Readers' Questions and Answers
About Music Theory

Question 1 Meaning of key in music.
Question 2 Meaning of circle of fifths. How to figure out which sharps or flats are in a given key.
Question 3 Meaning of augmented and diminished keys [sic].
Question 4 Meaning of numbers indicating chord inversions; figured bass in Baroque music.
Question 5 Primary and Secondary Triads.
Question 6 Meaning of "tonic sound" and "subdominant sound."
Question 7 Passive and active chords.
Question 8 Secondary dominants.
Question 9 Chord roots.
Question 10 Cut time.
Question 11 Difference between 4/4 and 2/2 time signatures.
Question 12 Accidentals.
Question 13 How to read notes.
Question 14 How to read sharps and flats.
Question 15 Two sets of half-notes connected by two lines in 4/4 time.


A chord is a group of notes that are related to form a specific sound. A triad is a three-note chord.

What does key mean?

I like to think of key (also called "key center" or "tonal center") as "home." That means that if you don't hear this at the end of the piece, you aren't convinced that the piece is over: you aren't "home" yet.

Try this experiment. Play "Mary Had a Little Lamb" in the key of C (melody starts on E). When you come to the last note, instead of playing C, play E-flat (or some other note). You are vastly dissatisfied because you wanted to hear C. C at the end is what the whole song was leading up to. That's because C is the key of this song (the way you started it here on E, that is; if you started on a different note, you'd be in a different key; starting a song on another note is called "transposition.")

If the composer goes into other keys, on the way back to C (or whatever the key is), these are just red herrings. Don't worry. The music will come back to the original key, and you'll be satisfied that it's the end of the piece. It's part of the composer's skill to see how far afield the music can range and still return smoothly to the original key.

There are exceptions, of course. There are some pieces which start in one key and end in another, but these account for about 2% of all music. In this case, I'm not counting pieces that start in, say, C Minor and end in C Major. I'm speaking of a piece that starts in C Major and ends, say, in A Major or B-flat Major. (A transformation from minor to the major, which is called a "parallel major" because both key names have the same letter, usually occurs on the last beat of the song and is called a "Picardy third." Bach does this all the time.

You didn't ask, but here it is, anyway: relative major and minor keys are keys which share the same key signatures and thus the same set of triads, such as A Minor and C Major. Composers love these sets of relative majors and minors; it's like the hidden passage way in the board game "Clue" where you can automatically be somewhere else with no effort.)

Another favorite composer device is to set up a big V - I cadence (the one that says "The End!") and then go V - vi instead. This is called a "deceptive cadence" (The....En- woops! Surprise!). A deceptive cadence is a key harmonic device of Mozart and Handel, to the point that it is chosen as a technique when the music of these composers is satirized.

It's basically a technique to extend the piece (not to modulate into another key). The composer then repeats the music, usually verbatim, and then gives you want you expected the first time: V - I, so you hear what you expected at the end: the I chord, the key name note.

Can you explain the "circle of fifths"?

Yes. Pull up a chair; or rather, print this out, find a pencil, and go to the piano.

The circle of fifths explains how keys relate to one another through shared triads. It is also a mechanism for modulation. Real helpful, huh?!

Draw a circle. Pretend it's a pie (chocolate, of course!) and draw a diameter through it (is that called a "cord?"; it's been a long time since I took geometry!). Ignore half of the circle. Draw a clock face on the upper half (9:00, 10:00, 11:00, 12:00, 1:00, 2:00, 3:00).

Isn't this easy in e-mail?

Now you've got a circle and 7 lines radiating out from the center. Another way: you have the top half of a clock face.

Label 12 noon as C. 1:00 is G; 2:00 is D; 3:00 is A. 11:00 is F; 10:00 is Bb; 9:00 is Eb.

Put this paper aside, and on a new sheet:

Write C D E F G A B C, spaced 1" apart, and below the C write the Roman numeral I (one). Below D, write II, and so on until you get to C again. You could call the new C "VIII," but since it's a duplication of the previous C, we'll also call it I. These numbers represent what is called "degrees of the scale." Since we started on C, it is a C scale. C is the first degree of a C scale. D is the second degree. If we started on A, A would be the first degree and B the second, etc.

Triads may be built on any degree of the scale.

Put aside this paper and these ideas for the moment. We must discuss how triads are built.

There are 4 kinds of triads: major, minor, augmented, and diminished. Each has a different "secret formula," which works to build that flavor of triad in any key, no matter how weird (like the key of Gb!).

The secret formula is measured in "half-steps." A half-step is the smallest distance possible on the piano keyboard, such as between C and C# or Ab and A.

The middle note of a major triad is 4 half-steps away from the bottom note, which is called the "root" in fancy restaurants. The middle note is called the "third" (as in -third degree- of the scale!) and the top note the fifth. To find the fifth of a major triad using the secret formula method, count 3 half-steps away from the third.

So, in gray-bearded pedantic verbiage: between the root and the third of any major triad is a distance of 4 half-steps, and between the third and the fifth is a distance of 3 half-steps. (Actually, graybeards would call these "the interval of a major third" "the interval of a minor third," respectively.)

A minor triad has a different secret formula: between root and third is 3 half-steps; between third and fifth is 4 half-steps. Note the distances (also called "intervals") are reversed from what they were for the major triad.

The augmented triad has a formula of 4 and 4; and the diminished 3 and 3.

Now let's go back to scales. Depending on which degree of a scale you use, the "flavor" of the triad is different.

In a major scale, triads built on I are major; triads on II are minor (and thus written in lower case: ii). Triads on other scale degrees of the major scale: iii, IV, V, vi, vii(. That "degree" symbol is how one designates a diminished triad.

So, in a major scale, triads built on the first, fourth, and fifth scale degrees are major. Those on the second, third, and sixth are minor. And we have a odd one, the seventh degree, which is a diminished triad.

Triads I, IV, and V are called (drum roll!) "the pillars of tonality." These immortal words were given to us by Jean-Phillip Rameau, French Baroque composer and music theorist. (Do look up his "Tambourin." It's a great piece! A tambour is a small two-headed French drum. A tambourin is an old dance from Provence done to the accompaniment of such a drum.)

With these 3 triads (I, IV, and V) you can easily establish a tonality. Actually, you can do it with just two of these: V - I. This "cadence" (ending formula") sort of says, "The End!"

Now return to your clock face. Write I above C in red pencil; put V above G and IV above F. Compare this to the degrees of the C scale. V was under G and IV was under F. Yes? Everything checks.

Get a blue pencil. Write I somewhere above G (a.k.a. 1:00) and IV above C (12 noon). Write V above D at 2:00.

Get a green pencil. Write I somewhere above D (a.k.a. 2:00) and V above A (3:00). Write IV above G (1:00).

Get a purple pencil. Write I above F (a.k.a. 11:00) and V above C (a.k.a. 12 noon). Yes, this is getting a bit crowded. Write IV above Bb (10:00).

Last pencil. Your choice of color! Write I above Bb (a.k.a. 10:00) and V above F. Write IV above Eb.

On this clock face, the key of C is in red pencil. C is I, G is V, and F is IV. You already knew this!

The key of G is in blue. G is I, D is V, and C is IV.

The key of D is in green. DG is I, A is V, and G is IV.

The key of F is purple. F is I, C is V, and Bb is IV.

The key of Bb is in your-color-choice. Bb is I, F is V, and Eb is IV.

Note that C, for example, functions three ways in three different keys. It can be the key (when C is I, the root). It can be V (called the "dominant," which is a carryover from Hebrew chant). And it can be IV (called the "subdominant," which is *not* because F is "under" G in the alphabet but because F is the "dominant from below"--that is, the distance ("interval") between F and C is the same as between C and G.....7 half-steps (a.k.a. interval of a "perfect 5th").

The firmest cadence is V -> I ("authentic cadence"), but a IV -> I is final-sounding, too ("plagal;" sometimes called the "amen cadence"). An even better cadence is IV -> V -> I (pillars of tonality, you see).

Back to the clock face.

Note that the G triad can also function as I, IV, and V, depending on what the key is (that is, which chord is I).

And so on.

Notice that between each of the notes at the points of the clock face there is a distance of a perfect 5th (P5). This is where the name "circle of fifths" comes from. I'll bet you thought we'd never here!

The circle of fifths is important because by using the chords which have a main function (I, IV, V) in two "adjacent" keys, we can modulate easily. These guys are called "pivot chords." Actually, any triad shared two keys can be used as a pivot chord.

And I'm really sorry they are not called "pivot triads." A triad is ONLY a 3-note chord, never more notes. A chord, on the other hand, can be any number of notes, from three on.

And there's also a thing called a "diad," which is only 2 notes, but it's not a kind of chord and the term is hardly ever used; forget about a diad if you want.

So! A composer can start in the key of A, for example, play A and then D, which is V -> I (the quickest way to establish a "tonality" or key, as you recall).

That D is V of G, so the composer can then write a G triad. Now we're in the key of G in just two moves! And so on around. In the space of a short time, the composer can modulate from A (3 sharps) to Eb (3 flats), which is pretty darn distant.

Hie thee to a piano and play this sequence of triads in root position; that is, the letter name as the bottom note. (I took them from the clock face, starting at 3:00 and moved counterclockwise to 9:00):

A - D - G - C - F - Bb - Eb

Do you hear how the triads "interlock?"

The whole circle of fifths has 15 "stations." Your homework is to use the 7 half-steps method and fill in the rest of the circle of fifths. The sharp keys go to C#. The flat keys go to Cb. Your circle comes to a crashing halt when Cb meets C#. If you don't arrive at this brick wall, you've made an error.

You didn't ask but I know you'll be keen to have this other information: sharps and flats are added to key signatures by the 7 half-steps method!

Count 7 half-steps away from F# (the first sharp) and you'll get C#. Any key with 2 sharps in the signature will have F# and C#. What is the third sharp? Count 7 half-steps away from C# and you get - - - G#!

Now do the flats. Count 7 half-steps from Bb (the first flat), and you find the second flat, which is Eb.

Isn't that all nice and tidy? Glad you like it. Now derive the names of the rest of the sharps and flats for other keys on your circle of fifths.

Congratulations! you've just completed a couple weeks of college freshman music theory!

What is an augmented or diminished key?

Only triads are augmented or diminished. Never keys.

It's an unfortunate redundancy that the words major and minor apply to keys as well as triads. To augment means to make larger; to diminish to make smaller. In augmented triads, the outside distance is larger by a half-step - - as compared to the outside distance in a major triad. In diminished triads, the outside distance is smaller by a half-step, as I mentioned above.

Ex: for triads built on the note C:
C - E - G - - plain old major triad
C - E - G# - - augmented
C - Eb - Gb - - diminished
C - Eb - Gb - - minor

I am wondering about the notation - - that is, the chord symbols - - for the inversions of seventh chords. In a book I have, they are notated as follows, for the inversions of the C7 chord: root position: C7; first inversion: C 6/5; second inversion: C 4/3; third inversion: C 4/2. Of course, they are not written as fractions, as I had to type them, but with the first number directly over the second and no line in between them. Can you tell me how these numbers were arrived at? I have thought of several possible answers, but the rules don't hold true for all three inversions.

The answer has to do with the Baroque accompaniment practice called figured bass. The accompaniment to a sonata (or whatever) was often given only as a bass line with some numerals ("figures") beneath it.

As you may know, accompaniment (called basso continuo) in this period was highly improvisatory, with the soloist and the continuo player tossing motifs, riffs, and ornaments back and forth for the sheer fun of it. Therefore, to write out a complete accompaniment (now called a realized figured bass) was not often done in the period; only modern editions do this, as most keyboard players cannot realize figured bass at sight; if they do play from a figured bass line, they often write out their ideas or at least write out a skeleton of what they are going to use, dependent on what the soloist throws them.

The numbers below the bass note tell the player the intervals above that written bass note at which pertinent notes of the chord will be found. Knowing this, the players would know what type of chord it is (major, diminished, etc.).

C7 is a C Major chord topped by a minor 7th interval. This is root position, as you noted: C, E, G, Bb. (I am not going to discuss here why it's a minor 7th interval.)

First inversion of a C7 chord is E, G, Bb, C. E is in the bass, by definition because to invert anything means "to make the bottom become the top" (like a glass of water). In a root position chord, the bottom (root) becomes the top, thus exposing the 3rd of the chord (in this case, an E).

Let's look at the numbers. A 6th above the bass note is one note we need; and a fifth above the bass note is the other note we need. Thus a C 6/5 is an E (because this is the note written in the bass), a Bb (a 5th above the bass note), and a C (a 6th above the bass note). Notice that the G (which would be a 3rd above the bass note) is not "discussed" in the figured bass. This is because the important notes are the root and the third; the quality of the chord is not altered by its 5th. Unspoken implication: a trained performer would know to add a G if desired.

Let me stop here and discuss intervals above a bass note, in the event that other readers are not sure what this means.

The easiest way to learn this is to write the alphabet, A through G, out twice:


If C is "one" (I, in theorists' jargon), then VI (six) is A. If E is I, then VI is C. Every time I changes, VI changes.

In essence, you count the starting letter as one and stop counting when you've gone as far as the number tells you to. E is I, F is II, G is III, A is IV, B is V, and C is VI.

That is why if E is I (1) B (Bb in this example) is V; and (2) C is VI. (As I said, I'm not doing to discuss here why it's Bb instead of plain B.)

On to second inversion, which is 4/3. If first inversion (of the C7 chord) has E on the bottom, second inversion has G on the bottom (the chord has been robbed of its bottom note again and the note next up in the stack is now exposed): G, Bb, C, E. Using the alphabet system of intervals, above, you can see that if G is I, then III is Bb, and IV is C.

Third inversion (4/2) has another robbery of the bottom note and a new note exposed on the bottom. By now the 7th is on the bottom (Bb). Therefore the chord, from bottom to top is: Bb, C, E, G. If Bb is I, then II is C and IV is E.

In all these, note that the 5th of the basic C triad (G) is never discussed, as it has no bearing on the quality of the chord. It's always plain G (never Gb or G#).

Also note that the first number always tells the location of the root of the chord, except for a 7th chord (which we have here). Pretend it's just a plain 6/5 and 4/3. In the first inversion (6/5), 6 tells the location of the root (C, in our example). In second inversion (4/3), 4 tells where to find the root.

Why this system falls to pieces for the third inversion (4/2), I don't know. Maybe it's an aesthetic thing: 2/4 looks dumb.

Let me make one more comment. In figured bass, you would never see C 4/2 under the bass line. Instead you'd see a Bb as the written bass note and the numbers 4/2 beneath it. As an knowledgeable basso continuo player, you'd recognize that 4/2 indicated 3rd inversion of a chord (that is, a 7th chord), and it would be easy to figure out what the root was.

My teacher said chords I, IV, and V are primary chords; and chords ii, iii, vi, and vii are secondary chords. Is this because they are major triads (for primary) and minor and diminished (for secondary), respectively? How about in minor scale? The same?

It just happens that I, IV, and V are major in a major key. In a minor key they are minor (exception: usually the V chord is a major triad in a minor key so the half-step relationship between the 7th and 8th degree of the scale can be preserved. The half-step "wants" to resolve upward to the VIII. VIII is just I an octave higher. Since they are the same letter name, VIII is not used, but I again. In the scale of C Major, there is a half-step between B and C. In a C Minor scale, there are three flats (Bb, E, and Ab), so the B would be Bb. The distance between Bb and C is a whole step, not a half. Therefore the instability between VII and I is gone. That VII doesn't necessarily want to go anywhere; it's stable. So, in a minor key, the 7th degree is raised a half-step. In the case of the C Minor scale, that Bb is raised to B-natural. Once more we have a half-step between VII and I. They are "primary" because the three of these determine, for all intents and purposes, which key a piece is in. When you hear a piece using these three chords, you know what key it is in. Jean-Phillippe Rameau (in 1744) called these three chords the "pillars of tonality" for just this reason.

In each key, of course, the letter names of the triads which represent I, III, and V will change, just as the letter name of the key will change.

Actually, you need only two chords to establish a key: V and I ("the end").

The secondary chords are there to provide variety, and, as you said, to provide alternate ways of moving between the primary chords.

As for the minor scale, yes, but the chord type will be different. Ex: tonic is minor (i), supertonic is diminished (ii dim), mediant is major (III), etc.

Why are the I, iii and vi chords said to have tonic sound? Why do the chords ii and IV said to have subdominant sound?

Let's use C Major key.

I is C: made of C E G

iii is em: made of E G B
vi is am: made of A C E

You can see that both iii and vi have two notes in common with I. Therefore, you can substitute these interchangeably if you want.

As to the ii and IV having a subdominant sound, you should know that IV really means the "dominant from below," rather than "the chord beneath the V". There are 7 half-steps between IV and I, just as there are 7 half-steps between I and V.

Now look at IV (again key of C)

ii: D F A

I'll bet you can answer the question yourself now.

What are active and passive chords? My teacher said they called active because they tend to move to tonic chords, and they called passive because they tend to move to nowhere. Do you have better explanation?

Not really.

I don't think in terms of active or passive, really. But I do think in terms of what the chords 'want to do' when they're in use in a progression. Some of them want to resolve (such as a V7), and some seem to want to "sit there." Perhaps this is what your teacher had in mind.

Would you please explain the concept of secondary dominant? I've heard that it is the dominant chord (V) of the secondary chord (ii iii vi vii). It can be built by taking the secondary chord as Tonic, then secondary dominant will be the V chord. Is this the correct definition?


Say you are in the key of C. V is G. V moves to I: that is, G moves to C.

Now, what moves to G? D moves to G (when G is I, D is V).

What relationship is D to C? It is ii. (In the key of G, it's a major chord, but in C is a minor chord.)

Therefore, a common progression would be:
dm moves to G moves to C
ii V I

ii is called a secondary dominant because it is the dominant of the dominant.

You also make make the ii chord into a major chord (it's now written as II). In this case, there is an accidental written in. For this example, we'd need an F# because the chord would be D Major, not D Minor. The presence of accidentals in music often means the composer is moving around to different keys temporarily. Using a secondary dominant and the circle of fifths is a good way to do this.

What is the root of a chord, and how do I determine what it is?

The root is the bottom note of the triad (a triad is a 3-note chord; "triad" is a subset of "chord") and the letter for which the triad is named. Rearrange the notes until the letters are a skip apart. Example:


You get:

B and D are a skip apart (see list of letters, above), but D and G are not. So we take the B off the bottom and put it on top. Now we have D G B. D and G are still not a skip apart but now G and B are. So we do the same thing: take D off the bottom and put it on top. Now we have B G D. These letters are a skip apart; that is, there is another letter between them.

G is the root of this triad.

What is cut time?

Short answer: An indication of a quick tempo. You'll see this a lot in Bach, especially his organ preludes/fugues and trio sonatas. See also the alla breve section in the first movement of Beethoven's Pathétique: page two takes off like a shot!

Long answer: This all goes back to medieval and Renaissance music notation (14th century).

Perfection was allied with the Holy Trinity. Therefore, church music was always in triple meter. Ah, "perfection!"

By contrast, non-sacred music (secular music, or, as the Catholic church put it, "profane music") could not be in triple meter. It was in duple.

Whether a piece were in duple or triple meter is called its "time" (tempus). Time was notated with a circle. Perfect time (triple meter) was a full circle; imperfect time (duple meter) was a circle open on the right side (like the letter C).

Now then, each note could be cut into two or three parts. This subdivision is called "prolation" (prolatio). Prolation was notated by the absence or presence of a | through the circle or C.

Still with me? This is complex, and the terms are arcane.

Perfect time with perfect prolation is today's 9/8 time signature (3 beats, each subdivided into 3). Perfect time and imperfect prolation is

Imperfect time with perfect prolation is 6/8 (2 beats, each subdivided into 3).

Imperfect time with perfect prolation is 3/4 (2 beats, each subdivided into 2).

Imperfect time with imperfect prolation is 2/4 (2 beats, each subdivided into 2).

This last is cut time.

In the time signature, then, cut time is written as C with a | through it. An easy way to think of it is as though the C were "cut" in half.

My dad calls an answer such as this "a drink from a fire hose."

What is the difference between 2/2 and 4/4 time? They look the very same: there are four quarter-notes per measure for each.

Which time signature to use is based on how many pulses there are in each measure. A pulse is a musical "pressure". If you were walking to this music, would you take four steps per measure, three, or two?

A march is in four pulses per measure. A waltz is in three pulses per measure.

In two-two time, there are only -two- pulses per measure. In four-four time, there are -four-.

The confusion begins when notes in two-two time are subdivided. This yields quarter-notes. Therefore, a measure with each pulse subdivided looks like four quarter-notes.

This is the same as four-four time with each pulse subdivided: the measure looks like eight eighth-notes.

Now, here's the rub. In four-four time, for any subdivided quarter-notes, the second eighth is unaccented. In two-two time, for any subdivided half-notes, the second quarter is unaccented. This is called a feminine ending.

So, this is why it's important to indicate in the time signature whether you want two or four pulses per measure: so you know which notes are accented (the musical pressure I mentioned above), and which are unaccented.

When I took piano lessons forty years ago, I learned that an accidental holds through the measure for any like notes anywhere on the staff. For example, a sharp sign in front of C means that any other C's in that measure will be sharped. But in reading recently published music it seems that the sharp only applies to that particular C on the staff. Has musical notation changed? or did I learn it wrong? If it has changed, what about works published 50 or 100 years ago?

The accidental (in modern usage) means it changes, within that measure, only the note (say, C) on THAT line or space. Other Cs in the measure (in same clef or in another clef) are not changed. Same for music 100 years ago. Change was during the Renaissance.

I came across a notation I've never seen before. It looked like two half notes connected by three lines (like thiry-second-notes would be connected), but not touching the stems. There are two sets of them in the measure! The piece is in 4/4 time, so it looks like there are too many counts!

This is tremolo. What it means is to go back and forth between the two notes for the duration of a half-note, using thirty-second-note values for each note. (You will have 16 of each note.) Do the same for the second set. You'll have the equivalent of two half-notes, which is perfect for 4/4 time. This method is easier to read that a bunch of thirty-second-notes!

copyright 1996-2003, Martha Beth Lewis, Ph.D.
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Last updated August 30, 2003.