from our times with that avuncular fondness that we currently observe for discredited ideas such as bodily humors and bloodletting.

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Why have dissonant intervals changed over time, and how? Briefly, the concepts of consonance and dissonance are a combination of objective, acoustic phenomena—what we might call "acoustic consonance/dissonance" or "concordance/discordance"—and subjective, cultural phenomena that may or may not correlate with acoustics (but often do). Cultures change; acoustic principles don't. Tenney (1988) offers the broadest look at the changing concept of consonance/dissonance across Western musical eras, but this work need not be taken as the final word; there are fine details within each period that underlie these larger shifts.

Now, on to some examples. The perfect fourth is the archetypal example of an interval that has experienced a change in consonance/dissonance category. This interval, bearing a 4:3 ratio of fundamental frequencies, is acoustically consonant—it has high harmonicity (likeliness to a single harmonic complex tone) and promotes perceptual fusion (hearing two sounds as one). But it also doesn't fit well within the concept of harmonic root: the P4 is an interval "standing on [its] head" (Thomson 1993, 127), in that a modern (chord-oriented) listener is prone to hear the P4 as an inversion of a P5, pointing toward the upper tone as the "root" (as in a six-four chord). So with the gradual change in Renaissance-era musical practice from tenor (middle) to bass (lowest) as structural voice, the P4 changes from stable (consonance) to less stable (dissonance). Basically, the change is with the system of consonance/dissonance, not with the P4: the P4 just happens to be standing on the fault line.

Regarding the distinction between augmented second and minor third—intervals that are enharmonically equivalent in equal temperament, which is implicit in your question—the "dissonance" ascribed to the A2 is one of melodic practice and musical expectation. Most works in the Western art-music tradition ("the canon") simply do not make use of the A2 as a melodic interval, though we may find one or two hidden instances, like the forbidden parallel fifths that we occasionally find in Romantic-era piano works. The essential instance of this interval is found in the so-called harmonic minor scale between the lowered sixth (le) and raised seventh (ti) scale degrees. Western music (and its listeners) have learned to treat and hear each of these scale degree as "gravitating" toward the more structural degrees that lie a semitone away—le above sol, ti below do. A jump from le down to ti, a diminished seventh, is culturally okay, because it honors the "gravitational pull" of le downward. (Check out Bach's Two-part Invention no. 4 in D Minor, BWV 775, for an example of this diminished-seventh interval in both directions.) But le up to ti, an augmented second, violates this gravitational pull; it "leaps a gap" that is often obscured when we practice writing the harmonic minor scale as starting on do and going up the octave. And while talk of "gravity" in music smacks of some of the more lofty "will of the tones" concepts found in musical treatises, Larson (2004) offers a reasonable, cognition-oriented description and analysis of gravity as a "melodic force."

Lastly, can an octave be dissonant? I'd offer a qualified "yes." Because the octave has high harmonicity with its 2:1 ratio, it will always be prone to harmonic fusion, and will "sound good" in a chord or a melodic leap. But it's possible to create a context in which octaves are considered syntactically unfit. The "free atonal" style of composition, most often associated with Schoenberg and like minds in their pre-serial works, rigorously avoids the octave in its chords. (Yes, we may find octave doubling in a piano melody, but this is just that—doubling a melody for purposes of dynamics and texture, not two independent voices.) Cramer (2003) theorizes that Schoenberg and Webern used the major seventh and minor ninth—the "altered octave"—as a replacement for the stable octave, and that these intervals may be heard as both acoustically (through their roughness) and practically (through their frequency of use) structural. Would an octave in this style of music therefore sound dissonant? The test would be in the ability to hear an octave as a non-harmonic ("non-chord") tone—such that the listener would expect it to resolve to a consonance. I'd suggest a suspension that resolves from octave to major seventh; I can't think of such an example in the literature, but perhaps we might find one, or compose one for that matter.

Similarly (but not directly on-topic), can we hear the "altered octave"—major seventh or minor ninth—as a consonance? I'll suggest a musical example, heard in "Angst und Hoffen," no. 7 from Schoenberg's Book of the Hanging Gardens (Op. 15, 1908–09). Listen to the first two three-tone chords in the piano—the first an augmented triad (Gb-Bb-D), the second a "Viennese trichord" of stacked augmented and perfect fourths (set class 016), spelled upwards Fb-Bb-Eb. Try to hear them as dominant to tonic within an Eb tonic. (Minor? Major? Let's just call it "E-flat.") Hear the melody in the voice rising from ti ("Angst") to do ("Hof-"), doubling the upper tones of each chord. Now listen to the final melodic passage of the work, at the Sehr langsam tempo marking, as Schoenberg begins an embellished scalar descent in the voice from sol (Bb) to a final redo (F to Eb); the piece ends with the same "dominant to tonic" pair of chords as are heard in the beginning. Is the final chord consonant? Can you learn to hear it as consonant? Could other listeners do this as well? I think so, but perhaps I ask too much of listeners.

Sources

Cramer, Alfred. 2003. "The Harmonic Function of the Altered Octave in Early Atonal Music of Schoenberg and Webern: Demonstrations Using Auditory Streaming." Music Theory Online 9 (2), http://www.mtosmt.org/issues/mto.03.9.2/mto.03.9.2.cramer.html

Larson, Steve. 2004. "Musical Forces and Melodic Expectations: Comparing Computer Models and Experimental Results." Music Perception 21 (4), 457–498.

Tenney, James. 1988. A History of Consonance and Dissonance. New York: Excelsior.

Thomson, William. "The Harmonic Root: A Fragile Marriage of Concept and Percept." Music Perception 10 (4), 385–415.

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Consonance/dissonance are not a binary thing that intervals naturally fall into. It's a gradient. What we can do is mathematically determine the ordering of the intervals along this gradient to show which intervals are more or less consonant/dissonant than other intervals. This is fairly objective.

But where you want to draw the line and classify everything on one side as consonant and on the other dissonant, is up to you. Whether it's wise to classify intervals in such a binary fashion in the first place is also up to you.

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If 'Intervals considered dissonant have changed since the Middle Ages' worries you (and it's only a statement by one author, not an established fact) perhaps you'd rather think of the subject this way. The spectrum of dissonance is unchanged - we still recognise that an octave is perfectly consonant, a 5th nearly as much, a 4th as somewhat unstable etc. etc. - we've just developed a taste for intervals further away from pure consonance. We haven't changed our opinion of what's consonant and what's dissonant, just our opinion of how much dissonance is acceptable.

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All the above answers are good. I'd like to add a different point of view. The terms "consonance and dissonance" as used (if not always explicitly described) in music are not equivalent (nor necessarily related) to harsh and sweet sounds or the terms concord and discord. Dissonance (as I understand it from some texts and looking at lots of music) normally means a sound that indicates musical movement; dissonance needs to be resolved. Consonance means that no movement is implied. These are "local" features. In a large musical structure (sonata movement, fugue, etc.) there are tonal regions that seem to call for something to follow. The opening of a sonata (in a major key, the tonic section followed by dominant or other key area) seems incomplete though no local dissonance need occur. Similarly for a fugue after only a couple of statements of the theme.

There are two points I'd like to make here. First that consonance and dissonance are most subjective terms and musical physics doesn't really explain everything. (Temperament choices may be of interest in this field though.) Second, consonance and dissonance are not fixed terms but their meaning depends strongly on musical style. Note that the fourth above a bass was (is?) considered dissonant in two part counterpoint and that this may be because it implies a 6-4 chords which in turn is considered a decoration of dominant chord. Non-cadential 6-4 chords don't sound dissonant. Over the centuries, styles have changed but this change is (I don't think) rooted in the dissonant or consonant character of the music.

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