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Are intervals like major 3rd, minor 3rd, and major 2nd all based on the scales, or are they based on how many semitones they have? I ask this question because if I play the C major scale, and I press down on the second note, D, I notice that it is two semitones away from the root, C. If intervals were based on scales, then the major 2nd interval would be on that second note, right? That's because the major 2nd would correspond to the second note on the major scale. But if I press the second note of the C minor scale, I notice that it is also two semitones away from the root. If that's the case, then how would the minor 2nd interval correspond to the second note of the minor scale? Isn't a minor 2nd supposed to only have one semitone? That may all sound like nonsense (it probably does) because I definitely don't have a strong understanding of all this. Could someone just try to explain the whole idea to me? Thank you!

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The Basics: Interval Quality and Quantity:

Every interval has a quality-type and a quantity-size: For example, a Major 2nd interval. Its quantity is a 2nd, its quality is major.


Interval Quantity: The interval quantity: 2nd, 3rd, 4th etc - is dependent on its spelling in the scale you are working with. Every complete scale in the western tonal system is comprised of 7 notes: A-B-C-D-E-F-G. You can start with any note and follow that same order to make a complete scale: C-D-E-F-G-A-B, etc. (Sharped or flatted notes still have the same names - Ab,Bb, etc -the # or b is simply an adjective - a modifier specifying a particular characteristic of that note name.)

The interval quantity is determined by its spelling in the scale - If you are going from A to C, that is a 3rd: Reckoning A as 1, C is the 3rd note you spell in the scale. If you are going from C to G, that is a 5th:Reckoning C as 1, G is the 5th note you spell in the scale. When naming an interval, that is first thing you must determine: its quantity - its position in the spelling of the scale you're dealing with.

Once you know its quantity - 2nd, 3rd, 4th etc - you can begin to determine an interval's quality - major, minor, etc.


Interval Quality: The first thing we need to know with respect to interval quality - the key to it all: Major in the musical context means large, not important. Minor means small, not unimportant. Likewise, Diminished and Augmented, as will be explained.

The interval quality - Major, Minor, Diminished, Augmented, Perfect - is determined by the number of chromatic steps (commonly called half-steps), that comprise the interval.

There are several rules about how that works. In the musical alphabet as we outlined it, there are 3 possible counts of chromatic steps/semitones between adjacent notes: 1,2, or 3. C to D - is 2 semitones, a large (relatively speaking) count/distance. C to Db is 1 semitone - a small count/distance. C to D# 3 semitones - a larger count/distance.

That count of semitones - the objective chromatic distance between 2 notes in the scale alphabet - is how we determine interval quality, after we know its quantity, as already explained. The general rule for 2 adjacent notes:

  • 1 semitone==minor - a small distance, for example C to Db;
  • 2 semitones==major -- a large distance, for example C to D;
  • 3 semitones==augmented -- a larger distance, for example C to D#;

The same formula applies when dealing with an interval spanning more that just 2 notes, for example F to A:

  • F A is 4 semitones or 2 major 2nds - a large distance, making it a major 3rd.
  • F to Ab is 3 semitones - a smaller distance, making it a minor 3rd.
  • F to A#is 5 semitones - a larger distance, making it an augmented 3rd.

The rules for when we call an interval major, minor, diminished or augmented are actually more complicated than that, and to get into the technical details of how and why that is will make this already very long answer excessively long. The short version:

When we decrease the size of a Major interval by one half step, the interval becomes Minor (small) - so from C to Db is a Minor 2nd - it is still a second, because it's spelled with D - its fundamental quantity has not changed. However, its quality has changed - it has become Minor - small - since it is now an interval of only one half step (one chromatic step).

Reduce a Minor or Perfect interval by one half step, it becomes Diminished - not just small (minor) but reduced: Diminished.

Increase a Major or Perfect interval by one half step, it becomes Augmented- not just large (major) but increased/expanded: Augmented.


Enharmonics:

Where most get confused is when it comes to enharmonics: Intervals that have the same objective quantity - X number of chromatic steps - but can be spelled in two different ways. To use @Dom 's example from the comments, C to D# and C to Eb are both intervals of three chromatic steps, however they are two different intervals - enharmonics:

C to D# is an Augmented 2nd. 2nd - is its quantity, because it is spelled with D. Augmented, its quality, is because we have increased the Major 2nd, C to D, by another half-step, thereby augmenting it.

C to Eb is an Minor 3rd. 3rd - its quantity, is because it is spelled with E, the third letter in the scale, counting C as 1. Minor, its quality, is because we have decreased the Major 3rd, C to E, by a half-step, thereby making it minor - small.

How do you know which enharmonic to use? Your scale spelling tells you: If there is a C and a D beforehand, you already have a 2nd, so the next note in your scale will be E - a 3rd of some sort. For example in the C Minor scale: C is 1 - the root, D is 2 - a 2nd, so the next note will be an E - a 3rd. But since we are spelling the minor scale, it is Eb - which happens to be a minor (small) 3rd, although not every interval in the natural minor scale is minor, as will be explain later in more detail.

However, let's say we are spelling a scale that called for an augmented 2nd following C (Offhand I cannot think of a conventional scale or mode that would have that configuration, but it there is one - or we just invented it...) Then you would write D# - D because you need a second, augmented because you have augmented/expanded the major 2nd to three chromatic steps.


With respect to your question about the 2nd note in "the plain vanilla" minor scale (generally referred to as the natural minor scale): The 2nd note in the natural minor scale has an interval of a MAJOR 2nd - A to B; C to D, etc. When we talk about the quality of a scale - major or minor, for example - we don't mean that every interval must be either major or minor. Generally the quality of the 3rd and 7th notes of a scale (in particular the 3rd) determine how we refer to a scale:

If a scale has a Minor 3rd and a Minor 7th, it is called a minor scale. If it has a Major 3rd and a Major 7th it is called a major scale. (Again - the 3rd is all important: The jazz scale called the melodic minor scale has a minor 3rd and a major 7th.)

The reason for that is because when we discuss any scale as a whole entity, we listen to its general tonality, not every interval in particular. Humans seem to be hard-wired such that in a scale of 7 notes, those two notes, the 3rd and the 7th, are the ones our ears hear as determining the tonality - the musical feel or mood - of a scale: We call Major Major because the sound of the scale to our ears is dominated by the Major (Larger) intervals, especially the Major 3rd and the Major 7th. We call Minor Minor because the sound of the scale to our ears is dominated by the Minor (Smaller) intervals, especially the Minor 3rd and the Minor 7th.


Bottom Line:

This is a quite a bit of material to digest. But after doing so, it becomes apparent that your original assumption - that a note is major or minor because of the quality of its scale - turns out be the opposite. The quality of a scale is determined by its notes:

Consider the C Minor Scale. A more technical description for the C Minor Scale is the 6th (Aeolian) mode of the the Eb Major Scale. Another way of saying that is the the C Minor Scale is derived from Aeolian mode of the the Eb Major Scale.

It works like this: The Eb Major scaled is spelled: Eb-F-G-Ab-Bb-C-D. Starting that scale on C, the 6th note of the scale, gives us C-D-Eb-F-G-Ab-Bb - the C Minor Scale. Because the the C Minor Scale is derived from Aeolian mode of the the Eb Major Scale - the 6th note of that scale - it must be spelled using Eb, and C to Eb is a minor 3rd. It must also be spelled using Bb, and C to Bb is a minor 7th.

The result is that the scale built on the 6th note of the Eb Major Scale - C - must contain a minor 3rd (Eb) and a minor 7th (Bb). Therefore that scale is minor in quality, as explained. So we call it the C Minor Scale: C because it starts with C (the root note) and minor because its 3rd, Eb is minor in quality, and its 7th, Bb is also minor in quality.


Sources:

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    An excellent answer, but I quibble over the 4th para. in 'enharmonics'. It alludes that minor intervals come from minor scales - which is part right only. That statement will be taken by some and run with - which is why we have questions such as these to answer. A bit of theory which anyone can expand upon, and build their own, wrong, ideas on it.Sorry to be picky, but I think it needs re-phrasing, to eliminate any potential anomalies, like 'it's a minor scale, so the 2nd note will be a minor 2nd'. And - intervals don't have to be 'part of a scale', they are merely the distance between 2 ..... – Tim Aug 13 '17 at 6:27
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    .....notes, but, as you rightly state, not only measured in semitones! I appreciate that you've explained it in later paragraphs - but some people never read to the end. Dangerous, but nevertheless true... +1. – Tim Aug 13 '17 at 6:28
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    I think the proper term is 'perfect cadence'... – Tim Aug 13 '17 at 6:46
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    Well, it wasn't imperfect, or plagal, and certainly not deceptive! True, it was interrupted by me. Go on, just take the compliment - I rarely give them! – Tim Aug 13 '17 at 7:04
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    O.K. but now there's a grey area evolving. You mention minor key/scale, only from the natural minor/Aeolian viewpoint, but consider the rising melodic minor scale. The only real minor part of that is the m3. The 6th and 7th notes are firmly major, so something could be construed as confusing at best, awry at worst! – Tim Aug 13 '17 at 17:41
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http://nbviewer.jupyter.org/gist/leftaroundabout/76f62e8702d65709d47e69d5fc94779c

Are intervals like major 3rd, minor 3rd, and major 2nd all based on the scales, or are they based on how many semitones they have?

Neither, really. They're based on frequency ratios. The consonant Western (Ptolemaic) intervals are defined as:

  • Octave (P8) = 2:1
  • Perfect fifth (P5) = 3:2
  • Perfect fourth (P4) = 4:3
  • Major third (M3) = 5:4
  • Major sixth (M6) = P4∗M3 = 5:3
  • Minor third (m3) = P5 ⁄ M3 = 6:5
  • Minor sixth (m6) = P8 ⁄ M3 = 8:5

No semitones nor scales anywhere to be seen.

The reason we chose simple integer ratios: frequencies that are at a small integer ratio give nice, consistent sound signals. For example, a major chord has the following waveform:

Waveform of a JI major chord composed of tweaked-triangular signals

which is much richer than the individual components, but still has a clear repetitive structure that the ear can latch onto. Compare this to a chord composed of random-ish frequencies in a similar range:

Waveform of a detuned chord composed of tweaked-triangular signals

which is just a noisy mess.

From these fundamental intervals, the Ptolemaic scale is constructed:

How the Ptolemaic scale is constructed

In this scale, there are now also other intervals apart from the consonant ones, in particular, the intervals between neighbouring scale degrees, which are called seconds.

Tone-steps in the Ptolemaic scale

Here it is obvious why we talk of minor and major seconds: e.g. the distance (i.e. cross-ratio) between F and G is clearly much more than the distance between E and F, but at first glance seems to be at least similar to the distance between C and D, between D and E etc..

Here are the numerical values:

Frequency-ratios in the Ptolemaic scale

Note that there are actually two different kinds of major second here: the major tone ⁹⁄₈ and the minor tone ¹⁰⁄₉.

They are similar enough that at least in a melody, the discrepancy is hardly noticed. In meantone temperaments, which include the 12-edo tuning used by modern pianos and guitars, minor and major tones are approximated by the same interval.

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The intervals are determined by their distance between notes, and they are a way of describing how far apart two notes are.

You can use the intervals to describe the position of notes in a scale. For example, the Minor scale may be described as containing above the tonic a major second, minor third, perfect fourth, perfect fifth, minor sixth, minor seventh and the octave. The intervals continue past the octave, so playing from C to D would be a major 2nd, playing C to D above the octave would be a ninth.

1|m2|M2|m3|M3|P4|tt|P5|m6|M6|m7|M7|8|m9|M9|m10|M10 &c.

The intervals may be augmented (raised a half step), or diminished (lowered a half step). In this case you generally use the scale or chord that is being built to determine what the interval is called. For example if I am building an Augmented chord, the chord would contain a Major 3rd and an Augmented 5th. Even though the Augmented 5th is the same tone as a minor 6th, in the context of the chord it is taking the place of a 5th, so wouldn't be called the minor 6th.

When describing chords with added intervals, the chord is built up by the thirds, and intervals are added after the main notes. So describing a C9 chord, (C E G Bb D) you would call the intervals a root, a major third, a perfect fifth, a minor seventh and a major ninth. Even if the note D is played in a different position (inversion) other than above the octave, the interval is still referred to in context as the Major 9th due to its position in the building of the chord.

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    An interval is not just based on number of semitones. Or else C to D# and C to Eb would be the same interval. – Dom Aug 13 '17 at 1:36
  • @dom I didn't say they were. To quote myself " intervals are determined by their distance between notes", and like Sarkreth, not trying to get too deep into it, and to quote the wiki: "In music theory, an interval is the difference between two pitches", I think that explaining that the interval name can be contextual to the chord or scale is explanation enough to help with understanding. A comment explaining the difference (or even an answer to the question) from you would be more helpful. I'll edit my answer to include enharmonic equivalents, though. – Alphonso Balvenie Aug 13 '17 at 3:32
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    actually, I don't need to add enharmonics, looks like Stinkfoot covered it in his answer while I was writing my comment. – Alphonso Balvenie Aug 13 '17 at 3:35
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The most uncomplicated way to answer this is to derive it from your major scales. Let's take a look at a major scale example:

G major: G A B C D E F# G

From any major keys, the intervals will be in either major or perfect.

Major: 2nd, 3rd, 6th, 7th Perfect: 4th, 5th, 8ve

If the interval is from G to any note in the scale, it will be major 2nd, 3rd, 6th and 7th.

G-A: major 2nd G-B: major 3rd G-E: major 6th G-F#: major 7th

The rest will be:

G-C: perfect 4th G-D: perfect 5th G-G: perfect 8ve

Subsequently, work on it a step further if the note is not in the scale.

If the distance is shortened, meaning the note moves closer toward your key note or vice versa, then major is reduced to minor.

If the distance is lengthened, meaning the distance is bigger, then major becomes augmented.

If the distance is further reduced from minor, then it becomes diminished.

2nd, 3rd, 6th, 7th: Augmented ~ Major ~ Minor ~ Diminished

4th, 5th, 8ve: Augmented ~ Perfect ~ Diminished

Let's take a look at the examples below:

1: G-E is major 6th (because E is in the G major scale) 2: G-E flat is minor 6th (because it's moving a step closer towards G) 3: G-E double flat is diminished 6th (because it's moving 2 steps closer towards G) 4: G-E# is augmented 6th because it's moving a step further away from G)

Or if the key is G# minor, assume that it's a G major.

1: G#-B (first ignore the #) 2: G-B is major 3rd (B is in the scale of G major) 3: G#-B is a step reduced, so it's a minor 3rd.

Or; 4: G#-B flat is another step reduced from minor third, so it is now diminished 3rd.

Same concept applies to the perfect intervals.

As long as you are clear with the concept, you are good to go. I wished that I can reply with a shorter answer, but you really do need to grasp this well so that it will never bug you anymore.

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The full interval names (two words, like "major" and then "2nd") so indeed correspond to a number of semitones, regardless of which scale or position they are in. The interval between the first two notes of C Minor is actually a "Major 2nd", the same as it is in C Major.

Likewise, the interval between the 3nd and 4th notes in C Minor (Eb and F) is a Major 2nd, while the interval between the 3rd and 4th in C Major (E and F) is a Minor 2nd.

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    An interval is not just based on number of semitones. Or else C to D# and C to Eb would be the same interval. – Dom Aug 13 '17 at 1:35
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    your definition is wrong then. simplification does not mean you can leave out the important parts of the definition. – Dom Aug 13 '17 at 3:34
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    @Sarkreth I don't think you are wrong. I know you are wrong. But that doesn't mean I have nothing better to do with my time than correct every mistake on this site. Oh, and by the way, if you stopped to think before claiming something is beyond the scope of the OP's question, you might have remembered that both major and minor scales contain augmented and diminished intervals! – user19146 Aug 13 '17 at 6:00
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    This is totally incorrect information. I don't often downvote, but if this is still here tomorrow, guess what. If you are correct, please explain why an interval of 3 semitones is a minor third and also an augmented second. Every interval of x semitones has two names. But wotse still - someone has agreed with the answer! – Tim Aug 13 '17 at 6:08
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    Dom doesn't think the rest of us are wrong - only your answer! – Tim Aug 13 '17 at 6:35

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