Interval exercise gone wrong?

Question

Reading "Music Theory for Guitar An Introduction to the Essentials", I’ve tried to do the interval exercises. I am pretty sure there are two errors in the following:

• It should be M7 not m7.
  It is a full 7th interval with no modification, so major.

• It should be m2 not 2+.
  The sharp reduces the interval, so + makes no sense.

Otherwise, then I just don't get it.

Can I get any help from the pros? Thanks for taking time to explaining where/if I am wrong.

Answer 1

I think there are two misunderstandings, and I think one of them hasn't been covered yet.

1. When naming intervals, some of them have a "perfect" version; these are unisons (same note), 4ths, 5ths, and octaves. The others have "major and minor" versions. So when you say "it is a full 7 interval with no modification," that's the misunderstanding. There's no "default" 7th; there's just the wider (major) and narrower (minor) versions. (And also "extra-wide" and "extra-narrow," i.e. augmented and diminished, but let's not worry about that until covering the majors/minors/perfects.)
2. With the F and G, the sharp belongs to the G, not the F. When they're close together like that, it can be hard to tell which accidental goes with what. But if you look closely, the "tic-tac-toe" shape of the # is centered on the same line as the G notehead; that's what counts. (If you were taught "the sharp applies to the note that it's in front of," that's an oversimplification that lets you down in this instance. Since the F and G share the same stem, you can't cram the sharp in "to the right of" the F and to the left of the G.)

Answer 2

The 1st one is indeed m7. Any D>C is a 7th of some sort. Imagine, for a moment, that we're in key D. For the C to be M7, it would need to be C♯. Another criterion is that m7 intervals have 10 semitones between them.

On to the augmented 2nd. F>G♯. F to any G is always a 2nd, of some sort. F♮ to G♮ is called M2. The interval shown is expanded by one semitone, therefore it becomes +2. If the F had been dropped by a semitone instead, (but only to F♭, NOT E), and the G was natural, that again would be +2. And - the space between an interval of +2 is 3 semitones. F>F♯, F♯>G, G>G♯. Also note, that were that G♯ to be called its enharmonic of A♭, then the interval would be called m3. Intervals have two criteria - note names and number of semitones between notes. With intervals, minor means smaller, major larger. Job done!

EDIT: your problem seems to be that you appear to think that intervals are only counted in letter names. Wrong!

Answer 3

it should be M7 not m7 it is a full 7 interval with no modification, so major

It is good that you are look for modifications, accidentals, but when there are no modifying accidentals, the term to use is diatonic rather than major. Let's hold off on further detail for a moment.

it should be m2 not 2+ the sharp reduces the interval, so + makes no sense

Sharps and flats can reduce or enlarge intervals, but it depends on whether the sharp or flat is on the lower or higher pitch. A sharp will reduce an interval when applied to the lower pitch. In this case I think you may have misread the placement of the sharp. It is on the line and applied to the higher pitch. So it enlarges the interval.

I think the main thing you need to understand is diatonic intervals. Diatonic means the pitches and intervals of any key signature without altering accidentals. Without writing a whole summary of interval theory, let's just look at the issue of the seventh.

The mistake you made was not recognizing that the interval D C is a minor seventh. Obviously, you would like to be able to read such intervals correctly and quickly.

All sevenths in C major will be minor sevenths, except those between C B and F E. We can chart that out to see it more easily...

M7       M7
B  C  D  E  F  G  A
C  D  E  F  G  A  B

You can abstract that, and apply it to any key signature, by using pitch degrees instead of specific pitch letters...

M7          M7
^7  ^1  ^2  ^3  ^4  ^5  ^6
^1  ^2  ^3  ^4  ^5  ^6  ^7

...that second example might be described in English like this: the interval between the tonic ^1 and leading tone ^7 is a major seventh, and the subdominant ^4 and the mediant ^3. When you do it in that generic way, you can apply it to any key signature!

Another way to extend this concept it to understand the inversion of intervals. When you invert major or minor intervals, their quality flips. For example if a major seventh is inverted it becomes a minor second. We can invert our general rule for diatonic sevenths to make a rule for seconds: all diatonic seconds are major except those between the leading tone ^7 and tonic ^1 etc.

Understand inversions it a useful short hand, because you can think of the intervallic relationship of two pitches as "two sides of the same coin." For example, in your question's third interval F G# in C major, we can immediately say that second starts as generic F G, and is categorically a diatonic major second. From there, the sharp raises the higher pitch G, and expanding a major second creates an augmented second.

Gaining a solid understand of interval theory is not easy. You won't learning it just a few days. But there is a method to the madness!

You can examine other intervals to find more short hand rules. For example, all diatonic fifths, and by extension their inverted fourths, are perfect fifths, except those between the leading tone and the subdominant (in C major the intervals between B and F.)

                  d5
G  A  B  C  D  E  F
C  D  E  F  G  A  B

                        d5
^5  ^6  ^7  ^1  ^2  ^3  ^4
^1  ^2  ^3  ^4  ^5  ^6  ^7

A rule for thirds/sixths is more of a 60/40 split. All diatonic thirds are minor except those above the tonic, subdominant, and dominant.

One of the take-aways of all this is the importance of relative relationships and the special attention given to relationships involving the two pitches of the tritone (the leading tone ^7 and subdominant ^4 scale degrees.) In your exercise we can quickly see the seventh/second are the unexceptional m7/M2 type and the fifths are the unexceptional P5 type, we then just reduce or enlarge according to the accidentals.

Answer 4

The book is correct in labeling them like that. The problem is that these intervals are labeled due to the number of semitones between them:

0   Perfect unison  
1   Minor second    
2   Major second    
3   Minor third     
4   Major third     
5   Perfect fourth  
6   Diminished fifth/Augmented fourth
7   Perfect fifth
8   Minor sixth
9   Major sixth
10  Minor seventh
11  Major seventh
12  Perfect octave

So essentially the things that can make these trip up are the irregular steps from E to F and from B to C. When starting from C your approach of reading the absence of accidentals as "major" or "perfect" works, but unfortunately that's rather the exception than the rule, elsewhere you need to pay more attention. Now excluding the unison, a seventh has 6 steps, 5 hole tone steps (2 semitones) and one half tone step (1 semitone, in C major from C the step from E to F). So the sum total of semitones is 11.

Now if you look at your example, you're not going from C to B but from D to C. So you have again 6 steps, but this time 4 whole tone steps (D/E, F/G, G/A, A/B) and 2 half tone steps (E/F, B/C). Meaning you're 1 semitone short of the 11 you'd expect for the major interval.

Or in other words you'd have 2 of those E/F, B/C crossings in the interval while the major scale just has 1. So in consequence the interval is minor

Same goes for the augmented second. The step from F to G is a whole tone (2 semitones) so a major second. But as the sharp adds an additional semitone, it becomes and augmented second. So Sharps and flats don't just decrease the interval you need to look at which of the two notes is changed with an accidental. Sharps increase by a semitone, so increasing the upper note enlarges the interval by a semitone, while increasing the lower note shrinks it by 1, vice versa for flats, where lowering the lower note enlarges the interval like the second example of the augmented fifth, while lowering the upper note shrinks the interval.

Answer 5

Errors in understanding

it should be M7 not m7 it is a full 7 interval with no modification, so major

The size of an interval is not determined by whether the notes do or do not include modification.

it should be m2 not 2+ the sharp reduces the interval, so + makes no sense

Sharps raise the pitch, flats lower the pitch. In this case, the sharp moves the upper note further away from the lower note.

Explanation

How intervals are defined

Two factors determine the interval: 1) the letter names involved (i.e., the "spelling") and 2) the absolute distance (i.e., number of half steps) between the notes.

The problems of interval naming

Unequal natural note distances

The core of the problem is that the natural notes (A B C D E F G) are not all the same distance apart. For A to B is an M2, while B to C is an m2. This can be confirmed on the guitar by playing A B C on a single string. Moving from A to B will be a move by two frets, but B to C will be a single fret.

"Enharmonic equivalence"

The situation is further complicated by the fact that the standard tuning system means that some notes with different spellings produce the same sound ("enharmonic equivalence"). F# and Gb are "the same note", as are G# and Ab. As a result, "the same notes" can be represented by different intervals, depending on the spelling. A to B is an M2, but A to Cb is a d3.

Interval naming strategies

Two ways to conceive of intervals

1. My preferred way to think about intervals is the number of half steps (a.k.a minor seconds) from the lower note to the higher note. That is, the number of guitar frets on a single string from the lower note to the higher one.
2. Another common way to think of intervals is in relation to the major scale of the lower note. If the lower note is C, then the C major scale is the basis for calculation. If the lower note is Ab, then the Ab major scale is the basis. The interval names correspond to the major scale, not the minor scale.

How to determine intervals: D to C

1. Using the "half-step" way of understanding intervals, count the frets from the open D string up to C. There will be a total of 10 frets (half steps), which is the definition of an m7. (Remember that spelling is key. D to B# is 6+ even though there are the same number of half steps/frets involved.)
2. For the "major scale" approach, since D is the lower note, we think in terms of the D major scale. In D major, Cs are sharp, so D to C# is M7. Since we're looking at a C natural — a half step lower than C# — this is m7.

How to determine intervals: F to G#

1. Counting half-steps/frets from F to G#, there are three, making this 2+. (Again, were this spelled F and Ab, it would be m3, even though the half-step/fret count is the same.)
2. Since F is the lower note, we consider the F major scale. In F major, G is a natural note, so F-G is an M2. Since the G here is raised one half step (one fret) by the sharp, we have 2+.

Answer 6

It is a full 7th interval with no modification, so major.

Minor 7th is correct.

You're sort of right about 'no modification'. But the baseline isn't 'no accidentals in the notation'. It's the major scale starting on the lower note. D major has C♯. So a major 7th above D is C♯. If it's C♮, that's one smaller, so a minor 7th.