Extended Tertian Harmony (Multiple Choice and Dictation)
Hindemithian Value and Root Progressions
Pc-set analysis and transformation
Hexachordal Inversional Combinatoriality
Addition and Inversion Vectors
Bc-set class transformation

Dictation

Extended Tertian Harmony
Funk Ostinato
Syncopated Melodic Fragments
1 trichord row
1 tetrachord row

***

Now, here’s what to do about it:

0. Finish all your assignments for the quarter, and revise any assignments that I’ve asked you to revise. Then you’ll be studying for your exam while you do work that actually earns you credit. Do some pre-compositional planning for your final project — it will take increase your familiarity with at least two of the concepts reviewed below.

1. [30 minutes] Review the rules for notation and analysis of extended tertian chords. Remember that:

a. A “seven” chord is named according to the quality of its third and seventh, if they exist. X7 has a major third and a minor seventh, in M7 both are major, and in m7 both are minor. If there is no seventh, then it’s just m or M. (If there is no third, it isn’t realy extended tertian harmony. But chords can leave out the fifth, and it doesn’t change their name, because the sound is still substantially the same.)

b. Sometimes the bass isn’t the root, but those cases are usually so ambiguous that I won’t test you on them.

c. In chords with 9ths, 11ths, and 13ths, when unmodified, are always the 2nd, 4th, and 6th scale degrees in the major scale of the root. In other words, they form a ii chord, in the octave above the basic chord.

d. The number after the root name of the chord comes from its highest UNaltered tertian factor, assuming that no factors other than the 5th are skipped. An unaltered stack of thirds in the C-major scale, starting from C, is:

— a C major chord if 3rd and the 5th are present

— a CM7 chord if the 3rd and 7th are present

— a CM9 chord if the 3rd, 7th, and 9th are present

— a CM11 if you add an 11 to a CM9, and a CM13 if you add a 13 to a CM11.

e. If you add a 13 to a chord that doesn’t have 7, 9, and 11, then the chord is named for the highest unaltered factor below the skipped factor. Thus, a C major chord with a flat 7, a 9, and a 13, is called a “C9add13”. Because it has no 11, the highest unaltered factor is the 9. (You might say that 13 isn’t really an unaltered factor if it can’t rest on the 11; think of the system as one that really relies on stacking.)

f. Any altered factors, whether they are above or below the highest unaltered factor, are indicated as “b5”, “#9”, etc.

g. X7 chords with extra tritones, like “#4”, “9#5”, and “b9”, are useful as examples of structural dissonance, when they use those tritones to strengthen or intensify their function as dominant chords. In the right context, the need for resolution feels stronger.

h. Dissonances added to relatively stable chords, such as those at ends or beginnings of phrases (and elsewhere) can be regarded as individuated dissonances. They can assume the structural role of consonance, while their dissonant quality contributes to the overall style, or representational purpose, of the piece. The bottom line is this: Individuated dissonances feel stable and do not need to be resolved, structural dissonances at least imply the need for resolution.

 2. [60 minutes] Using a page with plenty of space, make a reduction of a short phrase from a Bach chorale using Hindemithian Roman numerals to measure the value of each chord (including non-chord tones and passing tones!). Write the root progression beneath the left-hand staff, and identify the guide tones each time you see a tritone in a chord. (Recall that the guide tone is the tone that moves most easily to the root of the next chord — normally by step.) You may need to review Hindemith’s interval series (1 and 2) in order to complete this assignment.

3. [30 minutes] Using a deck of cards or a pair of dice, produce random sets of 4 pitch-classes and analyze each set.

a. For a deck of cards, let A = 1, J = 11, Q = 12 (i.e. 0). Discard Kings, or use them as an excuse to take a swig of a non-alcoholic beverage nearby. (Stay hydrated while you study!) When you get repeated values, put them aside for use in a later chord.

b. For a pair of 6-sided dice, let one die represent C#-F# (pitch-classes 1-6) if the second die comes up odd. If the second die comes up even, let the first die represent G-C (7-12 (i.e. 0)) instead, by adding 6 to the value. Thus, if the dice read 2, 3, the value is 2 (D). If the dice read 2, 4, the value is 2+6=8 (A flat), adding 6 because the second die was even.

c. Perform an interval vector of the chord. The vector (like all pitch-class interval vectors in mod-12) should have six positions for the six interval classes, and the total of the values in those positions (for a tetrachord) will always be six; tetrachords always have 6 intervals. Pentachords always have 10 intervals, and hexachords always have 15.

d. Find the chord’s normal form, Tn set class, and TnI set class. If you’ve forgotten how to do this, consult my tutorial.

4. [30 minutes] Produce a trichord, a hexachord, and a 12-tone row. Determine whether the row has hexachordal-inversional combinatorial properties.

a. Using the method described in step 3, find a trichord, and then duplicate the trichord with one of the same kind (a transposition of it), having no common tones with the original. (Do this intuitively.)

b. Determine the chord’s Tn set class. (Note, in past lectures, I’ve used the TnI set class in this stage, because the goal was to find properties of a basic trichord or tetrachord type, to use with its inversion in the style of Schoenberg’s Op. 19. However, each Tn set class within a larger TnI set class will produce its own unique matrix, and it will have its own properties. In general, use the Tn set class at this stage.)

c. Make an addition matrix of the chord (with zero in the upper-left hand corner, and the Tn set class in the left column and the top row).

d. If all values 0-e are in the matrix, then the chord’s complement is “Z-related”; it is not in any way a transformation of the original hexachord. (But what does it have in common with the chord?) If this is true, then start over from step a, and repeat, until you get to step e. (If not, then proceed to step e.)

e. If any value 0-e is missing, then create a “normal inversion” of the hexachord (turning its normal form upside-down around the axis of its first member, by counting down for the new chord, whereever the original chord counts up), and transpose it by that value. The result of this inversion+transposition should be the complement of the hexachord you started with, no matter which transposition of the hexachord you use as your starting point. All that matters is that you are transposing the “normal inversion” of that chord, by the missing value, and not trying to transpose some other chord by that value!

5. [15 minutes] Use the results of task 4 above, to create a beat-class set tesselation of four trichords. This won’t be easy until you learn exactly how your “complementary hexachord” (produced in step 4e) breaks down into two trichords that are members of the same TnI set class of the trichord you started with.

6. [15 minutes] Write one of the hexachords from task 4 above, as 16th-note beat classes in a 3/4 bar. Write a mod-4 interval vector of the resulting rhythm. Remember that beat-class interval vectors always involve forward motion, because time doesn’t move backward. The distance from one beat-class to the next, mod-4, will be calculated by counting forward in 16th-notes, like this: “1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3 …” etc. until you get to your destination. Is this beat-class set funky? It’s funky if it has fewer zeros than any other interval class.

7. [120 minutes] Meet with your dictation groups three times this week.

a. Play your three five-note extended tertian harmony chords for your peers, and hear theirs. If you didn’t get the right answer, were you at least correct about the presence or absence of semitone-class or tritone-class intervals?

b. Play your two rows for your peers, and hear theirs. Each of you should have one three-voice chorale made of four trichords, and one four-voice chorale, made of three tetrachords. Each should complete the aggregate without any redundant tones. Remember that you’ll be graded according not only to the accuracy of your pitches, but to the correctness of your chords with respect to the presence or absence of semitone-class and tritone-class intervals. So don’t withhold your guesses!

c. Practice writing syncopated melodic fragments from interesting rhythms (mostly from vocal lines) that you hear on the radio. You might be surprised at the syncopation in some of your favorite refrains.

Given the dimensions of Pitch, Time, Loudness, and Timbre, choose two that you will focus on as the bearers of some kind of technique that you learned in class this quarter.

1. For these two dimensions (your choice), plan your approach carefully using a technique described in class. These aspects of your piece should be analyzed thoroughly in the final draft.

2. Construct ways of using the other dimensions so that they are inutitive/impulsive, rather than planned. But bear in mind that intuitive/implive doesn’e keep you from making careful decisions. Examples given in class make frequent use of timbre and dynamic contrast in order to clarify relationships in the other voices?

And that’s it! Try to make your decisions based on what musically interests you rather than what you think is easy. This composition is an opportunity for you to put your learning into practice in a detailed, musical, and imaginative way.

I look forward to hearing what you come up with!

—-   —-   —-

*By the way, don’t skimp on the length of these pieces — I will grade projects down if they feel too sparse, incomplete, or haphazard! Need more length guidelines? Probably not — it’s more important to complete a full musical expression, consisting of at least 3 phrases, each of which feels well-rounded and meaningful. If you do that, my guidelines won’t matter, but when in doubt, consider the following three options:

- About 44-52 quarter-notes in length (e.g. 26 mm of 2/4) for a piece in the style of the Webern Op. 18. Think of highly active, and dynamic, ever-changing rhythms of sixteenth-note textures, dense two-handed chordal accompaniment with independent left and right hands, plus a sung melodic line. Other points of reference include Schoenberg’s pc-set saturation style (piano pieces Op. 19, songs Op. 15).

- About 60 quarter-notes (e.g. 20 mm of 6/8 or 3/4) in length for a piece in the style of the Stravinsky songs (including “Full Fadom Five”). This style consists of an 8th-note texture, with piano or instrumental lines are highly independent and contrapuntal. This is a good choice for beat-class set tesselation.

- About 80 quarter-notes in length (e.g. 20 mm in 4/4 or 27-30 mm of 3/4) for a piece in the style of the Berg “Der Wein” excerpt. This consists of a chorale-like setting of quarter notes in multiple (4-5) voices, with a few 8th-notes and syncopated passing tones. A good choice for the use of extended tertian harmony or the Hindemithian approach to harmony.

The only other requirement of this assignment is that there be two examples of one TnI beat-class set-class, appearing in two different transpositions (or inversions), in one of the ostinato lines. In other words, a particular rhythmic pattern of notes should appear in one metric position, and then also in another, contrasting position, within the same line.

To write a line like this 

1. Use your instincts about a rhythm that feels good, and then
2. Find 3-5 of the notes (any set, even if it isn’t a musically obvious grouping)
3. Analyze the bc-set that it forms. Then
4. Place a transposition of that set (preferably one that shifts weak notes to strong and vice versa) elsewhere in your 2-4 bar line. 
5. Find a creative way of linking the two instances of the pattern, and connecting them melodically. They do not need to make use of the same pitch-classes.

I mentioned two other things to consider, either of which will earn you extra credit.

A. Find a pattern with a mod-4 interval vector containing 0 elements in the i.c. “4” position (i.e. 1110, 1320, etc.) it appears that John Coltrane, James Brown, George Clinton, and Mishell Ndegeocello favor sets like this, so maybe they’re onto something.

B. Make the interaction of the lines semi-independent. That means give them a sense of separation and digression from one another, but not complete independence. Find a way to confuse us momentarily about which line is which.

Here are my takes on the two main serial techniques for rhythm, with brief explanations included.

In addition, your keyboard-improvisation assignment for next week makes use of the (less-common) beat-class set serialism method, along with a method of pitch-serialization devised by yours truly. :)

[ http://benleedscarson.com/feeling-about-hate-speech/ ]

Also, don’t forget to read the “Overview” page for answers to any specific questions that arise while you’re working. And feel free to email me too.

For more information on exactly what the bug was, see the comment below.

1. Two melodic Fragments. Compose two “16th-note syncopated” phrases of 2 mm each; one in 4/4 and another in 3/4. “16th-note syncopated” means, in this case, that notes are just as likely to fall on odd beat-classes (“e” or “a” in the “1 e & a” sequence of 16ths) as on even beat-classes. So, beat-class 0 is the down-beat of a bar, 2 is the next eighth, 4 is the next quarter, etc. Beat-classes 1, 3, 5, etc. are “off the eighth.” Compose fragments that contain about as many even and odd beat-classes.

2. Two twelve-tone chorales.

a. 4 x trichords, where the second pair of trichords makes a hexachord that is some inverted transposition of the first pair. (Hint: *most* 12-tone rows have this “hexachordal inversional combinatorial” characteristic! Just try writing a row, check to see if it’s kosher, and then if it’s not, make a change. Won’t take you long.)

b. 3 x tetrachords. (No explicit combinatoriality is required. In fact, since tetrachords are much harder to hear than trichords, it’s best if you try to give your dictation partners some sense of contrast between the three chords.)

EXTENDED COMPOSITION

1. Choosing either your “Hindemithian” chorale or your PC set-class saturation composition, perfect the work you’ve done so far.

2. Then add about 6-8 bars to the work, making use of beat-class set saturation (see lecture notes Wednesday 2/17) or beat-class set tesselation (see lecture notes 2/19). These terms are also explained in the first “comment” below this post. Use the “Matrices & Vectors” primer document to plan your work. Please note!: the document has two sheets, which you access with “tabs” at the bottom of the window. Please read my notes and step-by-step instructions on the first tab marked “OVERVIEW.”

3. Analyze your work in full. Hindemithian compositions need to include a full description of both harmonic value (Roman numerals and fluctuation diagrams), as well as an identification of your root progression. When analyzing your set-class composition, clearly identify (a) the basic characteristic set-classes you are using (b) each of their manifestations in the music itself, and (c) any points of obvious harmonic contrast.

1. Identify row forms throughout the work. Row forms are transformations of a basic series, or row*, which we identified in class. You may divide the initial identification task between yourself and a partner, but you will run into “snares” where Schoenberg’s distribution of the notes is imaginative, and not easy to see right away.

2. Graph the entire piece, describing how Schoenberg’s decisions shift from one passage to the next. In your graph, show row forms, rhythmic features, tendencies to favor particular vertical intervals and sonorities, as well as textural aspects. Emulate my graph of Opus 33a (distributed in class), but feel free to alter my method to suit your way of hearing the work.

*Series, row, and ordered aggregate are interchangeable terms, usually assumed to refer to an ordering of all available pitch classes. (All of these three are examples of “ordered pitch-class sets”.) In a 12-tone equal-tempered tuning system, the ordered aggregate is a “12-tone row”, or a 12-tone series. The method of composition we are studying here is called by many names, including “dodecaphonic music”, “twelve-tone music,” and “twelve-tone serialism.” Schoenberg, Webern, and Berg wrote extensively with this technique, beginning in roughly 1923. Dozens of composers in Europe, the United States, and East Asia made use of the system from 1945 and onward, expanding it to explore other dimensions like rhythm, duration, loudness, and timbre; in the 1960s extended the serial concept toward microtonal divisions of the octave.

2.  Compose a pair of melodic gestures (broadly defined) that together occupy approximately 3-5 bars. Together, they should make up a small musical sentence. Avoid composing anything that is purely diatonic; make sure that the pair cannot be encompassed by the major scale. Make sure that the rhythm is interesting and at least partially syncopated.

3. Emulate the style of Schoenberg’s Op. 15 song cycle and Op. 19 piano pieces (which we have discussed in class), to complete the composition. Note that Schoenberg’s opening two phrases in this work (mm 1-2) use at least one example of each TnI set class listed in step one. To complete your own phrase-pair, compose a supportive texture, beneath the melody, which changes character frequently. In composing the texture, make sure that you include at least 10 instances of the trichord characteristic you have chosen, voiced in a variety of transpositions and voicings, using both vertical and horizontal distributions. In doing so, also make sure that you use at least one example of all the possible TnI set classes that possess the characteristic you have chosen.

4. Analyze harmony in your composition by circling all instances of the “characteristic” trichords, and identifying any contrasting harmonic characteristics that you think are important. Use my Tn & TnI set class flowchart for a reminder of how to analyze the characteristics of your work.