Questions & explanations
1. Given a figured bass: a bass note C with figures 6 and 5. What chord is indicated?
The figures 6 and 5 above a bass note C mean the chord includes a sixth (A) and a fifth (G) above C. This is a C major seventh chord in first inversion? Wait: C-E-G-B: root position has 3,5,7. First inversion of Cmaj7 is E-G-B-C, so intervals above E: 3,5,6. But here bass is C, so figures 6 and 5 indicate a chord with notes C, E, G, A? Actually, common figures: 6 alone means first inversion triad (C-E-A? No, that's Am/C). Let's clarify: In figured bass, 6 means a sixth above the bass, so for bass C, a sixth is A, and the chord is typically a first inversion triad (C-E-A? That would be A minor? Actually, C-E-A is Am/C, but the third is E. Better: For a triad, 6 indicates first inversion: bass note is the third of the chord. So if bass is C, the chord is A minor (A-C-E) in first inversion? That would be C-E-A, yes. But 6 and 5 together? That's a seventh chord in second inversion? Actually, 6/5 indicates a first inversion seventh chord: bass is the third of the chord. For a dominant seventh chord on G (G-B-D-F), first inversion is B-D-F-G, so intervals above B: 3,5,6. So for bass C, 6/5
2. Give an example of a transformational network for the progression C major - A minor - F major.
C major to A minor is a Relative (R) transformation (C's relative minor is Am). A minor to F major is also R (A minor's relative major is C, but here it goes to F? Actually, Am's relative major is C, so Am to F is a Parallel (P) if F is major? Wait: Am to F is not a standard transformation. Correct: C to Am is R, Am to F is not a simple P, L, or R. A better example: C major to A minor (R), then A minor to C major (R again) would be a cycle. Or C to E major (L), E to G# minor? Let's use a clear one: C major - A minor - F major - D minor. C to Am is R, Am to F is R (F's relative minor is Dm, but Am to F? Actually, Am's relative major is C, not F. So Am to F is not R. A correct network: C (P) Cm (L) Ab major (P) Ab minor (L) E major, etc. For simplicity: C to Am (R), Am to F? No. Let's say C to Am (R), then Am to C (R) is a back-and-forth. I'll give a correct example: C major to E major (L), E major to C# minor (R), C# minor to A major (L), etc. But that's complex. Better: C major to A minor (R), A minor to F major? Actually, A minor's relative major is C, so to get F, you need a differ
3. How does a common-tone diminished seventh chord resolve?
A common-tone diminished seventh chord resolves by moving to a chord that shares at least one note with it. Usually, the shared note stays the same, and the other notes move by half step. For example, in C major, a common-tone diminished seventh chord built on C (Cdim7: C, Eb, Gb, A) resolves to C major (C, E, G). The note C is common, while Eb moves down to E, Gb moves down to G, and A moves down to G? Actually, A moves down to G? Wait, correct resolution: Eb goes to E, Gb goes to G, and A goes to G? That would be two notes moving to G. Better example: Cdim7 to C major: C stays, Eb goes to E, Gb goes to G, A goes to G? That's not standard. A common resolution is Cdim7 to C major with C common, Eb to E, Gb to G, A to G? Actually, A typically resolves to G? In common practice, the diminished seventh chord often resolves to a chord a half step above one of its notes. For common-tone, it's simpler: the chord resolves to the chord that contains the common tone.
4. Give an example of mixing second and third species in two voices.
Suppose the cantus firmus is in whole notes. The second voice is in second species (half notes) and the third voice is in third species (quarter notes). For example, cantus: C (whole). Second voice: half notes D and E. Third voice: quarter notes C, D, E, F. On beat 1, intervals: C-D (2nd) and C-C (unison) – okay. On beat 2, third voice has D, second voice has D? Actually careful: second voice half note D lasts beats 1-2, so on beat 2, third voice is D? That gives D-D unison, but then on beat 3, second voice E, third voice E? That would be parallel unisons? Actually, you need to avoid that. Better example: cantus C, second voice: G (half) then A (half), third voice: C, D, E, F. Beat1: C-G (5th) and C-C (8ve) – fine. Beat2: C-D (2nd) with G? Actually G still there, so C-G (5th) and D-G (4th) – fine. Beat3: C-E (3rd) with A? A-E (5th) – fine. Beat4: C-F (4th) with A? A-F (6th) – fine. No parallels.
5. What is the typical voice leading for a common-tone diminished seventh chord?
In a common-tone diminished seventh chord, the common tone is usually held in the same voice. The other three voices move by half step to the notes of the target chord. For example, in C major, the Cdim7 chord (C, Eb, Gb, A) resolves to C major (C, E, G). The note C is held. Eb moves up a half step to E, Gb moves up a half step to G, and A moves down a half step to G? Actually, A to G is a whole step, so that's not half step. A better resolution: Cdim7 to C major with C held, Eb to E (half step up), Gb to G (half step up), and A to G? That's a whole step down. In practice, sometimes the A goes to G, but it's not a half step. Alternatively, the chord might resolve to a different chord where all non-common tones move by half step. For instance, Cdim7 could resolve to Ebdim7? No. Actually, common-tone diminished seventh chords often resolve to a major or minor triad with the common tone as root.
6. Give an example of a Phrygian half-cadence in two voices.
In two voices, suppose the bottom voice goes from A down to G# (half step), and the top voice goes from B up to C (half step). The bottom voice ends on G#, which is the dominant of the mode (if the final is C#? Actually, in Phrygian on E, the dominant is B, but G# is the third of the dominant? Wait, better example: In E Phrygian, the bottom voice goes from A (fa) down to G# (mi), and the top voice goes from D (re) up to E (mi). The final interval is a sixth (G#-E) or octave? Actually, it's often a sixth that resolves? Let me correct: In a typical two-voice Phrygian half-cadence, the bottom voice moves from fa to mi (half step down), and the top voice moves from re to do (whole step down) or from ti to do (half step up). The final interval is a third or sixth. For example, bottom: A to G#, top: F# to E (whole step down) gives a sixth (G#-E) that is not resolved. That's the half-cadence.
7. In invertible counterpoint at the twelfth, what happens to a perfect fifth when inverted?
In invertible counterpoint at the twelfth, a perfect fifth becomes a perfect octave (since 5 + 8 = 13, and 13 - 12 = 1? Actually, the complement at the twelfth: a fifth becomes a seventh? Let's compute: the inversion of an interval at the twelfth is 12 minus the interval. So a fifth (5) becomes a seventh (12-5=7). But a perfect fifth becomes a perfect twelfth? Wait, I need to be careful: Invertible counterpoint at the twelfth means the voices are inverted by a twelfth (an octave plus a fifth). The intervals invert such that a perfect fifth becomes a perfect octave? Actually, let's think: if two voices are a fifth apart, and you move the lower voice up a twelfth and the upper voice down a twelfth, the new distance is a twelfth minus a fifth = a seventh. So a fifth becomes a seventh. But that seventh might be major or minor depending on the key. This type is less common.
8. What is the difference between son clave and rumba clave?
Son clave has a pattern of 3+2 or 2+3 across two bars. In 3-2 son clave, the first bar has three beats (on beats 1, 2.5, and 4 in 4/4) and the second bar has two beats (on beats 2 and 3). Rumba clave has a similar structure but the second beat in the three-beat bar is slightly shifted. In rumba clave, the three-beat bar hits on beats 1, 2.5, and 4.5? Actually, rumba clave's three-beat bar is on beats 1, 2.5, and 4, same as son? Wait, the difference is subtle: rumba clave has the second hit of the three-beat bar on the 'and' of 2, while son clave has it on the 'and' of 2? Let me be precise: In 4/4, 3-2 son clave: bar1: beat1, beat2's 'and', beat4; bar2: beat2, beat3. 3-2 rumba clave: bar1: beat1, beat2's 'and', beat4's 'and'; bar2: beat2, beat3. So rumba has the last hit on the 'and' of 4 instead of beat4. That makes it more syncopated.
9. Why does the 2-3 suspension resolve upward?
The 2-3 suspension is special because the second is above the bass, and it resolves downward to the unison? Actually, the 2-3 suspension: the second above the bass resolves to a third. But the second is a dissonance that usually resolves down by step to a third? Wait, in a 2-3 suspension, the suspension is the second, and it resolves to a third. However, the resolution is often upward? Let's clarify: In a 2-3 suspension, the bass note is the same, and the upper voice has a second above, which resolves to a third. For example, bass C, upper voice D (second) resolves to E (third). That is an upward resolution. This is an exception to the usual downward resolution. The reason is that the second is a step above the bass, and moving down would go to the unison, which is too bare. So it moves up to a third.
10. What are some common borrowed chords in a minor key?
In a minor key, you can borrow chords from the parallel major. For example, in A minor, you can use the major IV chord (D major) instead of the minor iv (D minor). Also, the major V chord (E major) is already common in minor, but you can also borrow the bVII chord (G major) from the natural minor? Actually, bVII is diatonic to natural minor. Common borrowed chords in minor include the major IV (from parallel major) and the major VI (F major? In A minor, parallel major is A major, so VI is F# major? That's not common. More typical: in A minor, you might use D major (IV) and E major (V) which are already used. Actually, the most common borrowed chord in minor is the Picardy third (major tonic at the end). But other borrowings include the Neapolitan chord (bII) which is from Phrygian mode.
11. Give an example of a diatonic sequence using the chords I-IV-V-I in C major.
A diatonic sequence repeats a chord pattern within the same key. In C major, the chords I (C), IV (F), V (G), I (C) are all diatonic. If you move this pattern up a step, you get ii (Dm), V (G), I (C), ii (Dm) – but that would be a different sequence. A true diatonic sequence repeats the same interval pattern. For example, the pattern C-F-G-C (I-IV-V-I) moved up a step would be Dm-G-Am-Dm, but Am is vi, not a diatonic chord in the original pattern? Actually, a simple diatonic sequence is I-ii-V-I in C: C-Dm-G-C. If you move it up a whole step, you get Dm-Em-A-Dm, which is still diatonic to C major? No, A is V of D minor, but in C major, A is vi. So it's not strictly diatonic. A better example: the descending thirds sequence: C-Am-F-Dm (I-vi-IV-ii) is diatonic in C major.
12. Why might an unequal fifth from diminished to perfect be problematic?
The problem is that the diminished fifth (a dissonance) resolves to a perfect fifth (a consonance) by both voices moving outward. For example, B-F (dim. fifth) to C-G (perf. fifth): B goes up to C, F goes down to G? Actually, to get C-G, B goes up to C (half step) and F goes up to G (whole step) – both up, not outward. Wait: B-F is a diminished fifth; to resolve inward, B goes up to C, F goes down to E (making a third). If it goes to C-G, both go up, which is not the typical resolution. This can sound forced. In strict counterpoint, it's often avoided unless the voices move in contrary motion? Actually, the standard resolution of a diminished fifth is inward to a third. So an unequal fifth to a perfect fifth is unusual.