# Do interval ratios consonance apply to a single pitch or a scale?

I have established the order of consonance to dissonance of a pitch as seen below in pitch C..

• Unison - C to C - 1/1
• Octave - C to C1 - 2/1
• Perfect Fifth - C to G - 3/2
• Perfect Fourth - C to F - 4/3
• Major Sixth - C to A - 5/3
• Major Third - C to E - 5/4
• Minor Third - C to Eb - 6/5
• Minor Sixth - C to Ab - 8/5
• Tritone - C to F# - 7/5
• Minor Seventh - C to Bb - 9/5
• Whole Tone - C to D - 9/8
• Major Seventh - C to B - 15/8
• Semitone - C to C# - 16/15

Visuals of interval ratios

My question is as follows..

When I compose a melody in the key of C for e.g. do these ratios of consnance/dissonance apply to the scale of C as a whole or do these ratios only apply solely to the note C?

Allow me to elaborate slightly..

Let's say my melody in the key of C begins with the note C and the following note is G.. Do these ratios for C no longer apply once I've reached the note G, but in turn I would need now to shift the ratios to that of G or could I use C's ratios throughout an entire composition once I remained in the key of C?

EDIT Through the responses you good folks have shared, I feel the question has bŕoadned a tad and I would like to add to this original question. In terms of physics, what is the adequate approach/understanding of melody composition? We have established the physics behind chord progressions, so what is the physics to melody composition?

Many thanks guys!

• Fascinated as to why this is important to you. Is it just the science behind it, or are you hoping to use any science to produce melodic lines? Just using ratios for each successive note based on the last one will most likely not work, as the melody, even if only consonant ratios are used, won't sound right. It'll sound as if the key is constantly changing, which in itself will produce dissonance, I think. – Tim Jun 25 '19 at 7:01
• I'm in the process of building up a system to produce music from it's absolute source as opposed to intuitive or music theory based. That came to mind about the key being neverending but I thought maybe that was the process of modulating from key to key. What formula would you suggest to use compose melodies not by vague or intuitive means but like I stated, in its purest form? Thank you Tim – Seery Jun 25 '19 at 8:24
• The ratio structure is not dependent on the key. The 5th of G is also 3/2. If you are in the key of C the 5th of the 5th is (3/2)(3/2) = 9/4. Bump it down an octave and 9/8 which is your maj 2. – ggcg Jun 25 '19 at 13:31
• @ggcg if the ratio structure is not dependent on the key, how would we continue utilizing ratios in our chord progression? If im in they key of C major and the bass note of my first chord is C and the following bass note G, how should i approach the ratios at this point? Should i now use the ratios of G? If so, would that not lead to modulating key or would i only use the ratios thats notes exist within my key of C and ignore the others? Thank you. – Seery Jul 2 '19 at 11:43
• @Seery, I am more confused by your comment. I cannot tell what you are asking. The ratios produce the notes of the major scale regardless of the starting point. What is it you are trying to do on G? – ggcg Jul 2 '19 at 12:24

If you're going to measure the consonance of intervals by looking at the simplicity of their frequency ratio, that only makes sense if you actually play the intervals with those ratios, i.e. not in 12TET tuning. So let's assume you've tuned your instrument with the ratios you listed, starting with C at e.g. 1080 Hz (for simplicity):

```C      1/1    1080 Hz
C#    16/15   1152 Hz
D      9/8    1215 Hz
Eb     6/5    1296 Hz
E      5/4    1350 Hz
F      4/3    1440 Hz
F#     7/5    1512 Hz
G      3/2    1620 Hz
Ab     8/5    1728 Hz
A      5/3    1800 Hz
Bb     9/5    1944 Hz
B     15/8    2025 Hz
C'     2/1    2160 Hz
```

Let's look at e.g. the major thirds. The major third from C to E has a frequency ratio of 5/4, and some other major thirds in the scale do too, but some have a different ratio:

```C  - E     5/4 :  1/1     5/4
C# - F     4/3 : 16/15    5/4
D  - F#    7/5 :  9/8    56/45    1.2444444...
Eb - G     3/2 :  6/5     5/4
E  - Ab    8/5 :  5/4    32/25    1.28
F  - A     5/3 :  4/3     5/4
F# - Bb    9/5 :  7/5     9/7     1.2857142...
G  - B    15/8 :  3/2     5/4
Ab - C     2/1 :  8/5     5/4
A  - C#   32/15:  5/3    32/25    1.28
Bb - D     9/4 :  9/5     5/4
B  - Eb   12/5 : 15/8    32/25    1.28
```

As you see, the ratio 5/4 says something about the major third C-E, and six other major thirds in this tuning, but not about all major thirds, and not about the major third in general.

So let's turn to the idea of changing the tuning with every note. If we play a C and then an E, we now switch to a tuning based on E. Seven notes retain their tuning, five change:

```E      1/1    1350 Hz
F     16/15   1440 Hz
F#     9/8    1518.75 Hz    (1512)
G      6/5    1620 Hz
Ab     5/4    1687.5 Hz     (1728)
A      4/3    1800 Hz
Bb     7/5    1890 Hz       (1944)
B      3/2    2025 Hz
C'     8/5    2160 Hz
C#'    5/3    2250 Hz       (2304)
D'     9/5    2430 Hz
Eb'   15/8    2531.25 Hz    (2592)
E'     2/1    2700 Hz
```

If we now play an Ab, and then switch to a new tuning based on Ab, we find that C now has this frequency:

```Ab     1/1    1687.5 Hz     (1728)
A     16/15   1800 Hz
Bb     9/8    1898.4375 Hz  (1944, 1890)
B      6/5    2025 Hz
C'     5/4    2109.375 Hz   (2160)
...
```

So we started at C=1080 and end at C'=2109.375 Hz instead of 2160 Hz. This means that C is now 39 cents flat from where we started, after playing C-E-Ab-C. If we played an arpeggio C-E-Ab-C-E-Ab-C... the pitch would go flatter and flatter.

Another problem would arise as soon as you start to consider polyphonic parts instead of just a melody. What if one instrument played C-E-Ab-C and ended up 39 cents flat, but another instrument took a different route to arrive at the same note, say E-A-B-E? This second instrument plays perfect fourths and a major second, which means it stays in the same tuning. So you'd start off with a C at 1080 Hz and an E at 1350 Hz, and end with a C' at 2109.375 Hz and an E' at 2700 Hz, which is not a 5/4 ratio. So the two instruments have now gone out of tune with each other.

I guess you could turn this into a usable theory if you confined the melody and harmony to those intervals which either stay in the same tuning, or return to the initial tuning after each phrase. Or if one instrument leads the tuning, and the others have to follow. But it would be fiddlier than 12-tone music and microtonal music combined, and the constantly changing tuning would make it a challenging listen.

Let's see what rules for consonant harmony we can come up with using the tuning with "perfect" intervals based on simple ratios. I'll be using a slight variation of your table of ratios, with 16/9 for interval 10, because of symmetry, so that intervals 2+10 equal an octave:

``````DEGR. NOTE  RATIO    FRACT.   (12TET)

0    C      1/1     1        1
1    C#    16/15    1.066    1.05946
2    D      9/8     1.125    1.12246
3    Eb     6/5     1.2      1.18921
4    E      5/4     1.25     1.25992
5    F      4/3     1.333    1.33484
6    F#     7/5     1.4      1.41421
7    G      3/2     1.5      1.49831
8    Ab     8/5     1.6      1.58740
9    A      5/3     1.666    1.68179
10    Bb    16/9     1.777    1.78180
11    B     15/8     1.875    1.88775
12    C'     2/1     2        2
``````

If we check which perfect intervals can be combined while still staying within the C-based tuning, we get this table:

``````     0  1  2  3  4  5  6  7  8  9 10 11

0   *  *  *  *  *  *  *  *  *  *  *  *
1   *  -  *  -  *  -  -  *  -  *  -  *
2   *  *  -  -  -  *  -  -  -  *  *  -
3   *  -  -  -  *  *  -  -  -  *  *  *
4   *  *  -  *  -  *  -  *  *  -  -  -
5   *  -  *  *  *  *  -  *  *  -  -  *
6   *  -  -  -  -  -  -  -  -  -  -  -
7   *  *  -  -  *  *  -  *  *  *  *  -
8   *  -  -  -  *  *  -  *  -  *  -  *
9   *  *  *  *  -  -  -  *  *  -  -  -
10   *  -  *  *  -  -  -  *  -  -  -  *
11   *  *  -  *  -  *  -  -  *  -  *  -
``````

If you want to check whether two notes will form a perfect interval, check whether the combination of their scale degree is marked with a star; e.g.:

From C to E is interval 4, from E to G is interval 3, 4 combines with 3, so E-G is a perfect interval. From C to D is interval 2, from D to F is interval 3, 2 does not combine with 3, so D-F is not a perfect interval.

To check whether you can add a third note to e.g. E-G, we need to check the intervals for both notes; so e.g. the intervals from C to A via E-G are 4+5 and 7+2; 4+5 is ok, but 7+2 is not. So we can not combine E-G-A. However, the intervals from C to B via E-G are 4+7 and 7+4, and 4+7 is ok, so we can combine E-G-B and have perfect intervals.

The note that the tuning is based on can be added to any combination of notes, so in this case, we can add C to get a C-E-G-B chord. Let's see if we can make that a major ninth C-E-G-B-D. From C to D via E-G-B is 4+10, 7+7 and 11+3; of these, 4+11 is not ok, so we can't make a Cmaj9 chord with perfect intervals. (We could use C-G-B-D without the E, though).

Equally, an Fmaj7 is ok, but an Fmaj9 is not: C to A via F is 5+4=ok, C to C via F-A is 5+7 and 9+3 which are ok (because of the ratio symmetry any two intervals that form an octave are ok, except 6+6) and C to E via F-A-C is 5+11, 9+7 and 0+4, which are all ok; then, C to G via F-A-C-E is 5+2, 9+10, 0+7 and 4+3, of which 9+10 is not ok.

So you can work out every combination of notes which contains only "perfect" intervals in this tuning. From the examples, we already have the chords C, Cmaj7, Em, G, F, Fmaj7, and Am. We also know that Cmaj9 and Fmaj9 contain imperfect intervals in this tuning (unless we drop the third). To use a chord that is imperfect in this tuning, you could move to another tuning in which the chord is made of perfect intervals.

With a tuning that uses the above list of intervals, the following combinations of scale degrees have only perfect intervals between them (with 0 being the note that the tuning is based on):

```0  1  3  8
0  1  5  8
0  1  5 10
0  2  3  7
0  2  7 11
0  3  7  8
0  4  5  7
0  4  5  9
0  4  7 11
0  4  9 11
0  5  7  8
0  5  9 10
0  6
```

This is of course also true for any two or three scale degrees from one of these combinations, like `3 7` or `0 5 9`.

When the tuning is based on C, these are some of the chords that can be played:

```NOTES         CHORDS

C-Db-Eb-Ab    Ab, Dbsus2
C-Db-F-Ab     Db, Dbmaj7, Fm
C-Db-F-Bb     Bbm, Fsus4
C-D-G-B       Csus2, G, Gsus4
C-Eb-G-Ab     Cm, Ab, Abmaj7
C-E-F-G       C, Csus4, Fsus2
C-E-F-A       F, Fmaj7, Am
C-E-G-B       C, Cmaj7, Em
C-E-A-B       Am, Esus4
C-F-A-Bb      F, Fsus4
C-F#          tritone
```

There are also quartal/quintal chords: `C-F-Bb`, `Eb-Ab-Db`, `G-C-F` and `B-E-A`. Some chord types are obviously missing, such as diminished and augmented triads, dominant sevenths, and any chord built with more than four notes.

When creating melodies, you can use notes from one of these combinations, and have perfect intervals between consecutive notes. You can also switch to a different combination via a common note. You'll see that you can never combine e.g. a G and an A without interjecting at least one other note, e.g. C, E, F or B.

• @Seery Whatever this sort of approach leads to, it's going to be experimental. Western musicians have tried out all kinds of tunings for centuries, and finally settled (mostly) on 12TET, because every system is an approximation anyway. And theories about consonance and dissonance that use physics are abstractions that ignore the physical reality of acoustic instruments. But electronic instruments offer possibilities, of course. I'd advise you to look into the "spectralist" music of Tristan Murail, if you want to mix physics with music. – Your Uncle Bob Jun 27 '19 at 1:00
• @Seery Btw, there's an inconsistency in your choice of ratios. Intervals 1+11, 3+9, 4+8 and 5+7 all add up to an octave, but 2+10 doesn't. It would do if you used 10/9 for 2 or 16/9 for 10. – Your Uncle Bob Jun 27 '19 at 1:05
• @Seery The thing is that, while there's some physics at the basis of music, the human ear and mind aren't precise frequency and phase measuring tools, and the link between music and physics isn't mathematically rigorous, and physics and mathematics can only describe by approximation how we hear music. If you scientifically analyze your favourite recording, you'll find that not a single note is actually completely in tune for any length of time, but it sounds good once it's passed through the imperfections of your ears and been interpreted by your brain. – Your Uncle Bob Jun 27 '19 at 1:19
• @Seery Music theory as we know it is actually more like engineering than like physics or mathematics. It's not all perfectly precise and mathematically rigorous, but it's good enough to be useful. And you have to ask yourself: do I like the sound of a piano? Because that's one big mess of out-of-tune harmonics. And do I like Chopin and Debussy? Their music probably doesn't strictly follow any rule based on physics or mathematics, but it sure is "good enough" :-) – Your Uncle Bob Jun 27 '19 at 1:39
• @Seery Bach definitely understood mathematics and acoustics. He understood very well that it is not possible to tune a keyboard purely in the ratios you mention in the question (for general use, at least). Instead, a keyboard must be "tempered," which in those days meant there would be one or more very out-of-tune "wolf" intervals. He apparently liked to tease organ builder Gottfried Silberman by playing his organs so as to emphasizing the wolf, but he avoids certain intervals in the Well Tempered Clavier, presumably because these were wolfish in the temperament he was using. – phoog Jun 27 '19 at 2:58

Most sources that give that kind of order of consonance will have generated it with simultaneously-sounding notes, rather than successive notes. So it's concerned initially with Harmonic dissonance, rather than Melodic dissonance. Its immediate utility would be enabling you to find a way to measure the consonance or dissonance of a chord.

But is consonance and dissonance not utilised in single note melodies as tension and release?

Yes - and accordingly, there is also the concept of melodic consonance and dissonance. This results from the way that human listeners remember previous notes, and hear later notes in relation to them.

Let's say my melody in the key of C begins with the note C and the following note is G.. Do these ratios for C no longer apply once I've reached the note G, but in turn I would need now to shift the ratios to that of G or could I use C's ratios throughout an entire composition once I remained in the key of C?

This is where it gets complicated. There are many different relationships in a piece of music - you might even consider that every note in the piece has a relationship with every other note in the piece. Of course we don't necessarily perceive things in terms of all these relationships. Instead, from hearing all these relationships between notes, we often come to sense an overall sense of the "home note" of the piece, and hear melodic notes with respect to this. However, this sense of 'home note' can change - which is why a single-note melody can still suggest a tonic, outline chords, and even have local tonicizations.

So I think the answer to your question is "both". As soon as you have three notes, there are three relationships happening - between note 1 & 2, note 2 & 3, and note 1 & 3. Which are most important will depend on the phrasing and accenting of the notes. As you get more and more notes in a piece, the number of relationships increases dramatically.

You might find the discussion on this answer interesting, as it talks about the different levels of context in which a note might be considered consonant or dissonant.

Much of this is subjective, too. if I'm listening to a jazz musician and they play a fragment of the melody that they're improvising on, my perception of the next part of the piece might depend on whether I recognised the fragment and started to hear other parts of the piece in its context. A dissonant note or chord that's 'expected' (either through familiarity with a particular piece, or with the genre) might seem less dissonant than one that surprises.

Taking all this into consideration, we can see that there are a lot of variables. You might have heard the phrase “an art is a science with more than seven variables”; Consideration of consonance and dissonance definitely gets into the realm of "art" pretty quickly!

• Thank you very much for your informative answer topo. I was thinking that if we look at chord progressions in the sense of tension and release, we could apply the same approach to a melody with the only difference being singular notes as there is obviously consonance and dissonance between two intervals even if they're played in succession as opposed to simultaneously. Would you agree with this statement? – Seery Jun 27 '19 at 0:31
• The question would then become, do we shift the ratios every new note of the melody but which would destroy a composition within a single scale or is it not common to compose outside of an original key of a song, called modulation? Or is it a better option to only use ratios of consonance and dissonance which it's notes only pertain to the key you're on? This approach would be similar to composing within the scales notes harmonic series instead of the ratios which I've done before but felt limited. What would you reckon? Many thanks again topo – Seery Jun 27 '19 at 0:35
• @Seery The difference between harmonic and melodic intervals is that dissonance is much more significant with the former. A perfect fifth of 40:27 sounds horrible harmonically but far less so methodically. – phoog Jun 27 '19 at 1:32
• @Seery when hearing melodies in terms of tension and release, I don't think I primarily hear the consonance/dissonance between successive notes. Personally, I think the melodic consonance/dissonance I hear is primarily the dissonance between a note and whatever I perceive to be the local tonic of the piece (which, as you say, may be due to a modulation). – topo Reinstate Monica Jun 27 '19 at 6:48
• So you could come up with a heuristic that determined what the local tonic was at any given time, and calculate the consonance of a given note with respect to that. Of course while a piece is modulating, it might be that the listener hears a note with respect to both the old and new tonic - which is why modulation sounds interesting. – topo Reinstate Monica Jun 27 '19 at 6:54

Consonance and dissonance refer to notes that sound simultaneously. A C and a B sounding simultaneously are relatively dissonant. A G and a B sounding simultaneously are relatively consonant. Therefore:

Let's say my melody in the key of C begins with the note C and the following note is G.. Do these ratios for C no longer apply once I've reached the note G?

Correct. For example, the interval between C and B is a major second, which is dissonant, while the interval between G and B is a major third, which is consonant.

or could I use C's ratios throughout an entire composition once I remained in the key of C?

No. For example, the ratio between C and A-flat is a minor sixth, which is relatively consonant, but between G and A-flat is a minor second, which is relatively dissonant.

• @Seery if you are composing a melody of single notes, then you do not need to worry about consonance or dissonance, because you will never have more than one note sounding at the same time. If you're working with chord progressions, it's not really that difficult. You just learn which intervals are consonant and which are dissonant. You're not constantly recalculating ratios. It's a bit like reading: you don't actually read every letter individually unless you encounter an unfamiliar word. Your second chord, G major, has the same shape as your first chord, C major. – phoog Jun 25 '19 at 2:09
• But is consonance and dissonance not utilised in single note melodies as tension and release? What other formula would serve in creating a beautiful sounding melody if it's not the interplay between tension and release? Maybe it is also worth noting that these melodies would be later combined with harmony and rhythm also if that has any relevance to the context. – Seery Jun 25 '19 at 2:24
• @Seery, I am very confused by your conclusions regarding the ratios. These ratios are based on harmonics. The relative ratio of the notes of the major scale do not change with note or degree, or key. However every note played produces a harmonic structure. For example, on the guitar every note plucked produces the major triad from the harmonics. This can (and does) create dissonance with the other notes in the key. – ggcg Jun 25 '19 at 13:35
• @Seery, but you don't have to consciously change key with every note (my interpretation of your post). – ggcg Jun 25 '19 at 13:36
• @Seery yes there is of course the issue of melodic tension, but I do not think it is so strongly bound to frequency ratios. For example, those kinds of music in which melody plays a much more important role than harmony tend to use so-called microtonal inflection to that end. In any event, the ratios are more complicated than what you present: a major second can also be 10:9, a half step can be 25:24, and a major third can be 81:64. – phoog Jun 25 '19 at 14:06

You did not mention a musical style so I will assume common practice along with some melody guidelines from Fux.

When I compose a melody in the key of C for e.g. do these ratios of consnance/dissonance apply

As other answer have pointed out these intervals are harmonic intervals for simultaneous tones. They don't really apply to melody.

(from comments) But is consonance and dissonance not utilised in single note melodies as tension and release?

Yes. Tension and resolution are aspects of Melody

...What other formula would serve in creating a beautiful sounding melody if it's not the interplay between tension and release? Maybe it is also worth noting that these melodies would be later combined with harmony and rhythm also if that has any relevance to the context

You use the "formulae" of harmony and melody. There is lots of pedagogy on both topic.

What I read in your question is you expect melody to operate in some way derived from the ranking of the consonance/dissonance of harmonic intervals. That is not how melody works. I think you can understand the problem in that expectation by considering how a melodic leap is perceived in harmonic context.

The follow isn't an exhaustive list, but should lay out the idea. When a melody leaps, for example up a minor sixth, it can work in different harmonic contexts:

• it could be a chord arpeggiation like `^3 ^1` over `I`
• it could be a leap between two chords using chord tones like `^1 b^6` in `i iv6`
• it could be a leap between two chords using an appoggiatura like `^1 ^6` in `I V6`

...the point here is that all melodic leaps are not equal and perception of chord tones (even those implied by an unaccompanied line) matter.

An arpeggiated leap does not have the tension and need for release that an appoggiatura does.

If we look at things from a more pure linear perspective and think of the guidelines of Fux, for the most part a leap is just a leap and they all are consider to carry some amount of "tension" that needs to be resolved by a step in the opposite direction of the leap. That melodic guideline applies regardless of the size of the interval. Leaps by third, fourth, fifth, sixth are essentially the same in that sense of a "tension" needing resolution.

But, as you might expect, there are exceptions in Fux. The largest leap he allowed was a minor sixth. And, various melodic outlining of a tritone were forbidden. I never really understood the leap size limit. It seems kind of arbitrary. But the avoidance of a melodic tritone seems clearly related to the implied harmonic sound of a tritone.

But the big problem with applying harmonic consonance/dissonance to melody comes when considering step-wise motion. The harmonic list puts whole tones and half tones at the bottom as harmonic dissonances. If you mis-apply that to melody, you would treat conjunct (step-wise) motion as loaded with tension. Melodically that in not the case. Conjunct motion is the basic melodic motion. Conjunct motion to flow freely without any sense of tension building.

More importantly mis-applying harmonic dissonances to melody overlooks the importance of tendency tones and especially the half steps. In tonal music some tones within a scale have a tendency to move to others. `FA` moves down to `MI` and `TI` moves up to `DO` are the essential tendencies. Those half step movements are considered dissonant or tensions (as they appear on the harmonic chart), instead they are resolutions to the tonic chord tones.

Do interval ratios consonance apply to a single pitch or a scale?

They apply only to the two tones being played.

(from comments) fast track to summarised, vague and extensively time consuming music theory ...Beethoven also wrote pieces while deaf so I'm set on the idea of some mechanics.

You hope this will be a fast track. But, if you hope to make music along the lines of Bach, Beethoven, etc. you will only make the path to understanding more difficult. Music isn't a physics application, it's the art of music. As an art it is subjective. That subjectivity is exactly why physics won't be the short cut you hope for.

Beethoven learned how to compose before he went deaf. He was able to keep composing, because had learned harmony and melody when he could hear!