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I've recently been learning about interval ratios and about why some intervals are more consonant or dissonant than others.

I was trying to find out about the order of intervals by consonance/dissonance. While there's discrepancies later in the list, I kept reading that it in order of consonant to dissonant, it goes: Unison, Octave, Perfect 5th, Perfect 4th... and so on.

What I'm confused about is that if you consider an interval, it depends which note you consider the root note. So a P5 can be considered a P4 if you switch the root note, and vice versa. Their frequency ratios even have a close relationship.

So why then is a P5 always considered "more consonant" than a P4 when they can be kind of considered having the same sound. Am I the only one who thinks they sound the same?

Any answers are greatly appreciated, thank you :)

Edit: Here's a little bit of context if anyone cares to read further.

I'm actually writing a program to try and demonstrate visually all the not relationships between notes in a chord and I was looking for a visual way to show which relationships are more dissonant than others (probably by color: blue = consonant, red = dissonant).

The only thing is, the program represents the notes in the chord using a graph, but each note in the graph is irrespective of its octave position. That is, each node on the graph only shows the note letter, for example. If a user selects C4 and C5, the graph will display a single node "C".

This is why it's important to me to figure out how to work out how dissonant intervals are when they can be considered differently by switching which note is the root.

Images below (I find these 2 to sound pretty much the same so that's why I'm unsure about why one should be more consonant than the other):

P5

P4

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    I wonder what the musical application of this dissonance thingy really is. :) Many people seem to be interested in it. I like "dissonant" intervals, and a lot of the time, adding a third note makes the harshness go away. C-B is "dissonant", but E-C-B is not. C-D is "dissonant", but C-D-B is a bit less so. Anyway, an interval of a fourth points to the higher pitch as a root candidate. An interval of a fifth points to the lower pitch as a root candidate. In C-F, F is root'ish, and in F-C, F is root'ish. This can be seen in chord inversions. G-C-E is a C major, and E-C-G is a C major too. May 17, 2023 at 9:26
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    Part of the problem with this area of discussion is that there are at a least two kinds of dissonance/consonance. One is acoustic, and has to do with frequency ratios, which is often discussed (but also waveform, which is rarely discussed). The other is musical, or perhaps better, functional, having to do with instability and stability, and that is cultural, so it varies. For example, 20th century music, especially jazz, treats chords that were formerly dissonant as consonant.
    – phoog
    May 17, 2023 at 12:57
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    Why write a harmony program before understanding basic harmony concepts? This comes up a lot on this forum and I don't know why. May 17, 2023 at 19:47
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    @MichaelCurtis I think it comes up a lot because even though it's Music Practice & Theory, it's still StackExchange and therefore draws a lot of people in computer science and related fields.
    – Theodore
    May 18, 2023 at 13:58
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    If by dissonance you mean "roughness" like Plomp & Levelt, and if you're trying to create some kind of a magic formula for making nice chords based on the assumption that "roughness = bad", there is quite a lot of discussion on it on this site. Read these questions and answers, and the source material referenced in them music.stackexchange.com/questions/4439/… music.stackexchange.com/questions/88564/… May 19, 2023 at 10:11

7 Answers 7

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It doesn't matter which tone is the "root" or the lower of the two.

An interval is just a measure of distance.

Using octave numbers with letters makes it a bit easier to talk about without staff notation.

C4 to G4 is a perfect fifth whether you measure from C4 up to G4 or G4 down to C4. It spans 5 letters CDEFG and so is a fifth of some kind. It is 7 half steps and so a perfect fifth.

G3 to C4 is a perfect fourth whether you measure up from G3 to C4 or C4 down to G3. It spans 4 letters GABC and so is a fourth of some kind. It is 5 half steps and so a perfect fourth.

I was trying to find out about the order of intervals by consonance/dissonance.

For some reason this is a fairly common interest with people learning about intervals. There are two main ways to consider it.

First, intervals expressed as ratios are considered more consonant for simpler, smaller number ratios with 1:1 - the unison - being most consonant, 2:1 - the octave - next most consonant, 3:2 - the perfect fifth next, etc. etc. You might call that the mathematical or acoustical approach.

The second approach is the stylistic approach that depends on aesthetics and the system of harmony used. In common practice era harmony you really have only two categories: consonant for major/minor thirds, sixths, and perfect unisons, fifths, and octaves, and dissonant for all other intervals. Beyond that you can distinguish perfect and imperfect intervals, or relative stability for things like chord inversions, but those get into the details of counterpoint, chord balance, etc. and strictly speaking those elements are not about consonance/dissonance.

This may seem contradictory, for example, if dissonance is unstable, isn't consonance stable? To some degree "yes", but there isn't a clear link. Consider that while an inverted major chord is less stable than a root position major chord, both are considered consonant. Could you place those two chords on a spectrum of consonance? Not really. There is no measure to do so. The distinction between the two chords would not be expressed in terms of consonance/dissonance, but in terms of stability.

From that perspective, in common practice style, you can't really says something objective like a sixth is less consonant than a third. Somethings like a minor second being more dissonant than a major second seem fairly obvious, but a lot has to do with context - a chromatic minor second will sound more "jarring" than a diatonic minor second - and contrapuntal treatment would not be different in terms or preparing/resolving a dissonance.

In the second, stylistic approach there isn't a gradation of dissonance, but two categories - consonance and dissonance - with certain part writing conventions associated with various aspects of the two categories.

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    "Could you place those two chords on a spectrum of consonance? Not really. There is no measure to do so." According to this accepted answer that currently has a whopping 79 upvotes, there indeed is a measure for placing any combination of pitches somewhere on a dissonance spectrum music.stackexchange.com/questions/4439/… :) For at least some meaning of "dissonance". I'm inclined to somewhat disagree on the usefulness of that "measure", but it's a measure and there's even published "science" or something on it. May 20, 2023 at 8:04
  • @piiperiReinstateMonica, are you talking about the answer regarding "roughness?" The article that says in its opening that dissonance is "hard to measure quantitatively?" There is good reason to distinguish between acoustical dissonance and musical dissonance. There are acoustical rankings, but they don't match up to common practice treatment. P4 versus M3/m3 is the obvious disparity. The point of my answer is to explain that disparity... and how to measure an interval. May 23, 2023 at 14:01
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The fourth and fifth are not the same interval; they are inversions of each other. Intervals are always computed from the lowest note.

A perfect fifth has a ratio of 3/2. A perfect fourth has a ratio of 4/3.

(I'm not sure why "perfect" but it's probably because the F-C and C-F intervals are bordered by half steps.) Perfect intervals have only perfect, augmented, and diminished forms.

The whole thing of consonance and dissonance is a bit complicated; it's not entirely related to ratios. The fourth is treated as a dissonance against a bass note but a consonance against an upper voice.

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  • Hey thanks for responding, I'm not sure I fully understand, but I need to read this again when my brain isn't so tired. Do you reckon I might be able to ask you further questions about it, maybe by email or discord or something? I saw you are a retired math professor, and I thought I could show you my program and you might be able to help me with it. Otherwise no worries.
    – John Smith
    May 17, 2023 at 1:35
  • @JohnSmith If you're looking for a more open-ended discussion, please consider The Practice Room, the primary chat room connect to this site.
    – Aaron
    May 17, 2023 at 7:42
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As suggested in a comment, there is an acoustic aspect to consonance and dissonance:

Harmonic partials

In (perhaps overly-) simple terms, the sound of a single note is composed of many individual frequencies. Collectively, these are called "partials". For most pitched musical instruments, the partials are (at least approximately) "harmonic", meaning they occur at frequencies that are whole-number multiples of the fundamental frequency. (For example, an A2 contains 110 Hz, 220 Hz, 330 Hz, 440 Hz, etc.) The partials generally decrease in amplitude with increasing frequency.†

Consonance

When two notes are played together, one aspect of the perceived consonance between them arises from the alignment of the partials in one note with those of the other.

Fewer partials align in a perfect fourth than in a perfect fifth.

Further details

The intervals analyzed for consonance below are based on just intonation (JI), which uses whole number ratios between pitches. The equal temperament (ET) system in wide use now is a compromise based on the uniform, irrational ratio of 21/12 between semitones. The ET equivalents are close, but the JI ratios make for neater presentation of the numbers.‡

An octave corresponds to a frequency ratio of 2/1. The perfect fourth corresponds to a ratio of 4/3 and a perfect fifth is 3/2.

Consider the fundamental of the root note to be 1. (It doesn't make a difference which basis frequency we use, so this just makes the numbers easier to read.)

The root note contains partials at these relative frequencies:

1, 2, 3, 4, 5, 6, ...

A note one octave higher contains these relative frequencies (all partials of the root note, each multiplied by 2):

2, 4, 6, 8, 10, 12, ...

You can see from the above that every partial of the octave corresponds exactly with a partial of the root.

A note a perfect fifth above the root contains these (each multiplied by 3/2):

3/2, 3, 9/2, 6, 15/2, 9, ...

A note a perfect fourth above the root contains these (each multiplied by 4/3):

4/3, 8/3, 4, 16/3, 20/3, 8, ...

I've bolded the partials of the perfect fourth and fifth that align with partials of the root.

Every second partial of the perfect fifth aligns with a partial of the root, while only every third partial of the perfect fourth aligns. The perfect fifth has more partials that align and sounds more consonant as a result.

For comparison, let's also look at another consonant interval, the major third (with a ratio of 5/4) and its less-consonant inverse, the minor sixth (with a ratio of 8/5):

A major third contains these (multiplied by 5/4):

5/4, 5/2, 15/2, 5, 25/4, 15/2, 35/4, 10, ... (Every fourth partial aligns.)

A minor sixth contains these (multiplied by 8/5):

8/5, 16/5, 24/5, 32/5, 8, ... (Every fifth partial aligns.)

In addition to the quantity of partials in alignment, consider that the higher-order partials are generally weaker in amplitude and contribute less to the perception of consonance or dissonance. The more-consonant intervals contain alignments of lower-order partials.

Notes

† Some instruments (e.g. pitched percussion) may have significant partials that are substantially inharmonic. Some instruments (e.g. piano) have partials that are slightly inharmonic. The relative amplitude of the partials varies over time and some may be entirely absent. The composition is a large part of the specific timbre of each instrument

‡ An ET perfect fourth has a ratio of about 1.3348, a bit sharp of JI. An ET perfect fifth has a ratio of about 1.4983, a bit flat of JI. The third and sixth are further out-of-tune in ET.

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I have some bad news for your app project: It's impossible to rank intervals as being more or less dissonant than others. "Consonant" and "dissonant" are subjective words,* equivalent simply to "sounds good" or "sounds bad." And people in different places and times in history haven't even agreed about what sounds good. And even within a single genre or practice, a given interval might sound "better" or "worse" depending on the chord it's used in and what's going on around it. So whatever source has been telling you that they can be ranked is perhaps something you shouldn't continue paying attention to.

As to why you hear a similarity between P4 and P5: First of all, you've discovered complementation! They are related, just as 2nds are to 7ths and 3rds are to 6ths. And as for why this pair seem even more closely related than others, they're the "most similar." If you split an octave exactly in half, you get a "tritone," which you can spell either as an augmented 4th or a diminished fifth; for instance, a C up to an F#. If you go up from the F# to the next C, you still have a tritone. You have a dim. 5th rather than an aug. 4th, but you have the same number of half steps. Well, aside from this "identical" inversion, the next step away from the "exact middle" of the octave gives you the P4/P5 pair. The "difference" between the two is the least of any pair, without being (enharmonically) identical.

P4 and P5 also have a relationship in triadic harmony. A simply triad, like C major, contains C, E, and G: a fifth, with a third in between it. If you move the C to the top, you have a P4 on top, but the E still helps you identify it as a C major chord.

enter image description here

Meanwhile, the C minor triad has the same fifth but a different third. Its E flat identifies it as minor no matter the inversion.

enter image description here

But if we delete the third from the chord entirely...

enter image description here

... then we can't say whether these are major or minor chords. We have "modal ambiguity." These "open chords" feature in a lot of contexts like various folk musics, or rock power chords. Judith Kuhn uses the term "fourthy-fifthiness" to describe Shostakovich's second string quartet and its "power chords" as three instruments play either fourths or fifths in the opening:

(Embedding a video with score, though I prefer the St. Petersburg Qt. performance.)


* At least the way we're using them here; see phoog's comment about a more scientific usage.

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Intervals are named from what 'space' there is between the lower and higher notes, along with the note names, which gives the number of those intervals. So C>G is P5, while G>C is P4.

The difference, though, is C>G is deemed consonant as G is a harmonic of C,(the 2nd, and audible when C is played on many instruments) whereas G>C is dissonant as C is not a harmonic of G. At least not a particularly close one to note G - inaudible on most instruments.

So, on the face of it, while they both sound the same two notes, C>G is P5 and consonant, G>C is P4 and dissonant.

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  • I don't think this holds up, since whether P4 is consonant or dissonant depends on context. Also, thirds were considered dissonant for a few centuries, and major thirds, at least, are fairly nearby in the harmonic series. Also, the question doesn't claim that P4 is dissonant — only that it less consonant than P5.
    – Aaron
    May 17, 2023 at 7:40
  • @Aaron - the two terms seem to have been bandied about for a long time, and probably these days don't really have the same meaning as original. In fact, they're, to me, meaningless. But I just thought this may be a non-scientific way to explain the difference the OP searches for.
    – Tim
    May 17, 2023 at 8:07
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Sometimes in music theory we can deal exclusively with pitch classes, a C is a C irrespective of octave. A perfect 4th and its inversion a perfect 5th are completely equivalent.

Sometimes we can't. It matters which note is on the bottom. C - G sounds like ^1 - ^5, the most stable interval there is, bar a unison or octave. G - C sounds like ^1 - ^4 or ^5 - ^1. Not a chord you'd choose as the final one of a piece.

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  • Yup! And now I have...
    – Laurence
    May 19, 2023 at 19:33
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I agree with Theodore's answer that we need to consider the harmonic series aspect of the 2 intervals. As you can see in this graph showing harmonic amplitudes of a flue organ pipe, we can see the decreasing amplitudes as the harmonic numbers become higher.

Here's a comparison between the first 6 harmonics of perfect 5th (root=C1) and perfect 4th (root=G1).

  • Perfect 5th (root = C1): c1 c2 g2 c3 e3 g3 and g1 g2 d3 g4 b4 d5
    The sorted harmonics we hear: c1 g1 c2 g2 g2 c3 d3 e3 g3 g4 b4 d5

  • Perfect 4th (root = G1): g1 g2 d3 g4 b4 d5 and c2 c3 g3 c4 e4 g4
    The sorted harmonics we hear: g1 c2 g2 c3 d3 g3 c4 e4 g4 g4 b4 d5

Notice how within the first 2 octaves of the harmonics:

  • in the perfect 5th (root = C1) all 6 notes are C's and G's, even with a doubling of g2.
  • in the perfect 4th (root = G1), only 5 notes are C's and G's, with a d3 interfering next to a c3! The doubling of g4 is in the 3rd octave above the root, so it doesn't help much.

Therefore, the above acoustic analysis should show how (in theory) the perfect 5th is more consonant than the perfect 4th, especially when the sound is produced by a an instrument with a "natural" tonal characteristics such as wooden flue pipes in a pipe organ.

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