# Lesson 4: Numbers

(We’re going to be in C major all day today.)

Back in lesson 2 we talked about triads. We defined a triad as being a chord made of three notes, each a third apart. Because of octave equivalence, triads can look like many things. Here’s what the close-packed version looks like:

But all these are also triads:

If you carefully take apart each instance, and reconstitute to the close-packed conception, you get the first example.

Sometimes, we like to be able to talk in detail about which members of the triad are where. That’s where numbers first come in. That first triad in the second example has the middle note of the close-packed version on the bottom, in the lowest voice. This makes it sound less stable than if the root[*] were in the lowest voice. Since this note is a third above the root, we call it the third of the chord, and label it with subscript 3 to indicate the bottom voice:

All the numbers in the today’s explanations are from the root (name) of the chord.

We can use either the chord root name (C) or the function (T). If we were working on something with lots of detail, like following voice leading specifics, we might want to know the specific order of all the other notes. We can also show those, with the numerals 1 and 5:

This is known as an inverted chord (or chord in inversion). Inversion simply means that a chord member other than the root is the lowest sounding note. Inversion can also be applied to intervals, in which the bottom note is moved up an octave to create the complimentary interval (thirds when inverted become sixths etc).

Most of the time, a triad with 1, 3, and 5 is our baseline assumption for a chord, so we don’t have to mark those notes as being present unless we have a specific reason. However, there are lots of times that we hear a function, say predominant, but upon closer inspection the chord isn’t 1, 3, and 5 (fa la do in solfège), but 1, 3, and 6 (fa la re). Then we can indicate this chord as being a predominant with a 6 (instead of a 5).

This chord also happens with both 6 and 5:

In general, an even number added to a chord (6, 4, 2) replaces the odd number a step below it.

One other number that often gets added to chords is 7. Now the chord has four notes instead of three, but still all stacked in thirds in a close-packed version:

The seventh chord[†] is most commonly found on the dominant, but occurs in many versions depending on the genre of music. The 7 of any chord is an unstable note, and has desire to resolve down (just like the leading tone has a desire to resolve up). Part of this is due to the dissonant interval between the 7 of the dominant and the leading tone – a tritone, diminished fifth, or augmented fourth. Try playing ti and fa together.

The resolution from dominant 7 to tonic looks like this, with all the extra numbers:

Notice that ti resolves up to do, fa resolves down to mi, and all voices besides the lowest move by step (or stay the same).

Other numbers sometimes get added; odd numbers 7 and higher (9, 11, 13) are in addition to the triad and not replacing any pitches.

Here’s a Prelude by JS Bach for keyboard recording: https://www.youtube.com/watch?v=PXMVkQ70I88

with a reduced score[‡] with the all numbers next to the triad letter names, and a further level of T P D T. We’ll talk more about the other chords’ relationship to T P and D next time. This is a very basic understanding, leaving out modulation and other prolongational concepts, but we’ll come back to those later too.

This type of very detailed analysis can help study voice leading – the idea that each of the five strands of music above have a coherent internal idea (most of the time). Studying voice leading helps us understand why certain notes tend to resolve in certain ways and lets us form expectations for how horizontal voices can effect vertical chords and chord progressions.

Next time: substitute functions

Main Ideas:

Root: The bottom note in the close-packed version of a triad, the letter name of the chord.

Inversion: Chords: When a chord has a chord member other than the root in the lowest voice.

Intervals: to invert an interval, move the bottom note up by octave (or the top note down by octave) and you’ll have the complimentary interval. Seconds become sevenths, thirds become sixths, fourths become fifths. Additionally, major intervals invert to minor intervals and vice versa.

Seventh Chord: Chord with four notes, all stacked in thirds so that the outside interval is a seventh.

Diminished fifth, Augmented fourth: also called tritone. Technically, a tritone is the augmented fourth, but due to equal temperament we don’t really differentiate between Augmented fourth and diminished fifth. A diminished fifth is one half step less than a perfect fifth (B up to F) and an augmented fourth is one half step more than a perfect fourth (F up to B). Highly dissonant in classical music. Often used in a symmetrical fashion in more chromatic genres.

How numbers work: Arabic numerals show interval from the root of a chord. Can be used with chord letter names or functional T P D. Numbers in subscript show bass or lowest note, numbers in superscript show upper voices. 1, 3, 5 are assumed unless other numbers show otherwise. Even numbers (2, 4, 6) replace the odd number below (1, 3, 5). Odd numbers (7, 9, 11, 13) add to the triad without replacing. Tho not shown here today, sharps and flats should be added to any number that is outside the key signature or expected triad: for example, the D major triads in mm. 6 and 10 of the Bach example might include a # 3 to show that the F# isn’t in the C major key signature. Or in m. 33 where the B♭ should be indicated with a ♭7. The application of # and ♭ changes when you’re using chord names or symbols that indicate the major- or minor-ness of the third, fifth or seventh.

[*] The bottom of the close-packed version, the name we call the triad

[†] This sound is called Dom7 in lead sheet symbols and jazz lingo, regardless of function.

[‡] A reduced score can come in a couple different versions, but this type condenses all the arpeggios into chords so that more music fits on one page. If you’d like to see the score with rhythm: http://imslp.org/wiki/Prelude_and_Fugue_in_C_major,_BWV_846_(Bach,_Johann_Sebastian)