HOMEWORK: Workbook Chapter 2 pgs 9,10, 11, 14.

Important: There will be a quiz covering Chapter 2 on 9/19/06. It will closely resemble the format of the the homework assignment above.

One great way to prepare is to go here: http://www.emusictheory.com/drillIntervals.html

Intervals

Recognizing and Constructing Intervals

We saw in chapter 1 that the interval size can be determined by simply counting the letter names representing the span between two notes. In addition to this quantitative identification we must have a qualitative identification to represent the difference between those tones that contain "accidentals"

Quality

If the interval between C and E is a third than what differentiates it from the interval between C and Eb? Certainly they are dramatically different sounding intervals. There are 5 common qualitative identifiers of interval relationships. They are:

Major (M)
Minor (m)
Perfect (P)
Augmented (A)
Diminished (d)

We will also categorize the quatitative interval identification into two groups"

Group 1) Unison, 4th, 5th & 8va

Group 1 qualitive identifier: If the upper tone of an interval is in the major key of the lower tone, the interval is Perfect. If it is a half step higher it is Augmented. If it is a half step lower, the interval is Diminished

Group 2) 2nds, 3rds, 6ths, 7ths

Group 2 qualitive identifier: If the upper tone of an interval is in the major key of the lower tone, the interval is Major. If it is a half step higher it is Augmented. If it is a half step lower, the interval is Minor. If it is a whole step lower, the interval is Diminished.

Please note: As your understanding of Music Theory develops concepts such as chords will, from time to time disolve. Interval identification, however, is one of the few constants in music theory and forms the basis of almost all harmonic and melodic analysis. It is extremely important that you be able to immediately recognize any interval.

IN CLASS: Construct all intervals above and below various Tonics.

Compound Intervals

An interval that is larger than an octave is called a compound interval. For our present purposes they will function exactly as their corresponding simple intervals. Be careful to understand that the Octave is really the first compound interval, the 9th is the 2nd, the 10th is the 3rd and so forth. (Add 7 to the simple interval to get the compound version)

IN CLASS: Construct various compound intervals above and below various Tonics.

Inverted Intervals

By displacing the lower tone of a simple interval 8va, it become an upper tone and the interval is aid to be inverted. The sum of an interval and its inversion will always = 9 and its qualtiy will be similarly inverted as follows:

Major <-> Minor
Augmented <-> Diminished
Perfect <-> Perfect

IN CLASS: Name various written intervals. Name the Major and minor keys in which intervals exist.

Overtone Series

Almost all tones we hear are not single pitches but the composite of many tones sounding simultaneously. As you ear matures, you will be able to hear these "other" tones called overtones. In orchestration class we learn that the series of these other notes provides the foundation for the mechanical production of sound for all of the instruments. Adler tends to dismiss the importance of these overtones in music theory. I tend to lean the other way. I find it hard to believe that there is no correlation between the first 5 paritals in the harmonic series and the fact that these 5 tones, by Adlers definition, are the 6 most stable tones in western music.

As noted, the
1st Partial of the series is the Unison = Most stable
2nd partial of the series is the Octave = Most stable
3rd Partial of the series is the P5 = Very stable
4th Partial of the series is the P4 = Originally perceived as very stable it now has a double function
5th Partial of the series is the M3 = Stable and forms the identifier for a Major chord.
6th Partial of the series is the m3 = Stable and forms the identifier for a minor chord.

Consonance and Dissonance

PLEASE NOTE: Aldwell & Schacter state that the 6th is a consonant interval. This seemingly contradicts my statement in lecture 01b that the 6th was originally considered dissonant. In fact, this is not a contradtion but a difference in historical perspective. To be sure, at the onset of written music, the 6th was considered dissonant. Its classification as a consonant element appeared only after the introduction of triadic structure into music theory. In order to stay consistant with the text and avoid confusion, we will adhere to A&S definition that the 6th is consonant.

Intervals tend to produce impressions of stability or activity. Stable intervals we call Consonances and active intervals we call dissonances.

Consonant Intervals: PUnison, POctave, P5, P4(someti mes) M3, m3, M6, m6
Dissonant Intervals All 2nds, all 7ths, all A, all d, P4(sometimes)

So whats up with the P4? Can it be both Consonant and Dissonant? Indeed it can.

Around the medieval period of sacred music the use of vocal lines moving in parallel P4ths was common. It was considered a very stable relationship. Around the 17th century as the concept of triads began to form the ear began to accept the Major or Minor 3rd as a stable element of the triad. Thus a P4th that was in close proximity to a tone functioning as the 3rd of a triad, would often be heard as leading to that tone and therefore be active, not stable. See Ex. 2-11 & 2-12

In general,

When a P4th is in close proximity to a tone functioning as the 3rd of a triad, it tends to be dissonant.
When a P4th is in close proximity to the 5th of which it is an inversion, it tends to be consonant.

Intervals in a key

If we look at any major key we see that the interval relationships from the tonic are: PUnision, M2, M3, P4, P5, M6, M7 POct. A consistant yet different relationship will hold for any minor key. In fact, a consistant relationship between intervals is what defines note only Major and Minor Keys but any modal key relationship.

Diminished 5th and Augmented 4th (Tritone) (Ex. 2-15)

Please note: Technically the d5th is not a tritone. However it is commonly called such and for our purposes will be called a tritone.

One of the most important functions of western music is defined by the active nature of the relationship between a tone and its d5th or A4th interval. We saw in chapter 1 that this tritone relationship between the 4th and 7th scale degree tends to gravitate very strongly to 3 and 1. The motion is so strong, in fact, that we call the resolution of the d5 and A4 a key-defining progression. As well, the tritone relationship of any two pitches occurs in one key and one key only thereby creating a very strong sense of pull towards that key.

Tritone in Minor (Ex. 2-16+17)

In natural minor the tritone exists between 2&6. However, the resolution of these to 3 & 5 respectively is much less convincing since the Leading tone is not present. As a result, when 1 is not present in the resolution, 3 is often heard as the root of the resolution. Raising 7 in harmonic and melodic minor creates an "artificial tritone" and clarifies the tonic triad at the resolution.

Diminished 7th and augmented 2nd (Ex 2-18)

In harmonic minor the interval between the raised 7th and 6th is a dim 7th interval and is thus dissonant. As a result, its resolution is to 1 and 5 respectively. Similarly the inversion between the 6th and the raised 7th is an A2 and most often resolves to a consonant P4. The gravitational pull of the d7 and a2 to resolve to scale degree 1 and 5 is so strong it can define a key like no other interval relationship.

Remaining Intervals

Two other intervals we will see are the A5 between the 3rd and raised 7th in minor (and d4 inverted) and the A6th between the 6th in minor and the raised 4th. For now we are not responsible for these.