# Defining intervals.

There seems to be a lot of confusion over naming intervals. Some seem to think it involves major intervals being from major keys/scales, thus minor intervals need to follow from that. Some seem to think the number of semitones difference defines the interval. Neither are spot on. What is a simple way to define intervals, that works for all of them?

• Haha, I was posting a similar question at the same time. Mine is about interval names as opposed to something like the amount of frequency separation. – jdjazz Aug 3 '17 at 15:05
• I thought mine was too. But there is confusion (on internet sites in particular, about whether say C>D# or C>Eb is a min3, et al...We both know the answer!, but a simple explanation would maybe help clear up this confusion.I don't look forward to answers which quote Hz, etc... or even percentages of same. – Tim Aug 3 '17 at 15:09
• Agreed--I also was thinking about what I actually wanted to ask, and I think the answers would be basically the same as those to this question. – jdjazz Aug 3 '17 at 15:10
• My comment was in another thread where the OP was specifically talking about the size of major and minor thirds, not augmented seconds! It wasn't intended to be a complete discussion on intervals, just a comment pointing out that the OP's question was based on logical deduction starting from an incorrect assumption! – user19146 Aug 3 '17 at 16:23

I think it helps a lot, at first, to separate the two parts of an interval name and look at them in isolation, and then go back to combining them. The context for interval names that I will consider in this answer is strictly "Western" tunings and scales, meaning based on the notes A B C D E F G (or Do Re Mi, etc. if you prefer).

## The Numerical Part

The most significant part of an interval name is the "numerical" part, e.g., third, fourth, ninth, fifteenth, etc. And here's something that I think confuses a lot of people: the name of this part of the interval name is based on the names used for the two notes that make up the interval.

To dive deeper into that last sentence, the interval name for the interval made up of C and E# is a different interval name from the one composed of C and F, even though E# and F are (generally) the same pitch (they are enharmonic).

That doesn't make a lot of sense when you are looking at the piano keyboard, because you are literally playing the same notes in both cases. It makes a lot more sense when you are looking at sheet music and arranging and transposing parts and generally working with music theory. Right off the bat, it's clear that writing E# in a score looks different from writing F.

So to determine what the numerical part of an interval name should be, you start by knowing/determining the note names of the two notes in the interval. Ignoring all accidentals (sharps, flats, double sharps, naturals, whatever), merely count the distance between the letters, starting with 1 for the same note (so C to the same C is 1, C to D is 2). When intervals are wider than an octave, you keep counting (it doesn't reset to 1), so that's how there are 9ths and 15ths, etc.

## The Quality Part

Once the numerical part is determined, that forms the basis for the "quality", but the quality is determined using the number of half steps. Here's a table of sorts showing the most common interval name and then a less common name and the half steps between them:

• 0 half steps - Unison
• 1 half step - Minor second, augmented unison
• 2 half steps - Major second, diminished third
• 3 half steps - Minor third, augmented second
• 4 half steps - Major third, diminished fourth
• 5 half steps - Perfect fourth, augmented third
• 6 half steps - Tritone, augmented fourth, diminished fifth
• 7 half steps - Perfect fifth, diminished sixth
• 8 half steps - Minor sixth, augmented fifth
• 9 half steps - Major sixth, diminished seventh
• 10 half steps - Minor seventh, augmented sixth
• 11 half steps - Major seventh (not sure I've ever heard of "diminished octave" before)
• 12 half steps - Octave, Augmented seventh, diminished ninth

After that, the quality pattern repeats but with the numbers continuing to increase.

• I guess, if an aug unison lives, then a dim octave ought to have a chance! – Tim Aug 3 '17 at 16:42
• When determining the two letter names (numerical part above) do we need notation on staff to get the right spelling? I think the answer is yes, otherwise we need to state assumptions about key signatures and spelling tertiary chords in thirds. Ex. D# v. Eb. In C minor Eb would be the normal spelling unless D# was actually written in the notation. – Michael Curtis Aug 3 '17 at 19:15
• @MichaelCurtis Good question. I think if I tried to dive into all the ways to know what the note names would be (e.g., just having a nominal key for a piece or section implies what note names should be used, even where there is no notation), this answer would have to be so long and involved that it would make the question seem too broad. – Todd Wilcox Aug 3 '17 at 19:19
• @ToddWilcox, it is hard to think of concise way to write it. I was trying to tease apart the 'confusion' some people have with the numerical part. I suppose that confusion is especially for folks who can't read the staff or don't know key signatures. – Michael Curtis Aug 3 '17 at 19:26
• After a re-read, a small comment. In the 1st para it says ABC etc, or do re mi.The two aren't substitutes, unless it reads 'C D E or do re mi'. A B C# would do, but may confuse! A slight edit? – Tim May 2 at 6:54

Intervals are two distinct measurements: distance in semitones and distance in letter name. The type of interval (unison, 2nd, 3rd, ect) is based off of the distance between the letter names. The quality (i.e Major, minor, perfect, ect) is based off of the "typical" space that the type of interval occupies. The following I have posted before on other questions asking about why a quality is named the way it is and I think it applies well to this question.

To better explain why the intervals ended up this way, let's look at this solely from a distance perspective at first. The distances from unison to octave are as follows in semitones:

``` 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12
```

In C these notes would map to:

``` C - Db - D - Eb - E - F - F#/Gb - G - Ab - A - Bb - B  - C
```

As you can see, both 0 and 12 map to C and the furthest you could be away from a C in semitones is 6. This leaves us with 5 notes on each side with 1 - 2 - 3 - 4 - 5 closer to 0 and 7 - 8 - 9 - 10 - 11 closer to 12.

Now let's look at the standard interval names. In this I will use M for major, m for minor, P for Perfect and tt for tritone (which is considered both an augmented 4th and a diminished 5th).

``` P1 - m2 - M2 - m3 - M3 - P4 - tt - P5 - m6 - M6 - m7 - M7 - P8
```

I'll also group the typical groupings. Notice how the tritone is in both the 4th and 5th grouping:

``` P1 - (m2 - M2) - (m3 - M3) - (P4 - [tt) - P5] - (m6 - M6) - (m7 - M7) - P8
```

Now looking at the whole interval spectrum, we notice

• The tritone (or 6 semitones away and equal distance from both 0 and 12) has a perfect interval above and below (P4 - tt - P5) and can be described as both an A4 and d5 for this reason (because they both are "contesting" the tritone).
• The note you are basing the name off of (C in this case which is both 0 and 12) is also perfect (P1 for unison P8 for octave) .
• The other notes are group into twos (because of the two semitones typical max in scales) with the smaller one being minor and the bigger one being major (m2 - M2 - m3 - M3) and (m6 - M6 - m7 - M7).
• The augmented and diminished intervals of the major and minor intervals are for when one of the intervals stretches out of its typical designation.

Now taking a step back, if there is one "typical" space for the note the normal quality is prefect and if there are two the bigger one (going upward) will be major and the smaller one will be minor. Anything bigger than the normal interval (i.e. C to F♯, C to D♯, ect) is considered augmented in quality. Anything lower (i.e. C to G♭, C to B♭♭, ect).

• Good answer, as ever - haven't seen you post for a while - but what reasons are there for the Perfect intervals not being called maj/min? – Tim Aug 3 '17 at 15:27
• @Tim because they have one defined space and one "contested" space. think of it like the 4th and 5th both can't claim it as there normal space. – Dom Aug 3 '17 at 15:30
• "Theoretically" there's infinite, but I've never seen anything above "Tripply" and even that was just a theory exercise. – Dom Aug 3 '17 at 15:39
• @Tim contemporary music notation sometimes includes triple-flat and triple-sharp accidentals on any note. An interval of Dbbb to A### should presumably be named as some sort of 5th, even though it's actually a semitone wider than an octave! – user19146 Aug 3 '17 at 16:26
• I have never been puzzled by the use of perfect for fourths and fifths. It is the minor second that puzzles me as the second is still major in a minor scale. – badjohn Aug 3 '17 at 16:36

There exist only three basis intervals in Western music: octaves, fiths and major thirds. Everything else is compound. Ignore the naming for now, those intervals can be defined as pure mathematical ratios:

• The octave is the simplest interval, created by doubling the frequency.
• The fifth is created by multiplying the frequency with 32.
• The (major) third is created by multiplying the frequency with 54.

All together, these form the classic narrow voicing of a 4-voice major chord, for example C3 - E3 - G3 - C4. That sounds nice but quickly gets boring; to enable leading actual melodies, we fill the gaps by stacking some of those intervals upon already-constructed notes. The conventional way to do this is to extend one extra major chord (the dominant) from the fifth, and one chord whose fifth is the original fundamental (the subdominant). This way, we end up with a diatonic scale.

This is a (Ptolemy-) diatonic scale, frequencies in log-domain:

If you plot all the ratios between two notes in that scale, you'll observe two pretty tight clusters, some rather diffuse blobs, and one lonely oddball:

About that loner we shall not speak, the other groups we give numbers (starting at, erm... 2), giving us the rough intervals. For the “blobs” we always still see a pretty clear division into two subgroups; these are the fine intervals.

We thus end up with the classification

• `E-F`, `B-c`: minor second
• `C-D`, `D-E`, `F-G`, `G-A`, `A-B`: major second
• `D-F`, `E-G`, `A-c`: minor third
• `C-E`, `F-A`, `G-B`: major third
• `C-F`, `D-G`, `E-A`, `G-c`: pure fifth
• `C-F`, `D-G`, `E-A`, `G-c`: pure fourth
• `E-c`: minor sixth
• `C-A`, `D-B`: major sixth
• `D-c`: minor seventh
• `C-B`: major seventh

Disclaimer: this derivation is historically debatable, because especially in the middle ages, the major third was not considered a basis interval, instead it was constructed by stacking four fifths on top of each other. But those Pythagorean thirds don't sound consonant at all, and the Pythagorean scale's ratios actually don't exhibit a “cloud patterns” structure (in fact, the Pythagorean minor third is closer to the major second than it is to the major third).

• Another interesting answer! Right now, I'm hard pushed to decide on an 'accepted' one. thanks. – Tim Aug 4 '17 at 7:08

If we write out the scales, we see where the intervals get their name (or at least the numerical part of it.) This is done below (using the scales of natural notes A minor and C major for convenience.) All the intervals are named as the difference between the root note and the number of the degree. For example the Major 6th is the difference between the root and 6th note of the major scale (C and A in the scale of C major.)

Where necessary, an interval which exceeds the normal bounds can be described as "augmented" (1 semitone more than a perfect or major) or "diminised" (1 semitone less than a perfect or minor.)

In fact, the only way to name the 6 semitone interval in this system is to call it an augmented 4th or diminished 5th, as this "tritone" or "devils interval" does not feature as an interval between the root and another note in either the major or minor scale.

`````` Semitones Aminor Cmajor

0    A     C       Unison
1                  Minor 2nd***
2    B     D       Major 2nd
3    C             Minor 3rd
4          E       Major 3rd
5    D     F       (Perfect) 4th
6
7    E     G       (Perfect) 5th
8    F             Minor 6th
9          A       Major 6th
10   G             Minor 7th
11         B       Major 7th
12   A     C       Octave
``````

But WAIT! the Second note of both the major and minor scale is 2 semitones! Why is the 1 semitone interval called a 2nd at all?!

This is the ONLY exception (unless you count the 9th, which is the same thing an octave higher). I have heard that the Phrygian mode was more popular in the past, and if we change the B in A minor to a Bb, we obtain A Phrygian and everything falls into place.

For a long time I thought this little oddity of naming was done simply to conveniently fill in the gap and find a name for the 1 semitione interval. After all, the Italian "Minore secondo" simply means "lesser second" and "Maggiore segundo" means "greater second."

With this in mind however, here's a look at the different intervals which exist in all 7 modes:

``````                                       1 1 1
Semitones          0 1 2 3 4 5 6 7 8 9 0 1 2

Name               P m M m M P   P m M m M P
1 2 2 3 3 4   5 6 6 7 7 8

Ionian (Major)     C - D - E F - G - A - B C
Dorian             D - E F - G - A - B C - D
Phrygian           E F - G - A - B C - D - E
Lydian             F - G - A - B C - D - E F
Mixolydian         G - A - B C - D - E F - G
Aeolian (Minor)    A - B C - D - E F - G - A
Locrian            B C - D - E F - G - A - B

Total ocurrences   7 2 5 4 3 6 2 6 3 4 5 2 7
``````

Looking at this, we see the 2-semitone interval occurs in 5 of the modes, functioning as a second, and the 1-semitone interval occurs in the other 2 modes, also functioning as a second. Hence major and minor second. If we check all possible modes, we see both the major and minor forms of the Second in use!

Similarly the 4-semitone interval occurs in 3 modes, functioning as a third, and the 3-semitone interval occurs in 4 modes, also functioning as a third. Hence major and minor third.

A similar thing happens with the 6ths and 7ths. Hence we have two equally valid standard forms of the 2nd, 3rd, 6th and 7th: a minor one, and a major one.

Note that the 6-semitone interval (the tritone) occurs twice. In the Lydian mode it functions as an augmented 4th, while in the Locrian mode it functions as a diminished 5th. Therefore this interval cannot be readily classified as a typical variety of 5th or a typical variety of 4th (though the term "diminished 5th is rather more often used than "augmented 4th", due to the way in which chords are constructed.)

So we see also why there is therefore only one "standard" form of the 4th and the 5th, given the name "perfect", with the tritone being an oddball.

• Do you have any sources for this meaning behind interval names? As far as I know, some of this is incorrect. – Todd Wilcox Aug 4 '17 at 0:41
• @ToddWilcox Thanks for your comment. I think I had it backwards (but for years I didn't care because, with the exception of the minor second, it was working for me.) The truth is, the major and minor scale are most likely named after the major and minor intervals, and not the other way round. But as I say, the Major mode contains all four Major intervals, whereas the Phrygian (not the Minor) is the one that contains all four minor intervals. Checking all 7 modes reveals plenty of seconds of both the major and minor variety. – Level River St Aug 4 '17 at 22:16
• My initial intention was to provide a simple answer defining the more basic intervals (see my first table) as I thought it was missing from the page, but it has grown into something bigger! – Level River St Aug 4 '17 at 22:51