Tonal Harmony (book notes)

Created: 2017-12-19 14:49:57 -0800 Modified: 2018-06-03 12:33:03 -0700

Background

These are notes that I’m taking as I read through Tonal Harmony (With an Introduction to Twentieth-Century Music). I have the sixth edition.

Much of this note will have overlap with the Music Theory note. I should probably rearrange it all at some point, but I want notes from the book to mostly be in one place for now.

Scales

Major scale

Pattern of whole and half steps: W W H W W W H

The “W W H” substring shows up twice. The four-note patterns formed by both are called tetrachords.

Half steps only occur between scale degrees 3-4 and 7-1.

Minor scales

Natural minor: major scale with lowered scale degrees 3, 6, and 7
Harmonic minor: major scale with lowered 3 and 6 (or natural minor with raised 7)
Melodic minor
- Ascending: major scale with lowered 3 or natural minor with raised 6 and 7
- Descending: same as natural minor

Scale degrees 1 through 5 are the same for all three forms above, meaning only 6 and 7 can vary.

The key signature for any of the minor scales is written as though the scale were always the natural minor, meaning that for harmonic minor, the raised 7 is denoted by an accidental.

According to page 60, the melodic minor is what composers tended to confirm to.

Parallel major scale

A parallel major scale has the same tonic, so the parallel major of C minor is C major.

Relative major scale

A particular minor scale has a relative major scale that is a minor third above it. For example, C minor has a relative major of Eb major. It’s helpful to remember this because some minor scales (e.g. g#, d#, and a#) have no corresponding parallel major forms.

Memorizing scales

I like this quote from the book: “It is very important to practice faithfully all the major and minor scales on an instrument until they become memorized patterns. An intellectual understanding of scales cannot substitute for the secure tactile and aural familiarity that will result from those hours of practice.”

There are 15 key signatures in all, with one major and one minor scale associated with each (although in order for that second part to be true, enharmonics have to be employed).

Triads and seventh chords

Triad types

Major, minor, augmented, and diminished triads are prominent in classical music because tonal harmony makes use of tertian chords (as mentioned elsewhere in this note), which means they’re built of 3rds. The reason we end up with these four qualities is because there are only two kinds of thirds (m3 and M3), and since a triad has three notes, that means there are only two possible intervals that we can use (2 intervals for 2 possibilities each leads to 4 combinations).

Figured bass

The table that they use on page 47 of the book has pictures of a G major triad and a G7 chord, but I don’t have a quick way to do that, so I’ll modify the table a bit for these notes:

Notes in chord	GBD	BDG	DGB	GBDF	BDFG	DFGB	FGBD
Complete figured bass symbol	5 3	6 3	6 4	7 5 3	6 5 3	6 4 3	6 4 2
Symbol most often used		6	6 4	7	6 5	4 3	4 2 or just 2
How to find the root	Bass note	6th above bass	4th above bass	Bass note	6th above bass	4th above bass	2nd above bass
Inversion number	Root	1st	2nd	Root	1st	2nd	3rd

The “complete figured bass symbol” is just the numeric intervals above the bass note, meaning DFGB shows 6 4 3 because D-B represents an interval of a 6th, D-G is a 4th, and D-F is a 3rd. The “symbol most often used” is basically what makes a particular symbol unique. For example, a 2nd inversion triad and a 2nd inversion 7th chord both start with “6 4”, but the triad uses “6 4” alone to commonly represent it, so the seventh chord uses “4 3” since that makes it unique. I don’t know why they didn’t just use “3” to make it even simpler.

Someone playing just from figured bass would typically follow the key signature, meaning a root position triad may be major, minor, or diminished depending on the key signature. To specify differences, they may do one of three things:

“7b” - an accidental next to an Arabic numeral indicates to play that particular note should be raised or lowered. In this case, it represents that you should play a seventh chord in root position (no inversion) but with a minor seventh rather than a major seventh.
- E.g. “♮6” with a C written in the bass in Ab major would be “C Eb A♮” because the 6th above C is Ab and then you make it natural.

♭ <— 6 over a ♭ symbol means to play a 6 4, but the 4 is flat. Sometimes this would appear after a ♮6 just to “reset” to the key signature.

”#” - an accidental by itself always refers to the 3rd above the bass, e.g. if the key is C, the note on the staff is E, and there’s a ”#” beneath it, it means to play E major in root position because the third above E is G, and that’s sharp.
”6 4+ 3” - a slash or a plus sign in connection with an Arabic numeral is meant to raise that note. In this case, if the key is C and the note on the staff is A, then “6 4 3” tells you to play it in 2nd inversion and the slashed 6 and the 4+ tell you to raise both of those notes, giving you A C D# F#. The D and F are sharped because they’re a 4th and 6th above A.

Since figured bass uses the major scale, a dominant seventh in first inversion just looks like V6 5 (i.e. the fact it’s major tells you that it’ll be a dominant seventh, and the 6 5 tells you it’ll be in first inversion).

For figured bass symbols on suspensions, look at page 191.

Chord inversions

A chord’s inversion is only based on its bass note. For example, if there is a G in the bass and a bunch of Es and Cs above it, it doesn’t matter how many Es or Cs they are or where; as long as it’s a C major chord and G is in the bass, it’s C 6 4 (second inversion).

Because of this, when analyzing chords, you can just stack all of the notes from multiple staves or instruments onto a single staff to analyze it.

First-inversion triads in composition (figured bass symbol “6” or “63”)

In general, inversions in composition can serve to vary a bass line (pg. 123-124). Being in first inversion means that the third of the chord is going to be in the bass, which is the quality-defining note in a triad. The book lists three main reasons for why triads are used in first inversion (pg. 125):

To improve the contour of the bass line
To provide a greater variety of pitches in the bass line
To lessen the weight of V and I chords that do not serve as the goals of harmonic motion. I.e. when a V or I chord appears but it isn’t supposed to be a highlight, so it’s lessened by making it be in first inversion.

They include some specific “rules” (pg. 126):

The diminished triad almost always appears in first inversion so that the intervals from the bass are consonant and do not contain a tritone (even though every diminished triad obviously contains a tritone, the tritone here is not from the bass).
A V-vi progression should not have the vi in first inversion since it will sound like a mistake (it sounds like it should be V-I instead of V-vi).

They call first-inversion triads sixth chords (pg. 128) and mention that they can sometimes be used in a series of parallel sixth chords.

For four-part textures, they include some suggestions based on the fact that a triad only has three tones and triads in first inversion tend to be complete (meaning all three tones show up), so one tone will have to be doubled (pg. 130):

When writing for counterpoint, double the one that gives the best voice leading.
For a homophonic texture, double the one that gives the best sonority (i.e. play them all and figure out which one you like best). They have guidelines for which are most common on pg. 131 (which are typically both outer voices (soprano/bass) or both inner voices (alto/tenor)).
In any texture, try not to double the leading tone since it will result in parallel octaves.

When writing for three parts, the triads are almost always complete, although omitting the fifth is possible (you can’t omit the third since it’s a first-inversion triad, so it has to show up in the bass, and you typically don’t omit the root so that there’s no confusion about what chord you’re playing, although you could omit the root if it’s either obvious what the chord will be (e.g. V-I) or if it doesn’t matter (e.g. ii or IV both have the same predominant function in a major key)).

Second-inversion triads in composition (figured bass symbol “64”)

Second-inversion triads are not used as substitutes for root-position triads. For one, the interval of a P4 had been considered a dissonance if it appeared in the lowest two voices (pg. 143).

There are essentially four correct uses of a second-inversion triad in tonal harmony:

Bass arpeggiation and the melodic bass
- Bass arpeggiation is when the bass is outlining a chord, e.g. an E major chord going from E down to B down to G# down to E over four measures. If, in each of those four measures, you would analyze an E major chord in each measure, then the time when the bass has a B would put it in second inversion (i.e. B - E - G# is second inversion).
- A melodic bass is when the melody is in the bass as opposed to one of the upper voices. It would be kind of silly to keep analyzing how the bass incidentally interacts as figured bass in this case since the bass is functioning as a melody instead of a harmony.
Cadential six-four
- This is the usage of a tonic chord in second inversion followed by a dominant chord in root position followed by a tonic chord in root position. This I64 - V - I should not be analyzed as I-V-I (pg. 145) as it’s really just V - I. In a cadential six-four, the scale degree 1 usually moves to 7 and 3 to 2. Typically, the second-inversion triad will appear on the stronger beat or stronger portion of the beat (pg. 145). The only exception to that last bit is apparently in triple meter when the I64 can show up on the second beat.
The passing six-four
- In a three-note figure that follows a scale (AKA “a three-note scalar figure”) in the bass, there may be a second-inversion triad that results on a weak beat. Based on the example they use on page 148, this sort of seems like an instance of “melodic bass” from the first “correct” use of second-inversion triads. Like with the cadential six-four, some theorists do not assign a roman numeral to this because of its weak harmonic function.
  - There are a couple of musical features that commonly appear in this pattern:
    - Voice exchange (originally discussed on pg. 110-111) where something like a soprano may have A B C and the bass may have C B A. I guess the transition from A in the soprano to the bass (over three notes) and C from the bass to the soprano (over three notes) is the exchange.
    - Parallel sixths (i.e. just minor/major sixths that show up in parallel). Note that these are not a bad feature of a passing six-four, just a feature of them.
  - You will typically find a I64 or V64 as a passing six-four
The pedal six-four (AKA “embellishing six-four” or “stationary six-four”)
- When you have a static root position triad that you want to embellish, you can move the 3rd and 5th up by step and then back down by step to their original positions. This results in a six-four chord temporarily.
- Like with the passing six-four, you typically have a I64 or V64 as a pedal six-four

Voicing (for second-inversion triads)

Almost all voicing should be as smooth as possible leading into and out of a second-inversion triad.

In a three-part texture, usually the complete triad is there. Obviously the 5th can’t be omitted or it couldn’t be a second-inversion triad, so if you’re going to omit something, omit the root or the third and double the fifth.

In a four-part texture, the bass (the 5th of the chord) should be doubled. Exceptions to this are rarely encountered in tonal music (pg. 151).

In either case, the reason why you don’t double the root or the third is because of the tendency for 1 to go to 7 and 3 to go to 2 (when doing I64 - V), and you don’t want to end up with parallel octaves.

Lead sheets

Figured bass provides the bass line with symbols indicating the chords to be constructed above it. Lead sheets tell you which chords to construct below a melody.

Example symbols:

E (E major triad - going to omit the “E” from now on since they’re all based on E)
Em (minor triad)
Edim or E° (diminished triad)
E+ (augmented triad)
EM7 (major seventh)
E7 (dominant seventh)
Em7 (minor seventh)
Em7b5 or Eø7 (half diminished seventh)
- Note: this does not mean to “make the 5th flat”, it means to “lower the fifth”, so F#m7b5 is F# A C E.
Edim7 or E°7 (diminished seventh)

Lead sheet symbols occasionally specify which note to put in the base, e.g. C/G, indicating to play C major over a G in the base (which would be a triad in second inversion).

Seventh chord types and the dominant 7th

Tonal harmony focuses on the five most common:

Major seventh (major triad with a major 7th)
Major minor seventh (AKA “dominant seventh”) (major triad with a minor 7th)
Minor seventh (minor triad with a minor 7th)
Half-diminished seventh (diminished triad with a minor 7th)
Diminished seventh (diminished triad with a diminished 7th)

Quick refresher on figured bass for inversions (pg. 221-223 discusses how to resolve these):

7: root position
6-5: 3rd is in the bass
4-3: 5th is in the bass
4-2 (or just 2): 7th is in the bass

Chapter thirteen (pg. 211) is entirely about the V7 chord. The most important “rule” is that the seventh should resolve downward and the third should resolve upward, especially when in an outer voice (pg. 212). E.g. a C7 should have the E go to F and the Bb go down to A (or Ab if we’re going to f-minor). This means that it’s not necessary that the triad that you’re resolving to be complete (i.e. in C7 —> F, we could omit the fifth from the F chord so that we just have a tripled root and a third) (pg. 213).

The one exception to this is a V4-3 resolving to an I6 — the 7th may move up by step to a 5 (pg. 227).

Much of how to resolve a V7 to a tonic triad is discussed on pg. 214-215. The important thing seems to be not to resolve to a triad in 6-4 position because of the implied parallel octaves (pg. 218), unless you’re coming from a V4-2 chord (pg. 223), in which case dropping the 7th will almost always get you to an I6 chord.

Other brief V7 guidelines:

The V7 almost always resolves to the tonic or deceptively to the VI or vi (pg. 217).
Avoid approaching a 7th by a descending leap (pg. 228).
Almost all chord 7ths are going to resolve down by step. This means that the following chord needs the resolution note, which is why some resolutions are “delayed” by involving an intermediate chord (page 255).

II7 and VII7 (chapter 14, pg. 229)

The most commonly used seventh chord is by far the dominant seventh chord, V7. After that, ii7 (supertonic seventh) and then vii half-dim 7 (leading-tone seventh) are the most frequently seen in a major mode (pg. 229).

In general, the chapter says that the voice-leading concerns aren’t difficult and boil down to the following principles (pg. 229):

The 7th of the chord almost always resolve down by step
The 7th of the chord may be approached in various ways (pg. 224-225): suspension, passing tone, neighbor tone, and appoggiatura (in that order of frequency).
Incomplete chords must contain at least the root and the 7th. These are what make them seventh chords, after all! The fifth is more commonly omitted than the 3rd, and from what I’ve personally seen (i.e. not what’s written in the book), it seems like you’d only really omit the 3rd when it’s already represented in another voice (see Chopin waltzes).
Doubled tones should not be the chord 7th or the leading tone (since these lead to parallel octaves).

The supertonic seventh usually appears as a minor seventh chord in major modes and a half-diminished seventh in minor modes. Just like with triads, the supertonic seventh usually moves to V. It can sort of function as a IV chord because it contains the IV chord, and while that’s true, the book also says that it’s essentially a IV with an added 6th in some cases where II7 appears to function as IV (pg. 232).

The VII7 functions almost exactly like a dominant seventh would because of how many tones it has in common with a dominant chord already. You just need to be careful about avoiding parallel 5ths (pg. 233).

In the major mode, VII7 naturally appears as a half-diminished seventh, whereas in the minor mode, it’s a fully diminished seventh. Regardless, it tends to resolve to the tonic because it’s performing a dominant function.

The one clarification they make in the book on pg. 235 is that a diminished 7th is composed of two tritones, and their resolution depends on whether it’s spelled as a diminished fifth (in which case resolve inward) or an augmented fourth (in which case resolve outward). For example, in e minor, a d#°7 in root position is composed of +5s and would resolve to an e minor triad in root position, but if the fifth is in the bass (4-3 in figured bass), then it would be composed of °4s and would resolve to triads in 1st inversion (6-3 in figured bass).

IV7 (subdominant) (chapter 15, pages 243-245)

Like IV itself (without the 7), it typically moves to V (or vii°6) often passing through some form of the ii chord on the way. Resolving to an inverted ii7 is easy because they share all but the 7th.

In a minor mode with a raised 6th scale degree, the IV7 will sound like a dominant seventh, but its function is different since it will typically move to V, vii, or ii, not the tonic.

VI7 (submediant) (chapter 15, pages 245-247)

As shown on page 245, this takes one of three forms:

vi7 in a major key
VI major seventh in a minor key
(Infrequent) #vi half-dim 7 in a minor key

The first two act like their parent triads, which move toward V, passing through subdominant (which is the fourth scale degree) or supertonic (2nd scale degree) or both.

Let’s examine the vi7 in a major key going through the ii to V. For example, if we’re in F, then vi is going to be dm, ii is going to be gm, and V is going to be C. dm7 —> gm —> C is a 2-5-1 progression that is very common in jazz. However, “1” implies that the C is our tonic, which it isn’t since I said we’re in F, so we would just go back to F at the end. The book demonstrates this with Chopin’s second ballade on page 246.

I7 (chapter 15, pages 247 - 249)

This appears as an M7 in major modes and m7 in minor modes. Adding the 7th “deprives the tonic triad of its tonal stability” (pg. 248), so it is the only real exception to the property that adding a seventh doesn’t change the function of a triad. The seventh has to resolve down, so what follows an I7 chord has to contain the sixth scale degree.

III7 (mediant) (chapter 15, pages 249-250)

This is typically iii7 in major and IIIM7 in minor. It typically progresses to a VI7, but it could be followed by a IV.

Diatonic chords

Chords found in a particular scale (e.g. playing ii in G is A minor) are called diatonic. All other chords are either called altered or chromatic.

When it comes to the three minor scale forms, remember that only scale degrees 6 and 7 can vary. For diatonic chords, we essentially consider there to be a single minor scale with variable 6 and 7 degrees, so the raised 6 and 7 or the ones in the natural minor will be diatonic. E.g. in E minor, E, F#, G, A, and B are the unvarying scale degrees and thus are always diatonic. C and D appear in the natural minor, and C# and D# appear in the melodic minor, so the set of E F# G A B C C# D D# are all diatonic in E minor.

For a major scale, the diatonic triads are: I, ii, iii, IV, V, vi, vii°

This is pretty easy to remember by just visualizing a C major scale: C major, d minor, e minor, F major, G major, a minor, b diminished.

Because the 6th and 7th scale degrees in minor vary, the book lists all possible chords but circles the most common ones. I’ll bold the common ones here:

i
ii°
ii
III
III+
iv
IV
v
V
VI
#vi°
VII
vii°

Listed out without having to look at bolded text: i ii° III iv V VI vii°

For seventh chords, you write a major 7th like IM7, a minor 7th like ii7, a dominant 7th like V7, and diminished and half diminished as shown below:

Diatonic seventh chords in a major scale: IM7, ii7, iii7, IVM7, V7, vi7, viiø7.

Diatonic chords in a minor scale: i7, iiø7, IIIM7, iv7, V7, VIM7, vii°7

Voicing

Voicing is how the chord is to be distributed or spaced. Voicing for a four-part harmony generally follows SATB (soprano, alto, tenor, bass, in that order).

Close structure: less than an octave between soprano and tenor

Open structure: an octave or more between soprano and tenor

Composing guidelines for voicing:

Crossed voices: do not allow any part to cross above the soprano or below the bass because the essential soprano/bass counterpoint might become unclear. The alto and tenor lines may cross briefly if there is a musical reason to do so.
Spacing: when writing for three or more parts, avoid overlay spacious sonorities by keeping upper parts (excluding the bass) within an octave of each other, i.e. essentially write in close structure.
Stick to the range of SATB voices, e.g. soprano is C4 to G5. These can be looked up online.
- Also, I assume that transpositions to and from concert pitch can just be done by programs now, so I don’t know if much has to be memorized here in the modern age. However, it’s probably worth noting that some detail on why this process is needed is covered on pages 99 and 100. In short, it boils down to the histories of individual instruments and how they were created. The book writes on page 100: remember that a transposing instrument “sees a C but sounds its key.” This means that a horn player who sees a C will sound an F because the French horn is pitched in F.

Remember that these are just guidelines and they exist for a reason, but all “rules” can be broken.

Also, it’s important to note that when you see a note with two stems (one up and one down), it means that two voices are occupying the same note (rather than one voice disappearing for some reason).

Parallel motion (pg. 81-85)

This whole section is a relatively simple set of rules followed by exception after exception. I put a summary of this section at the end since it may be somewhat confusing. Also note: so that I don’t need to include sheet music here, I write out a set of voices playing simultaneously as “B-T-A-S”, where B is the bass, T is the tenor, etc. meaning “A-C to B-E” would be the bass going from A to B and the soprano going from C to E.

It’s important to consider the relationship between any two voices at a particular time in a piece. There are five kinds of motion between two voices:

Static: neither part moves
Oblique: only one part moves
Contrary: both move but in opposite directions
Similar: both move in the same direction but by different intervals
Parallel: both move in the same direction by the same interval

Parallel motion in particular is worth looking at because it can voices of their independence from one another. In general, avoid parallel fifths and octaves (and their equivalents, e.g. unisons and perfect 12ths). The exceptions to this are:

Duplicating an entire line by an octave (e.g. having two instruments play exactly the same thing while being separated by an octave)
20th+ century music (where parallel motion regained acceptance)

The reason why perfect fifths and octaves specifically are avoided for parallel motion is because they’re the most stable intervals, so having two voices stuck in those stable intervals while moving is what removes their independence.

To clarify the above “rule” without really adding new information:

Static “motion” where the voices stay at a parallel 5th or octave is fine because it’s not really motion.
This is only between two given voices. This means that the motion of a parallel fifth is fine as long as it’s not between the same two voices. For example, imagine C-E-G going to G-D-B. In the first harmony, the soprano (G) and tenor (C) form a perfect fifth, and in the second, the alto (D) and tenor (G) do, so this is a parallel fifth, but it’s not considering the same two voices each time, so this is fine.

Also, consecutive perfect 5ths and 8ves were generally avoided even in contrary motion in vocal music (pg. 82).

Unequal 5ths (pg. 83) are when a perfect fifth and a diminished fifth follow one another (e.g. °5-P5 or P5-°5). These are acceptable in parallel motion because the “unequalness” doesn’t ruin the independence of the voices. The only time they’re not allowed is when it’s a °5-P5 involving the bass voice (e.g. B-F to A-E; both are fifths, but the first is diminished, so it’s a °5-P5).

Direct (or hidden) 5th or 8ve is when the outer parts move via similar motion into a P5 or P8 with a leap in the soprano part, e.g. A-C to D-A. This is not a parallel 5th even though we do end up with a 5th between the two voices, but it’s still bad because it involves a leap in the highest voice (the soprano). If it had been A-B to D-A, it would be fine since the soprano wouldn’t be doing a leap.

This entire section about parallel motion is less strict in instrumental music as compared to vocal music.

Summary of parallel motion (from the book): “Parallel 5ths and 8ves are avoided in most contexts in tonal music because they undermine the relative independence of the individual parts. Also generally avoided are consecutive 5ths and 8ves by contrary motion and, in certain circumstances, unequal 5ths and direct 5ths and 8ves.”

Secondary chords

The most common sort of altered chord in tonal music is the secondary function: a chord whose function belongs more closely to a key other than the main key of the passage. For example, if we’re in the key of C and we start seeing F#s, we may think that we’ve shifted keys. However, the F#s may be a result of a D chord tonicizing the G, giving it special emphasis without changing the tonic (page 260). Tonic triads are always either major or minor, so only major or minor triads can be tonicized in this way (this means that a vii° cannot be tonicized by a secondary dominant (p. 275)).

Secondary dominants resolve just like primary dominants do (p. 264) except when it comes to the 7th. Note that the “chord of resolution” that they talk about in the book is the non-secondary chord, e.g. V7/V in C would be D resolving to G. The “G” is the chord of resolution. The D chord has an F# in it whereas the G7 will have an F, so the F# needs to resolve down a half step to F.

Secondary dominant chords (chapter 16)

Secondary dominant chords are a V or V7 of a scale degree that forms by itself a major or minor triad. For example, in C major, the second scale degree is d, and its dominant is A, so we would analyze an A chord as V/ii or V7/ii.

V/IV is special because it sounds the same as the tonic (e.g. IV in C is F, and V in F is C, so V/IV in any key is really just itself), so composers will typically write V/IV as V7/IV, that way it is distinct from I.

The most common secondary dominant in V7/V (p. 264).

Spelling secondary dominants (p. 262)

Find the root of the chord to be tonicized.
Go up a P5.
Using the note from step #2 as a root, spell a major triad or a major-minor seventh chord.

For example, suppose we want to spell V/vi in Eb:

The scale degree to find for the root is vi in Eb, which is C.
A P5 above C is G.
A major triad formed on G is G-B♮-D, so this would appear with a B♮ written in.

Another example of spelling V7/V in b minor:

The scale degree to find for the root is V in b minor, which is F#.
A P5 above F# is C#.
A dominant 7 formed on C# is C#-E#-G#-B. When writing this in, the key signature of b minor already has C# in it, so we just put sharps on E and G.

Recognizing secondary dominants (p. 262)

They’re the most common altered chord, so there’s a good chance that it’s going to be a secondary dominant, but here are the guidelines:

It has to be either a major triad or dominant seventh chord, otherwise it can’t be a secondary dominant.
If the note a P5 below the root is diatonic in the key, then it’s a secondary dominant.

Intervals

Definitions

Harmonic: notes played at the same time

Melodic: notes played successively

Simple: intervals smaller than an octave

Compound: intervals that are an octave or more

Inverting intervals

I already wrote about this here. TL;DR: nine minus the original interval gives you the new numerical name, then also invert the quality (m —> M, dim —> aug, P —> P).

Qualities

”Perfect” only applies to unison, fourths, fifths, octaves, and their compounds (11ths and so on). Major and minor apply to seconds, thirds, sixths, and sevenths 9and their compounds).

Expanding a major or perfect interval by a half step [without changing the numerical name] makes it augmented. Likewise, contracting a minor or perfect interval by a half step makes it diminished.

Consonance and dissonance

The book says that these can roughly be defined as “pleasing [or not pleasing] to the ear”, but that depends on context. Usually dissonances resolve to consonances.

Basics of consonance and dissonance with respect to intervals:

Consonant: major 3rd, minor 3rd, major 6th, minor 6th, perfect 5th, perfect octave.
Dissonant: all others

Exceptions:

Unisons (which I assume are consonant and just weren’t listed in the book)
P4: this is only dissonant when it occurs above the lowest voice (also called the bass).

Rhythm

Meters

Grouping	Meter type	Meter accent pattern
Two-beat measure	Duple	Strong-weak
Three-beat measure	Triple	Strong-weak-weak
Four-beat measure	Quadruple	Strong-weak-less strong-weak

This means that the bottom number in a time signature doesn’t really affect meter; if Jingle Bells were written in 2 8 time instead of 2 4 time, it wouldn’t change the meter from duple.

Also, the table can apply to simple and compound meters. 6 8 time is still a two-beat measure, so it’s compound duple meter.

You can determine the beat and meter types just by listening, but you can’t determine the note value that represents the beat just by listening.

Hypermeter

This refers to a regular grouping of measures that is analogous to meter. For example, in Amazing Grace, the measures form groups of four, creating what’s called “quadruple hypermeter”. The measures themselves have their own meters, so hypermeter is just talking about the meter as the division instead of beats.

According to the book, scherzi (or potentially just fast scherzi; the wording is a bit ambiguous) are almost always notated as simple triple (usually 3 4 time), but the aural effect is of one beat per measure, for which we might use the term compound single. If you didn’t know the metric convention of such pieces, you would probably assume when hearing them that they were in compound duple because the measures tend to group into hypermetric pairs.

Remember, this means that two the measures themselves are forming groupings of two.

I can sort of see this with Chopin’s first scherzo.

Compound time signatures

When you subdivide a beat into three equal parts, you get a compound beat (which needs a compound time signature). For example, if the beat note is a dotted quarter note (as in 6 8 time), then you would subdivide it into three eighth notes.

Because we’re dividing into three equal parts in compound time signatures, a dotted note is used for the beat value.

Compound time signatures violate the rule that most people learn that “the top number tells how many beats are in a measure, and the bottom number tells what note gets the beat”.

Notating rhythms correctly

The rule of thumb is to emphasize the individual beats rather than obscure them. For example, if you’re in 3 4 time and you have an eighth note at the beginning and end of a measure, you do not notate it as an eighth note, a half rest, and another eighth note; that obscures the start of the second and third beats. Instead, you’d write it as an eighth note, an eighth rest, a quarter rest, an eighth rest, and an eighth note. You may violate this “rule” when you are purposely dealing with syncopation.

Probably useless

”M.M.” stands for “Maelzel’s metronome” and is a numerical indication of tempo, e.g. ♩=65. This is named after Johann Maelzel, who promoted the metronome during the early 19th century.

Composing

Beginning advice

In chapter four, they advocate that you start small (specifically having longer notes in the rhythm, having the melody notes be part of the harmony chords, and making most melodic motion stepwise (conjunct)). They also suggest that your melody be clear and simple and work toward the highest note of the melody as a focal point.

Specific advice (again, just to give some guidance):

Leaps

Avoid augmented intervals, 7ths, and intervals larger than a P8. Diminished intervals may be used if the melody changes direction by step immediately after the interval.
A melodic interval larger than a P4 is usually best approached and left in the direction opposite to the leap. This leads to a melodic line that may look like B down to E, then up to C, then back down to B. We leapt a perfect fifth down (and then leapt back up a minor sixth), then we ended up back down at B.
When smaller leaps are used consecutively in the same direction, they should outline a triad.

Tendency tones

The 7th scale degree has a strong tendency to move to the tonic. An exception to this is the scalewise line descending from the tonic: 1-7-6-5.
The only other tendency tone that needs to be considered is the 4th scale degree which often moves down to the 3rd (but this isn’t as regular a tendency as 7 going to 1).
Both the 7th and 3rd can be found in a dominant chord, so something like a G7 should resolve the B to a C and the F to an E.
Apparently, the leading tone (7th scale degree) doesn’t have to resolve to the tonic if it appears in an inner voice in a major mode.

Root position part writing (starts on pg. 89)

The book reduces the number of different intervals that can separate the roots of any two chords to four:

2nd apart (same as 7th)
3rd apart (same as 6th)
4th apart (same as 5th)
Same roots - a repeated chord (and it has to be repeated rather than being a different chord because we’re in root position).

General guidelines:

All members of a triad are usually present
- The fifth is sometimes omitted (e.g. when you have only three voices or when you’re writing the final I chord in a four-voice texture)
The root is frequently doubled or tripled
The leading tone (i.e. the tone a half step beneath the tonic) is almost never doubled because it has a strong tendency to resolve to the root (and I’m assuming this would be bad because it would lead to parallel octaves).

Movement

4th or 5th apart: there’s a bunch of guidelines around movement (pg. 91-92) specifically for when the roots are a 4th or 5th apart. For four-part textures, I don’t know how important it’s going to be, so I won’t write out the guidelines here. For three-part textures, they simply say to make sure you at least have a root and a third, and observe conventions concerning spacing and parallelism. It says to prioritize smooth voice leading instead of complete chords. Remember though: this is all for root-position chords.

3rd or 6th apart: for 3rd and 6ths, you’ll always have two pitch classes in common, so there isn’t as much movement needed. However, you need to make sure that do not omit the 5th so that it’s obvious that there’s a new chord, otherwise it’s difficult to discern. For example, if you have an F major chord (F A C) and you want to progress to an a minor chord (A C E), the E is important so that you know that you’re not just writing out an I 6 chord (i.e. F first inversion).

2nd or 7th apart: there won’t be any pitch classes in common here, so every voice must move between the two chords.

For four-part textures with a doubled root, you move the upper voices opposite to where you move the bass voice.

This kind of movement can lead to a deceptive cadence, which is either V-vi or V-VI. Be careful in a minor mode not to end up with an augmented second between the first and second chords if you do this.

When you only have three voices, you’ll get the smoothest voice leading by shifting either from a complete —> incomplete triad or vice versa (and “complete” just means that you have the root, third, and fifth all present). I.e. basically it would probably sound pretty bad to have something like G B D —> A C E, probably because there would be parallel motion (since you may move all voices up or down by a whole step) or wild motion (because the voices would get very spread apart).

Cadences, phrases, and periods

Cadences

Cadences are harmonic goals, as mentioned in the Definitions section. This means that not every V-I is going to be an authentic cadence since it may not be a harmonic goal. In that case, it could be called an authentic progression even though it’s not an authentic cadence.

Authentic cadence: V-I or V7-I (and in minor: V-i or V7-i)
- Perfect authentic cadence: an authentic cadence where both chords are in root position. This means that the highest voice (the soprano) of the final chord should be the tonic. This is the most final sounding cadence.
- Imperfect authentic cadence: any authentic cadence that isn’t perfect. There are three subcategories.
  - Root position IAC: like a PAC, but the 3rd of 5th scale degree is in the melody over the final I chord.
  - Inverted IAC: either or both of the V or I chords are inverted.
  - Leading-tone IAC: some form of vii°-I, the vii° substituting for a V chord.
  - Of the above three IACs, the root position IAC is the most final sounding, and the others are mostly used for less important interior cadences.
Deceptive cadence: when the ear expects an authentic cadence but gets V-? Instead. The ? Is usually a submediant triad (6th scale degree), but others are possible. This feels very unstable. V-vi involves special part-writing problems mentioned on page 96. A special deceptive cadence involving secondary dominants is discussed on page 266: it’s a dominant seventh going to a V7/IV, which looks like the tonic, but it’s an altered chord.
Half cadence: ?-V. Another unstable/progressive cadence. It ends in a V chord.
- Phrygian half cadence: a special HC that is iv6-V in minor.
Plagal cadence: IV-I progression. They’re not as final sounding as a PAC, so they’re sort of added on as an “amen” after a PAC and are commonly found in religious music.

Generally, authentic and plagal cadences are conclusive, whereas deceptive and half cadences are progressive.

Motives

A motive is the smallest identifiable musical idea. It can consist of a pitch pattern, a rhythmic pattern, or both (e.g. the iconic four opening notes of Beethoven’s symphony 5 is an example of having both a pitch/rhythmic pattern). You would typically only talk about a motive for a musical idea that gets developed further.

Periods (ch. 10)

Phrases are described in the Definitions section - they are musical ideas terminated in a cadence. A period is a larger structural unit that typically consists of two phrases in an antecedent-consequent (or question-answer) relationship. The second phrase usually has a stronger cadence, so the periods may be HC —> [PAC|IAC] or IAC —> PAC. They say on page 170 that this property of weak-strong cadences is the most important distinguishing feature of a period.

If the second phrase doesn’t have a different ending, then it is not a period but a repeated phrase.

A parallel period has the same starting in both phrases (or close to the same starting; having the second phrase be a third lower is still worthy of being parallel). A contrasting period has different phrase beginnings.

There are phrases that consist of three phrases, but a three-phrase period would have to have three different phrases (and would either be question-question-answer (“two antecedents and a consequent”) or question-answer-answer).

A double period is typically four phrases in two pairs, where the second phrase of the first period has a weaker cadence than the second phrase of the second period. According to the book on page 168, it says “the first two phrases will probably not form a period according to our original definition.” I imagine this is because a period has increasingly conclusive cadences between its usual two phrases, and a double period also has increasingly conclusive cadences, but this time it has four total phrases, so it’s not really possible to keep increasing the “conclusivity” of the cadences since you can really only go HC —> IAC —> PAC with conclusive cadences. Instead, in their example, they show HC —> HC —> HC —> PAC and call that a parallel double period.

A repeated period is not the same as a double period because the double period has contrasting cadences.

Non-chord tones

Classification of nonchord tones (and great examples on pg. 182)

NCT name (and abbeviation)	Approached by	Left by
Passing tone (p)	Step	Step in same direction
Neighboring tone (n)	Step	Step in opposite direction
Suspension (s)	Same tone	Step down
Retardation (r)	Same tone	Step up
Appoggiatura (app)	Leap	Step (not necessarily in the opposite direction as per pg. 197)
Escape tone (e)	Step	Leap in opposite direction
Neighbor group	See p. 199
Anticipation (ant)	Step or leap	Same tone (or leap)
Pedal point (ped)	See p. 202-203

We may additionally modify the NCT with qualities: diatonic vs. chromatic, accented vs. unaccented, ascending vs. descending, and upper vs. lower.

I like what the author says on page 189. A Bach chorale is shown, then it’s simplified into its harmony. The author says that the chorale sounds like music, but the reduction sounds like a theory exercise. The difference between the two is the embellishments.

Try to avoid parallel fifths and octaves created by adding neighboring and passing tones (pg. 190).

Suspensions (pg. 186)

A suspension can be split into its preparation (which usually starts off as a chord tone due to the harmony employed, then changes to a non-chord tone once the harmony changes) and its resolution, which is the tone following the suspension and lying a 2nd below it.

Suspensions are usually labeled according to intervals above the bass, so they’ll normal be something like “7-6” or “4-3” (note: the “4-3” suspension is notated as something like D7sus since it’s the most common suspension). The only time the interval will increase is when the bass is what’s suspended (meaning it’s getting farther from the next highest voice).

A suspension can happen in an upper voice alongside a change in the bass, which is simply called a suspension with change of bass.

A retardation is just a suspension that resolves up.

Neighbor group (pg. 199) (AKA “cambiata” / “changing tones”)

It’s two NCTs in succession, the first being an escape tone, the second an appoggiatura. To spell those out: it means that the first is approached by step and left by leap, and the second is approached by leap and left by step. For example: C D B C. The escape tone is C—>D, then the appoggiatura is D—>B, then the appoggiatura is left by going back to C.

Anticipations (pg. 200)

These anticipate a chord that has not yet been reached; the NCT moves by step or leap to a pitch that is contained in the anticipated chord but is not present in the current one (or else it wouldn’t be an NCT).

As mentioned on page 201, an anticipation can be approached and left by a leap (a free anticipation).

Pedal point (pg. 202)

A pedal point begins as a chord tone, then becomes an NCT as the harmonies around it change, and finally ends up as a chord tone when the harmony is once more in agreement with it. The other NCTs seem to be dependent on the harmony, but the pedal point has such tonal strength that the harmony seems to be the embellishment rather than the pedal point.

When analyzing with Roman numerals, you do not indicate inversions above pedal points.

Analyzing when NCTs are involved (the latter half of chapter 12 ending on pg. 208)

They basically say that sometimes it’s tough to analyze pieces with unusual passages. They say “The analysis of chords and NCTs must always be carried out simultaneously. Although most NCTs are clearly recognizable as embellishments of the basic harmony, ambiguous cases will be encountered occasionally.”

Counterpoint

Basics

Counterpoint is “the combining of relatively independent musical lines” (pg. 132). Motion, as mentioned elsewhere in this note, is either parallel, similar, or contrary, and for counterpoint, we mostly want similar or contrary motion (since parallel motion would make the voices sound more dependent).

The book says that post-Baroque music mostly only focused on soprano/bass counterpoint (pg. 135) and left the inner voices as “filler”.

Species

First species (pg. 134): both parts move with identical rhythms (AKA one-to-one counterpoint)

Harmonic series and functions

Keep in mind that this section apparently only deals with sequences as in “patterns that are immediately repeated”.

V —> I is the strongest of all root movements. V serves as a dominant chord. If work backward through the circle of fifths from V (by ascending a fifth), we end up with ii, which serves as a pre-dominant chord. The pre-dominant leads naturally to a dominant chord, which leads naturally back to the tonic. Similarly, we can ascend another fifth to get to the vi chord, which leads back to the ii chord (because it is “v - i” movement if you consider ii to be the tonic). Again, going another fifth backward gives us iii, which can lead to vi. The iii chord doesn’t appear in major modes as frequently as III appears in minor modes. Going back one final time gives us vii°. It could resolve to iii, but more frequently it serves as a substitute for V (to the point where moving between V and vii is not even considered to be a progression).

The only chord not covered above is IV, which is a P5 below the tonic. Most often, IV has a pre-dominant function, moving directly to V or vii°. However, it may expand its pre-dominant function by first moving to ii or ii in first inversion. It can also go directly back to I to form a plagal cadence.

The tonic chord itself can precede basically any harmony. Same thing with if “vi” is used as a substitute for I as in a deceptive cadence.

In general, this leads to this set of “normative harmonic functions” in major keys (pg. 113). In the list below, the “key” is the chord that we’re on, and the set of items that follows contains the typical progressions that you’d find.

iii:
- vi
- IV
vi:
- One of [IV ii]
[IV ii]:
- IV itself can go to ii
- IV itself can go to I (plagal cadence)
- Either can go to one of [vii° V]
[vii° V]
- V itself can go to vi (deceptive cadence)
- I

In minor modes, there are some important differences:

The mediant triad, III, is found much more frequently in a minor mode since it represents the relative major key.
The variability of scale degrees 6 and 7 lead to different qualities and functions
- The subtonic, VII, sounds like the V in the key of the relative major (i.e. the V of III).
- The minor v, usually v in first inversion, will usually resolve its lowered 7th scale degree to the lowered 6th scale degree (as part of an iv first inversion) so that it does not function as a dominant chord. For example, if we’re in A minor, then the minor v would be an E minor triad (notice the G is the lowered 7th scale degree of A minor). This would typically resolve to the iv of A (which is a D minor triad), but it’s in first inversion, so it’s F-A-D.

Those rules lead to this set of normative harmonic functions for minor keys (g. 113):

VII:
- III
III
- VI
- iv
VI
- One of [iv ii°]
[iv ii°]
- iv itself can go to ii°
- iv itself can go to i
- Either can go to one of [vi° V]
[vii° V]
- V itself can go to VI
- Either can go to i

Probably very useful

This section has notes that I think are repeated a decent amount throughout the book:

Be really careful with the rules of parallel motion (e.g. the whole “avoid parallel fifths and octaves”, although there’s slightly more to it than that). Mentioned in a lot of spots after introduction, e.g. pg. 129.
Be careful about tendency tones (and how they may end up implying parallel motion even if it’s not explicitly present). Generally, the leading tone (the tone a half step beneath the tonic) wants to go to the tonic, and the 4th scale degree wants to go to the 3rd. If you double up a leading tone in a three- or four-part harmony, you’ll likely end up with parallel octaves, which is bad.

Probably useless

Definitions

Harmony is the sound that results when two or more pitches are performed simultaneously. It is the vertical aspect of music produced by the combination of the components of the horizontal aspect.

Tonal harmony refers to the harmonic style of music composed from about 1650 to 1900. The book uses this definition on page xi: “Tonal harmony refers to music with a tonal center, based on major and/or minor scales, and using tertian chords that are related to one another and to the tonal center in various ways.”

Tertian chords are chords that a built of thirds, e.g. C-E-G, not the nontertian C-F-G.

Functional harmony refers to how chords tend to have “standard” roles or functions within a key, e.g. a dominant seven chord resolving to the tonic.

Pitch class refers to all notes of a particular pitch (including enharmonic equivalents like C and B#) regardless of octave, meaning there are only twelve.

Middle C is C4.

Octave register is every note with the same number next to it in keyboard notation, e.g. C4 and D4 are in the same octave register, but B3 and C4 aren’t despite being a half step away from one another.

Scale degrees are typically written with a circumflex over the number, but that’s super annoying to type, so I’m going to omit that from my notes.

Minor pentachord is the pattern of whole/half steps formed by w-h-w-w-w.

Enharmonics are notes that are spelled differently but sound the same (e.g. B# and C).

Counterpoint refers to the combining of relatively independent musical lines

A sequence is a pattern that is repeated immediately in the same voice but that begins on a different pitch class.

A tonal sequence will keep the pattern in a single key, which means that modifiers of the intervals (major, minor, etc.) will probably change

A real sequence (or modulating sequence) transposes the pattern to a new key

A modified sequence is one where the pattern isn’t exactly replicated, e.g. a series of notes where almost all are the same the second time around except a perfect fourth is instead a major third

This is all contrasted with imitation, which is a sequence but with a different voice.

A canon is a contrapuntal where two voices have the same rhythm and contour but are offset by a measure. For example, singing “Row, Row, Row Your Boat” offset by a measure from someone else would be a canon if it had an end, although in that example, it’s actually a round because it has no end. Canons and rounds make are both examples of imitative counterpoint.

A cadence is a harmonic goal, specifically the chords used at the goal (pg. 155). They say “goal” in the sense that the ultimate harmonic goal of a tonal composition is the final tonic triad.

A phrase is a relatively independent musical idea terminated by a cadence. A subphrase is a distinct portion of a phrase, but it is not a phrase either because it is not terminated by a cadence or because it seems too short to be relatively independent (pg. 160).

An elision is when the last note of one phrase serves as the first note of the next one (pg. 162).

A sentence is a musical unit consisting of an initial musical idea, a repetition or variation of that idea, and a subsequent passage that moves to a cadence. Sentences are typically organized as a single phrase (of variable length), although long sentences may contain more than one phrase (pg. 180).

A non-chord tone (NCT) is a tone, either diatonic or chromatic, that is not a member of the chord. An NCT might switch to being a chord tone throughout it’s duration if the harmony changes.

Information specific to the book

The exercises can have multiple answers and don’t necessarily list them all in the back of the book, but it doesn’t mean they’re wrong. No answers are given to the checkpoint questions/exercises though.

I remember reading in the Amazon review that there are some mistakes in the book and that they may trip up a new learner; it’s a good idea to try to find and then ignore/correct those.

Site: the site is “out of print”, but it was mhhe.com/tonalharmony6