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Why does an authentic cadence sound pleasing to the ear? What makes an interval "Consonant" or "Dissonant" and why are there only two categories for intervals?

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The notion of consonance/dissonance depends on the tradition or style used.

In European common practice perfect unions, octaves, fifths and major/minor thirds and sixths are consonant while seconds, fourths, tritones, sevenths and imperfect intervals are dissonant.

Some try to explain that arrangement acoustically by calling simpler ratios being more consonant. So an octave's ratio is 2:1 simpler and more consonant than a minor sixth with ratio 8:5.

The perfect fourth is interesting in this context, because sometimes it is considered dissonant other times consonant.

This doesn't explain the part of your question about perfect cadences. But that is a different question.

I would caution against thinking of intervals fitting into only two categories. There are several other categories/descriptors which can be reviewed here https://en.wikipedia.org/wiki/Interval_(music).

Perhaps you meant: why is there only a two-part consonant/dissonant duality? Some things can be though of in a dualistic way - light/dark, inhale/exhale, up/down, etc. - but sometimes you can or should reject dualistic thinking. There is music that doesn't work around notions of consonance/dissonance.

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Intervals aren't necessarily consonant or dissonant in themselves, it depends to an extent on where they are and what they're doing in the music.

There are not only two intervals per se. There are major, minor, augmented and diminished in most music.

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    I think his two categories are 'consonant' and 'dissonant', not major, minor, augmented... – Laurence Payne Oct 23 '18 at 17:26
  • You may well be right. To get an accurate answer, I've found one needs to ask accurate questions... – Tim Oct 23 '18 at 17:28
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It all depends on mathematical relationships. An octave is easy on the ear since it is a 2:1 relationship. A fifth likewise is an easy fraction. A third is more complex and thirds were for a long time considered dissonant. Then along came the 'consonance Angloise' and thirds became acceptable as consonances. Minor seconds have a relationship that is hard for the auditory system to analyse or relate mathematically so are heard as dissonant. It's all about the harmonic series.

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The "consonance" and "dissonance" relates to how the interval sounds like.

Here are perfect consonance intervals. They have a very good and a bright sound.

  • Perfect unison
  • Perfect 4th
  • Perfect 5th
  • Perfect octave

Here are imperfect consonance intervals. They have a "mild" or a "neutral" sound.

  • Major 3rd
  • Minor 3rd
  • Major 6th
  • Minor 6th

Here are dissonant intervals. They basically do not have a good sound. Please note that ALL augmented and diminished intervals are dissonant. Any chord that includes one or more of the following intervals are considered to be dissonant.

  • Major 2nd
  • Minor 2nd
  • Major 7th
  • Minor 7th
  • Augmented unison
  • Augmented 2nd
  • Augmented 3rd
  • Augmented 4th (Tritone)
  • Augmented 5th
  • Augmented 6th
  • Augmented 7th
  • Augmented octave
  • Diminished unison
  • Diminished 2nd
  • Diminished 3rd
  • Diminished 4th
  • Diminished 5th (Tritone)
  • Diminished 6th
  • Diminished 7th
  • Diminished octave
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    If your definition of dissonant is a displeasing sound, how are ALL augmented and diminished intervals dissonant? An augmented 7th sounds like a perfect octave. That cannot both be dissonant and consonant according to your definition. There are other problems as well, such as the diminished 4th, which sounds like a major 3rd. – Heather S. Dec 7 '18 at 12:25
  • Good question! An augmented 7th, the diminished 4th, and some other augmented and diminished intervals do not sound that bad as they are enharmonically equivalent to consonant intervals. However, they still are considered dissonant because they are augmented and diminished intervals. – RailroadHill Dec 7 '18 at 13:28
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    I'm sorry, but your explanation does not jive with your definition about what dissonance and consonance IS. You mention that it has to do with how the intervals sound, and that a P8, for example, has a "very good and bright" sound. Well, so does an augmented 7, regardless of the fact that it is an augmented interval and only an enharmonic equivalent. The enharmonic equivalence only affects what is written on the paper, not it's sound, assuming 12-tone equal temperament. The quality of an interval only explains it's theoretical function, not it's sound. – Heather S. Dec 7 '18 at 16:31
  • RailroadHill, I believe the controversy here is that your examples are of intervals that "look dissonant" (not a proper term but people know what I mean), but your definition of the term is based on sound rather than actual definedc ratios. So an augmented 7th could be argued to be a dissonance, but you'd have to clearly define dissonance in a way that doesn't mention sound or allow enharmonicity. This is the crux of what I believe @HeatherS. is saying. – user45266 Dec 7 '18 at 16:51
  • No matter of how they sound like, all 2nd, 7th, augmented, and diminished intervals are considered to be dissonant. It has nothing to do with enharmonic equivalence or 12-tone equal temperament at all. – RailroadHill 2 days ago

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