david bard-schwarz

    everthing you always needed to know about music theory but were afraid to ask

    In this book / online application I will teach you (and you will acquire) the skills and knowledge you need in order to thrive in music theory classes at any English-speaking graduate course of study. Here's who I think you are: you have made a serious life decision to be a musician; you have chosen to attend graduate school to attain a professional degree in music; perhaps you recently graduated from an undergraduate curriculum; perhaps you have been out in the world for some years making a living, raising a family and are returning to graduate school with a new sense of urgency and commitment to a career in music; perhaps you are from a foreign country and either belong to one of the previously-mentioned groups, or simply need to translate the ways you have learned to understand music in your native culture and language into the varous cultures here, in English.

    I am a teacher, scholar, writer, who has studied English, Comparative Literature, the violin, German, Interactive Telecommunications, and Music Theory (doctorate). I have read in psychoanalysis for twenty years and have published on my research specialty--music and psychoanalysis. I have taught English as a Second Language in Germany and at the University of Texas, Austin (total of 8 years). I have taught at the College of Music for 11 years.

    I mentioned "skills" and "knowledge" above. By "skills" I refer to the various systems of terms of describing and analyzing notes, chords, harmonies, and keys in tonal music, and various pitch relations that occur when tonal music yields to something new in central and western European culture around the turn of the twentieth century. By "knowldge" I mean how various theory course that you may or will take approach music.

    Be aware that there are three kinds of music theory: 1) pedagogy, 2) analysis, 3) speculation. In this course we will emphasize the second category ("analysis") exploring exactly what analysis means, what it is and isn't, but we'll touch on pedagogical and speculative issues as well. And I will place the entire course content in a cultural context as much as possible. This means that despite the fact that I may have to say from time to time "it is this way because it is this way", as often as possbile, I'll suggest motivating factors in culture that help shape how I'm suggesting we hear music here, at the College of Music.

    Presenting material like this poses a problem in my experience; to some of you the material will seem too little, too obvious, too slow; to others the same material will seem too much, too obscure, too fast. And most of you will have one reaction during one class and the other reaction during another class. It's like the problem of learning a foreign language well that one knows only imperfectly. Where to start? I'll do my best to negotiate this threshold between the too slow, too obviious, too little and the too fast, the too obscure, and the too much.

    In order for you to get anything out of this book / online application, you must know one thing: you must learn all the material and make it your own. It's not going to be enough to work through all of the handouts, do all of the assignments, write all the papers. For example, there's a simple pdf document on the neapolitan harmony. That document shows you what the neapolitan is and how it can be used in one piece of music. You should play the explanation at the piano and speak its structure out loud, as if teaching the idea to someone else. You should play the musical example so you can hear how a composer used the neapolitan, and you should speak out loud, as if teaching the piece to someone else, what the musical effect of that use of the neapolitan harmony involves. Then you should make up your own simple illustration of, say, a four-part harmonic progression using the neapolitan harmony, transposing it to a number of keys. Then you should listen for neapolitan harmonies in other pieces of music. Then you will begin to have internalized for yourself what the neapolitan harmony is. And you need to work like this for every single idea presented in this course. This is the only way for the course to be valuable to you.

      • pitches and pitch classes

        Pitches are notes that you play, sing, hear; pitch classes are silent. A pitch class is the abstract totality of all octave transpositions up and down of a pitch. I can't overstate the importance of this distinction. Pitches are sounds; pitch classes are silent abstractions. Musicians refer to pitches in a wide variety of ways (there is no standard agreement on how to refer to pitches); musicians refer to pitch classes more simply, either using generic letter names, or the numbers 1 through 12 or 0 through 11. So there are 12 pitch classes in the tonal universe, right? (Yes, but more can be said on this point; save this little open door for later). How many pitches are in the universe? An infinite number (unless there's a physicist out there who can prove that at some point low notes will flatten out to a dead line and high notes will granulate into tiny dots). We can of course only "hear" a small number of pitches; I put "hear" in quotation marks because at a certain point we "feel" rather than hear low notes and the same thing occurs for high notes.

        Here are examples of pitch and pitch class names: pdf

      • enharmonic equivalence

        If you played a note on your instrument (or sang one) and a child came up and asked you what note you played, what would you answer? If you said something like "C" you'd only be 1/2 right. Actually (even to a child) it would (in my opinion) be better if you said "well, that depends on the other notes around it!". In some contexts it could be a "C" (as in the root of a tonic triad in C major) or it could be "B-sharp" the leading tone in C-sharp minor (or major). The names and meanings of all notes depend on the context in which they occur. Enharmonic Equivalence refers to this phenomenon: C and B-sharp, C-sharp and D-flat, D-natural and E-double-flat, D-sharp and E-flat, E-natural and F-flat, F-sharp and G-flat, G-natural and A-double-flat, G-sharp and A-flat, A-natural and B-double-flat, A-sharp and B-flat, and B-natural and C-flat, are all enharmonically equivalent. And there are other equivalences; D-natural and C=double-sharp for example.

      • transposing instruments

        When you look at music and play it on your instrument either the sound that is produced will correspond to the written notes or else it will not correspond to the written notes. Notes that sound as they are written simply sound as written (how's that for a circular sentence)! Notes that do not sound as they are written must be transposed. I studied with Kent Kennan who was fond of defining transposing instruments as follows: "The key of the instrument is the sound that is produced when written C is played." Here is the very beginning of Beethoven's Eighth Symphony: pdf with the sounds provided that are produced when the written pitches are played. Notice the B-flat clarinet part and remember Kennan's definition. If B-flat is the sound that is produced when written C is played then all written pitches for the B-flat clarinet will sound a major second lower than written. Confirm this principle by comparing the key signature of the B-flat clarinet part (G major) with the key signatures of other instruments that do not transpose (F major). Since written B-flat clarinet pitches sound a major second lower than written, in order for F major to sound, G major must be written!

      • major and minor scales

        For centuries musicians made music from modes--series of whole steps and half-steps in various combinations. Around 1600 (more or less) two modes became more common than the others, or two modes were seen as the source of all the others and musicians began to focus on those two modes--major and minor. The major mode, or scale, has whole step, whole step, half step, whole step, whole step, whole step, half step. The minor mode or scale has whole step, half step, whole step, whole step, half step, whole step, whole step (the natural minor scale). Various adjustments have been made over the years to make the minor mode sound good in various contexts resulting in the melodic minor (raised 6th and 7th ascending / lowered 6th and 7th descending) and harmonic minor (lowered 6th and raised 7th).

      • keys and the circle of fifths

        the circle of fifths: pdf

        It turns out the major and minor modes interlock with one another. You can produce the A minor scale in C major by starting on the 6th scale degree of C major. So we say that A minor is the relative minor of C major and C major s the relative major of A minor. In order to maintain the intervallic relations above and start on different notes, the circle of fifths was invented. Moving clockwise by perfect fifth from C, you add leading tones to each new key. G major adds F-sharp. Continuing, D major adds C-sharp, A major adds G-sharp, E major adds D-sharp, B major adds A-sharp, F-sharp major adds E-sharp. We could keep going but then we'd spiral into very sharp regions! So we begin again back at C and move counterclockwise. When this happens, we don't add leading tones, we flatten the fourth scale degree. F major flattens B-natural to B-flat, B-flat major flattens E-natural to E-flat, E-flat major flattens A-natural to A-flat, A-flat major flattens D-natural to D-flat, D-flat major flattens G-natural to G-flat, G-flat major flattens C-natural to C-flat. If we kept going we'd get into extremely flat zones! So here's how the circle of fifths works; it's a compressed version of two spirals that could infinitely move in opposite directions--the clockwise spiral of sharps and the counterclockwise spiral of flats. At some point (often G-flat major and F-sharp minor) we enharmonically re-spell, or jump from one spiral to the other. Here is list of common relative key relations: (going clockwise from C) C major / A minor; G major / E minor; D major / B minor; A major / F-sharp minor; E major / C-sharp minor; B major / G-sharp minor; F-sharp major / D-sharp minor. (Going counterclockwise from C) F major / D minor; B-flat major / G minor; E-flat major / C minor; A-flat major / F minor; D-flat major / B-flat minor; G-flat major / E-flat minor.

        Here is a list of major keys and their relative minor keys keys: pdfThe upper key signatures move clockwise around the sharp side of the circle of fifths; the lower key signatures move counterclockwise around the flat side of the circle of fifths; notice how the two "arcs" moving in opposite directions overlap!

      • the overtone series

        the overtone series: pdf

        The overtone series (although mediated through complex tuning adjustments) is a fact of nature. When you flick your finger on a crystal glass, you hear various partials of the overtone series--usually partials 2, and sometimes, 3, 4, and even 5. Play the overtone series on C on your instrument and you can hear how theorists understood the "natural" quality of major; it comes right out of the early partials of the overtone series! The overtone series also helps us understand the enormous power of the dominant in western, classical music; the first non-tonic note we hear in the overtone series (partial 3) is a perfect fifth above the fundamental.

      • intervals

        Intervals are consonant or dissonant or neither consonant nor dissonant depending on the context of their use in a piece, the nature of the piece, and the time and culture in which it was written. In tonal music, intervals are either consonant or dissonant; in atonal music, intervals are neither consonant nor dissonant. Tonal intervals have names that suggest the consonant / dissonant nature of the interval (as in "augmented fourth" or "diminished fifth" for the tritone); in atonal music, intervals have, simply, numbers. Let's do tonal intervals first.

        For tonal music, take the letter names of pitches A, B, C, D, E, F, G. The interval between any adjacent pair of notes is some kind of second; the interval between notes that skip one note is some kind of third; the interval between notes that skip two notes is some kind of fourth; the interval between two notes that skip three notes is some kind of fifth; the interval between notes that skip four notes is some kind of sixth; the interval between notes that skip five notes is some kind of seventh.

        Refining "some kind of" in the previous language, a minor second = 1 half step between notes one letter name apart; major second = 2 half steps between notes two letter names apart; minor third = 3 half steps between notes three letter names apart; major third = 4 half steps between note three letter names apart; perfect fourth = 5 half steps between notes four letter names apart; tritone (augmented fourth or diminished fifth) = 6 half steps between notes four letter names apart (for the augmented fourth) or five letter names apart (for the diminished fifth); perfect fifth = 7 half steps between notes five letter names apart; minor sixth = 8 half steps between notes six letter names apart; major sixth = 9 half steps between notes six letter names apart; minor seventh = 10 half steps between notes seven letter names apart; major seventh = 11 half steps between notes seven letter names apart. Of course there are other intervals which may be enharmonically equivalent to these. For example, an augmented second = 3 half steps, and an augmented sixth = 10 half steps. Confirm these definitions with the following examples of tonal intervals: pdf.


        The Renaissance was a large-scale cultural movement that spread across Europe from the 14th to the 17th Centuries (roughly). All aspects of culture, politics, the arts, and education felt the influence of its re-discovery of classical sources of art, philosophy, and science. Artists found new ways of achieving a mastery and control of their materials, received from the past--colors and shapes (painting), sounds (music), physical objects such as marble and alabaster (sculpture), words and images (poetry).

        In sculpture artists carved powerful, elegant, and proportionally correct statues of classical subjects out of alabaster and marble; here is a reproduction of Michelangelo's David: (1504): jpg

        In painting artists developed the ability to paint in such a way that the image painted resembled the image as captured by the naked eye. The discovery resulted in paintings with one or more vanishing points as the following reproduction of Leonardo Da Vinci's Last Supper: jpg and a graphic representation of the vanishing point show.

        In Renaissance music, composers took small fragments of Gregorian Chant and composed new music above and below, in a strict system of consonances and dissonances. These principles (though varied through the ages) penetrated music from the 16th Century clear through the 19th Century and to the eventual dissolution of tonality at the turn of the 20th Century. Listen to Victoria, O Magnum Misterioso (1572)

      • consonance and dissonance

        Consonance is a conventional sense that notes are happy right where they are; a consonance does not need to be prepared or resolved; it just is. Dissonance, on the other hand, is a conventional sense that notes in certain contexts need to be prepared, sounded, and resolved in very specific ways. Conventional notions of consonance and dissonance are not absolute; they have changed from style to style and from culture to culture. There are melodic conventions of consonance and dissonance and harmonic conventions of consonances and dissonances. Let's start with harmonic conventions in the canonic, western european tradition of the common practice era.

        Perfect consonances are perfect octaves and perfect fifths; imperfect consonances are major or minor thirds and major or minor sixths. The perfect fourth is a consonance unless the lower note is the lowest note in a harmonic texture of more than two voices. Dissonances include major and minor seconds, perfect fourths (unless the lower note is not the lowest note in a harmonic texture, major and minor 7ths, and all augmented and diminished intervals. The best way to illustrate the relationship between consonance and dissonance is to refer to sixteenth century, two-voice counterpoint. In these examples consonances and dissonances are vertical or harmonic; there are also horizontal or melodic notions of consonance and dissonance.

      • conterpoint: first species

        No one knows exactly how Renaissance composers thought of the music they wrote nor how they would have articulated their principles of consonance and dissonance. The 18th Century composer and theorist Johann Joseph Fux examined the music of the great Renaissance composers and retrospectively suggested rules of setting notes (newly composed by the Renaissance masters) against notes (the fragments of Gregorian Chant called Cantus Firmus. This practice of writing notes against notes is known as counterpoint (from Punktum (note) kontra (against) Punkum (note)). These principles are called species and we're going to work with them now.

        first species counterpoint (note against note): pdf. In first species counterpoint, composers write only consonances (perfect unisons, thirds, perfect fifths, sixths, and perfect octaves; perfect intervals are reserved for final cadences (about which much more can and will be said in sixteenth-century counterpoint classes).

      • conterpoint: second species

        second species counterpoint (two notes against one note--the passing tone): pdf

      • counterpoint: third species

        third species counterpoint (four notes against one note): pdf

      • counterpoint: fourth species

        fourth species counterpoint (the suspension): pdf

      • counterpoint in music

        the species in music: allegri miserere: pdf You may study sixteenth and / or eighteenth century counterpoint in graduate school as a compositional matter; counterpoint also lies deeply within Schenkerian analysis.

        For much more on counterpoint, see Alfred Mann, The Study of Counterpoint (New York: W.W. Norton and Company, 1971); Robert Gauldin, Sixteenth Century Counterpoint (Long Grove, Illinois: Waveland Press, 1985); Peter Schubert, Modal Counterpoint (New York: Oxford University Press, 2008); see other texts on counterpoint as they are published.

      • chord tones and non-chord tones in tonal music

        As diatonic tonality (based on major and minor scales and the triads and functions their harmonies support) comes into focus after renaissance counterpoint of the sixteenth century, the distinction between chord tones (members of the major minor scale and triads that harmonize them) and non-chord tones (notes a step or more away from chord tones that embellish them) becomes clear. The aural perception of the distinction between chord tones and non-chord tones is essential to the understanding of tonal music; it is just as important to music perception as seeing the difference between the person to whom you are speaking at any given moment and everything else in the environment at hand. Imagine talking to someone and not being able to tell the difference between that person and the objects in the environment behind, above, below, all around that person! That would be the visual equivalent of listening to a piece of tonal music and having no idea which notes were chord tones and which notes were non-chord tones. Here is how they work:

      • non-chord tones: neighbor notes

      • non-chord tones: suspensions


        The music of the classical era (the late 18th century through the early 19th century) reflects a general philosophical, aesthetic, and social approach to public and private life in central European culture--the Enlightenment. The Enlightenment produced many things: the notion of the modern individual, democratic institutions of government, a sense of order and balance in the arts (including the garden: jpg--a major embodiment of Enlightenment ideals of balance and clarity). The rich clarity and beauty of the speeches of Dr. Martin Luther King ("Letter from Birmingham Jail": pdf and other great modern speakers owes a debt, in part, to the rhetoric of British essayists such as Addison, Steele, and Swift who perfected techniques of the parallel structure, and thesis driven argument. Jacques-Louis David painted the Oath of the Horatii: jpg as a representation of balance and clarity. Notice the lighted action in the foreground of the painting that is placed, contextualized by the darkened space among pillars in the background. Notice also the perfectly balanced triangles described by the mens' bodies, arms, swords.

        In music, the sonata developed as a method composers such as Haydn, Mozart, and Beethoven used to create structures of balance and clarity--first by establishing and then departing from an original key, theme, motive, and then resolving everything in tonic harmony. I don't mean to suggest that all of the forms discussed below are manifestations of the Enlightenment; to the extent that the composers create a sense of balance, harmony, and clarity however, it's hard not to understand them in terms of the dominant philosophical, aesthetic, and social ideas of their age. Listen to Mozart, Symphony no. 40 in g minor, I

      • part-writing

        When you part-write, you are composing progressions that are at the heart of tonal music. There are four voices: soprano, alto (on the treble staff), tenor and bass (on the bass staff). Stems go up on soprano and tenor parts; stems go down on the alto and bass parts. Here's how to set up four-part part-writing: pdf. Typically, you are provided a bass line with numbers (figures) underneath. The numbers indicate the intervals above the bass note in the key signature. A slash through the number indicates to raise the note a half-step. An accidental without a number means to apply that accidental to the third above the bass. a sample figured bass in B minor:pdf; a sample figured bass realization in B minor: pdf

      • structural levels

        Let's begin by listening to Chopin, Prelude Opus 28, no. 4 Theorists hear tonal music in structural levels. You might think of the surface of a sentence as each of its words and grammatical units; deeper levels move into simpler structures, such as the subject and verb at the heart of every English sentence. In tonal music, a single dominant - tonic motion is like the subject - verb deep structure of a sentence. Composers expand that single dominant - tonic motion (that we hear decisively at the final cadence) through various modulations, melodic and harmonic expansions of progressions, chord tone and non-chord tones. Take a look at and listen to the following exploration of the structural levels of the first 12 measures of the Chopin Prelude Opus 28, no. 4

        Chopin Prelude Opus 28 no. 4, transcription of left hand mm. 1-12: pdf

        Moving from the deepest level "up" to this surface, listen to the simple diatonic structure that underlies Chopin Prelude in E minor Opus 28, no. 4 mm. 1-12: pdf

        Moving closer to the surface, you can hear the parallel 6ths in the Chopin Prelude in E minor Opus 28, no. 4 mm. 1-12: pdf

        And moving up to the surface even more closely, you can hear the chain of 7-6 suspensions in Chopin Prelude in E minor Opus 28, no. 4 mm. 1-12: pdf

      • sequences

        Think of a sequence as certain kind of series (the literal meaning of the word)--things follow one another in time (as in a sequence of games that lead to a championship). In music, think of a sequence as some entity marked off by delimiters--as in [musical entity]. One delimiter points to a beginning; one delimiter points to an ending. The musical entity can be a chord, two chords, three chords, a phrase, even an entire longer portion of music. A composer transposes that entity (let's call it [x]) either up or down to fill musical space, to prolong a harmony, to prepare a big dominant (most commonly). The principle of transposition is either diatonic (moving up or down a major or minor scale), or chromatic (moving up or down by a series of identical intervals).

        In order for a sequence to be heard as such, a minimum of one transposition is necessary.

        Sequences are like little musical machines that are set in motion; how do they end? If left to run their course, they will either spiral infinitely out of earshot moving clockwise around the circle of fifths (actually two asymmetrical spirals as we discovered), or counter-clockwise around the circle of fifths. Or, given equal temperament and enharmonic re-spelling, they will cycle back upon themselves, ending up at their beginnings, like a snake eating its tail. Most often, composers don't let sequences spin away (either into infinite sharpness, infinite flatness, or as complete journeys around the circle of fifths). Most often, the composer jumps off the sequence at a certain point. Hearing and thinking about this point makes music and music analysis exciting. There are two sequences in Schubert's "Der Wegweiser." Find each one, determine what the [x] is, figure out the technique of transposition, and determine where the "jumping off" point is.

        Schubert, Der Wegweiser, sequence: pdf

        Schubert, Der Wegweiser, transcription of mm. 57-64: pdf and Schubert, Der Wegweiser, mm. 57-64 a reduction and hypothetical completion of the sequence: pdf

        Schubert, Der Wegweiser: pdf

      • diatonic harmony

        Diatonic Harmony directly and completely maps onto a diatonic scale in major:pdf or minor: pdf. The scales are shown with each pitch having an arabic number + caret; these are the scale degrees in major and minor. Each scale degree forms the root of a triad and seventh chord. Triads have upper case when major and lower case when minor (there are some theory practices in which all triads built on all scale degrees are in upper case). Diminished triads have a superscript degree sign after the roman numeral; augmented triads have a plus sign after the roman numeral.

        Here are some common names for triads built on scale degrees:

        • tonic = triad on scale degree 1
        • supertonic = triad on scale degree 2
        • mediant = triad on scale degree 3
        • subdominant = triad on scale degree 4
        • dominant = triad on scale degree 5
        • submediant = triad on scale degree 6
        • subtonic = triad on scale degree 7 in natural minor
        • leading tone diminished triad = triad built on the leading tone scale degree 7 (built into the major mode; it needs to be manually added in minor)

        Composers use dominant seventh chords in myriad ways in their music; for that reason it is a good idea to memorize all of them: Dominant Seventh Chords: pdf

      • root position triads

        If the chord member root is in the lowest voice (usually called, generically, the "bass") then the chord is in root position; no arabic numbers are necessary to the right of the roman numeral.

      • first inversion triads

        If the chord member third of a triad is in the lowest voice, the chord is in first inversion: pdf; write a superscript 6 to the right of the roman numeral.

      • second inversion triads

        If the chord member fifth is in the lowest voice the chord is in second inversion: pdf; write a superscript 6 to the right of the roman numeral and a 4 under the 6.

        In classical, common practice harmony (upon which we are focusing), root position chords are the most stable; pieces almost always begin and end with them. First inversion chords are stable, but not as stable as root position chords. It is very unusual for a piece to begin or end with a first inversion chord. Second inversion chords (except in a few special cases and in a few stylistic features of some common practice pieces) are dissonant; second inversion chords are most of time dealt with in one of three ways: 1) passing: pdf, 2) pedal: pdf, 3) cadential: pdf.

        Mendelssohn writes one of these second inversion chords on the first page of the Mendelssohn Violin Concerto: pdf. Which one is it? Where is it? To what musical effect?

      • root position seventh chords
      • first inversion seventh chords
      • second inversion seventh chords
      • third inversion seventh chords
      • diatonic harmony: description and analysis

        Bach, WTC Prelude no. 1 in C major: pdf

      • diatonic harmony: description and analysis continued

        Do as thorough a roman numeral description of Mozart, Symphony no. 40 in G minor, mm. 1-20: pdf as you can.


        Romanticism is a period of exploring ambiguities latent in the assumptions of the Enlightenment. The threshold between the Enlightenment and its quintessential expression classicism and the Romantic period is very fluid. Roughly the Enlightenment dominated central European and American thought, art, and institutions in the late eighteenth century; roughly Romanticism penetrated European and American thought and art in the early nineteenth century. If classicism is about clarity, serenity, unity; then romanticism is about ambiguity, yearning, and the fragment. Writing in the late eighteenth century Immanuel Kant described beauty (perhaps the quintessential aesthetic of the Enlightenment) as a representation of pure form as if in the object that must be communicated to others in social space; he described the sublime (perhaps the quintessential aesthetic of Romanticism) as a representation of extraordinary intensity, mixing joy with terror, as if locked within the subject that could not be communicated to others. He describes the dynamical sublime as the immense and unimaginable power of nature; he describes the mathematical sublime as the unimaginalbly small and large numbers.

        In painting, Caspar David Friedrich painted human subjects often dwarfed by nature as in Monk by the Sea: jpg. In poetry take a look at Shelley's Ozymandias: pdf with its powerful evocation of the tentative nature of representation through art and the fragmentary nature of knowledge.

      • diatonic and chromatic thirds

        Composers (particularly in the nineteenth century) often used third relations that are chromatic and not diatonic.

        In this diagram of diatonic and chromatic third relations: pdf notice that C major is a reference point. To the upper-right is E minor--the diatonic mediant; to the lower right is A minor--the diatonic submediant. To the upper left is E major--a chromatic mediant; the lower left is A major--a chromatic submediant. Taking C minor as a referent, to its upper right is E-flat major--the diatonic mediant; to its lower right is A-flat major--its diatonic submediant. To its upper left if E-flat minor its chromatic mediant; to its lower right is A-flat minor--its chromatic submediant. Make a map of third relations like this one, using B major and B minor as referents. What does this map of third relations show you about the third relations of B minor / D-sharp minor at work in Schubert's "Der Doppelgaenger"? In addition to knowing whether third relations in music are diatonic or chromatic, you need to know how they sound; does a composer at a given point bring out the strong contrast, for example, between chromatically related keys? How audible is the contrast between B minor and D-sharp minor in "Der Doppelgaenger"? What is the nature of third relations in Schubert's "Der Atlas": pdf

      • mode mixture

        composers often substitute a chord from major in a minor mode context, or substituting a chord from major in a minor mode context. This can be seen and heard in Mendelssohn's Overture to a Midsummer Night's Dream: pdf.

      • the augmented sixth chords

        The augmented sixth chord: pdf is a dominant preparation chord that usually occurs in minor; the submediant scale degree is usually in the lowest voice (in major you must lower the submediant so it's a half step above scale degree five with a different letter name), sharp subdominant scale degree, and here's the strange part--tonic (Italian Augmented Sixth), tonic + supertonic (French Augmented Sixth), tonic + mediant (German Augmented Sixth). In major you must lower the third scale degree so it's a half step above supertonic with a different letter name. In the diagram above, I've given you at the top staff all the augmented sixth intervals. On the staff below that I've shown you how the augmented interval resolves in context; each augmented sixth expands outward by semitones in contrary motion to an octave--the dominant of the key. In the notation the pitches of the augmented sixth are note heads only showing ties to the stemmed notes of the octave--outlining the dominant. This notation indicates the powerful tendency of the augmented interval to expand outwards to the dominant. The next three systems fill-out the expansion of the augmented sixth interval in fully-voiced four-part harmony using the German Augmented Sixth chord that resolves to the dominant in 8 / 6 / 4 position. There is an augmented sixth chord that prepares the dominant at the end of the opening piano introduction to Schubert's "Wegweiser." Listen to how it sounds.

        The German Augmented Sixth chord has the unique property in that it can be enharmonically interpreted as a dominant seventh chord of a key a half-step high than the key in which it would function as an augmented sixth chord. Here is an illustration of the enharmonic interpretation of an augmented sixth chord (in C minor) and dominant seventh chord (in D-flat major): pdf.

        Beethoven wrote a spectacular augmented sixth chord in the second movement of his Symphony no. 5 in C minor: pdf. The excerpt starts in A-flat major--the key of the movement. In mm. 27-28, Beethoven writes a fully-diminished seventh chord that could go to a number of sonorities in A-flat major--to the supertonic B-flat minor, to the subdominant D-flat major, for example. On the downbeat of measure 29, Beethoven enharmonically re-spells G-flat as F-sharp and the chord could either be a dominant seventh chord going to the subdominant or a German Augmented Sixth chord going to the chromatic mediant C major! And this is where Beethoven takes the music.

        The augmented sixth chord is a local dominant-preparation chord only; composers cannot prolong the harmony over a long span of time. You can think of the augmented sixth chord as an altered version of a number of diatonic triads. In functional harmony the augmented sixth chord is an altered dominant of the dominant with a flattened fifth, a missing root, and / or a ninth.

      • the neapolitan chords

        Think of the neapolitan: pdf as either an altered supertonic chord (particularly when it is in root position as in the Chopin Prelude no. 20 in C minor, mm. 9-13: pdf or an altered subdominant chord as in Beethoven Piano Sonata Opus 27 no.2:pdf. Composers can use the neapolitan locally as a dominant preparation chord; composers can also composes longer stretches of music in a neapolitan harmony as in the following: Beethoven, Piano Sonata Opus 110, Adagio (excerpt): pdf. In this piece the first measure clearly establishes tonic B-flat minor. In measure 2 Beethoven puts C-flat superscript 2 underneath pitches G-flat superscript 5 and E-flat superscript 5 sounding already in measure 1 to move the music to the neapolitan--the secondary tonal level of measure 2. In measure 3, listen to Beethoven's subtle modulation to a-flat minor!

      • tonicization

        Composers can emphasize a major or minor triad by writing its dominant right before it sounds; this is tonicization--a two chord progression in which a stable sonority in major or minor (in major: ii, iii, IV, V, and vi; in minor: III, iv, v, VI, and VII) are preceded by their dominant triad, a dominant seventh chord, a their leading-tone diminished triads, their fully-diminished seventh chord, or in unusual cases, their half-diminished seventh chords. In the following illustration of tonicization, I have provided you at the top with the triads in C major that can be tonicized; the half note rests stand for the place that will be filled with dominant seventh chords. On the second staff you will see the tonicizations--a dominant seventh chord followed by the tonicized triad. The arrow stands for "of" or "resolving to". You will notice that accidentals need to be added in some cases; this is because even for the brief task of tonicization, composers need to think and hear in the keys of the triads they are tonicizing. On the third system, I have provided the triads in minor that can be tonicized and on the fourth system those toniciations proper: toniciation in major and minor: pdf

        Note: there are part-writing issues that arise when you tonicize (as I did) a major or minor triad with a dominant seventh chord. If your dominant seventh chord is fully-voiced (one note per voice in SATB texture) then you will either have parallel perfect fifths, or you will double the chord member third of the triad being tonicized. The former is deadly; the latter not so bad really. I have chosen a common solution--omitting the chord member fifth of the dominant seventh chord. This avoid parallels and produces a fully-voiced triad being tonicized.

        There are two examples of tonicization in the American National Anthem: pdf. From measure 2 to 3 the composer tonicizes the submediant D minor; from measure 3 to 4 the composer tonicizes the dominant. Together, these two tonicizations intensify the move of the first phrase to the dominant of F major.

      • secondary tonal levels

        Composers write secondary tonal levels when they wish to emphasize a triad in a key (diatonic or chromatic) a bit more than a tonicization but not so much that they loose a sense that the home key is the home key; I showed you a secondary tonal level in the neapolitan of the Beethoven Piano Sonata Opus 110, slow movement in B-flat minor. For secondary tonal levels you draw (as on that example) a horizontal line; beneath the line, write the function of the passage in relation to the home key; above the line you write how the "local" harmonies function. On page 1 of Schubert's "Die Nebensonnen" there is a secondary tonal level within the A major home key. Where is it? What is it?

      • modulation

        A composer modulates to another key to create large-scale contrast or structural dissonance with the home key. Pivot modulations are quite common; a composer uses a pivot like a hand-off in a baton race; there's a moment at which both runners are holding the baton as it passes from one runner to another; the musical equivalent of this moment is the pivot itself and it relies on a sonority being common to both the home and new key, though of course functioning differently in each. In this list of chord functions relating relative major / minor keys and tonic and dominant keys, you can see that between a minor key (F-sharp minor in the example) and its relative major (A major in the example) every chord can function in either key, so each triad can function as a pivot; between a tonic chord (A major in the example) and its dominant, there are fourtriads that function differently in each key: those are the potential pivot chords between a tonic, major key and its dominant. Composers very typically modulate in minor to the relative major; composers also often modulate from a tonic major key to its dominant as in the following example from Haydn, String Quartet Opus 76 no. 4 minuet: pdf In this piece, Haydn clearly unfolds B-flat major in the initial measures of the piece; at measure 7, however, we know we are modulating to a new key (the E-naturals signal that move decisively). Frequently then the chord before the chord that signals the new key is / was the pivot (as shown in the example; the roman numeral above the bracket shows the function of the chord in the "old" key; the roman numeral below the bracket shows the function of the chord in the "new key." When we hear measure 6 we hear tonic B-flat major; when we hear measure 7 we retrospectively reinterpret that B-flat tonic sonority as having been the subdominant of F major awaiting the new key.

        Get in the habit of working out modulations by setting up a key with a module that you often use and another module as a cadence; then compose a pivot between them, like in this Modules for setting up keys, cadences, and pivot chords: pdf

        Write simple four-part progressions in which you modulate using pivot chords from C major to G major; from G-sharp minor to B major; from D-flat major to B-flat minor; from G major to D major; now (assuming you used triads), write simple four-part progressions using pivot seventh chords to modulate from E major to C-sharp minor; from F major to C major; from D minor to F major. Those are diatonic modulations that are quite common; now try some less common modulations. Write simple four-part progressions that modulate from C major to E minor; from F-sharp minor to B minor; from E-flat major to F minor. Can you modulate using a pivot chord from C major to A-flat major? If so, how? If not, why not?

      • cadences

        In order to prolong tonic harmonies, composers prolong those harmonies by moving to the dominant, and they prolong that move to the dominant by a number of progressive and regressive harmonic motions--to other triads and seventh chords in the major or minor of tonic, through tonicization and / or modulation to other keys. Harmonic motions that conclude phrases are cadences: pdf. In the illustration above, remember that there is only one form of a PAC: 1) the tonic chord at the cadence is in root position; 2) it is preceded by a root position dominant, and 3) the tonic chord supports scale degree 1 in the uppermost voice. If a dominant to tonic cadence does not fulfill all three of these conditions, then the cadence is an IAC (of which there are several varieties).

        The Phrygian Half Cadence is a particular progression: first inversion subdominant to a dominant. Composers usually write a dominant submediant progression as a deceptive cadence, but there are others. Notice that in a deceptive cadence the chord member third and seventh (forming a "tritone") resolve properly.

        Composers conclude a phrase on the dominant in a half cadence. Opinions vary on whether a half cadence can include a dominant seventh. The Plagal Cadence moves from the subdominant to the dominant--the well-known "Amen" cadence. There's a plagal harmonic motion in the opening measures of Schubert's "Die Nebensonnen." And I think the plagal motion in this work evokes the world of church music to represent the social sense of community for which the narrator yearns.

      • phrases

        A phrase is a short piece of music that ends in a cadence. Phrases by Haydn (for example) are often very short and modular; they are often (but not always) four measures long. Phrases by Mozart (for example) are often quite long. A period is two phrases--an antecedent phrase (the first one) and a consequent phrase (the second one). Most people agree that if the two phrases begin with similar or even identical harmonic and / or melodic material they are parallel; if the two phrases begin with different harmonic and / or melodic material, they are contrasting. Quite often an antecedent phrase ends on the dominant and a consequent phrase ends with a perfect authentic cadence--sometimes in a new key (the dominant if the piece is in major or the mediant if the piece is in minor).

        Haydn, String Quartet Opus 76 no. 1, minuet: pdf

      • one-part forms

        Chopin wrote a simple one-part form in his Chopin, Prelude Opus 28, no. 1:pdf. I hear this piece as a single, parallel period. Measures 1- 8 are the antecedent phrase establishing tonic C major and moving to an interrupted dominant in measures 7-8; measures 9-12 reinitialize the tonic harmony of measures 1-4 leading to the work's big dominant in measures 23-24 with the final tonic arrival at measure 25. Measures 25-34 bring melodic, metric, and motivic issues to rest in tonic C major following the final cadence. You may find this interpretation counter-intuitive based on the continuous flow of the music. For me (and others) continuous flow of surface detail does not lessen the musical structure of an interruption.

        Schubert wrote a one-part strophic song in his Schubert, Der Morgengruss: pdf.

      • two-part forms

        Bach wrote a two-part form in his Bach, Partita no. 2 for Unaccompanied Violin, Giga: pdf.

      • three-part forms

        Schubert wrote a three-part form in his Schubert, Das Fischermaedchen: pdf.

      • the sonata

        Beethoven wrote a sonata movement that sounds at once like a three-part form, and, perhaps paradoxically, as a two-part form in his Beethoven, Symphony no. 5 in C minor, I: pdf You can hear this movement as a three-part sonata form: 1) exposition, 2) development, 3) recapitulation + coda, or as a two-part form: 1) exposition + development, 2) recapitulation + coda. Which makes more sense to you?

      • music analysis: an introduction

        For me, music analysis involves saying anything about a piece above the level of description. When you apply a one-to-one label for a sonority across an entire piece, you are describing it. When you begin to connect and relate these descriptive elements with one another you are beginning the process of analysis. Some people believe that analysis is objective; "x, y, and z" exist in a piece; "such and such" is, simply, there. Others (like myself) believe that analysis breathes with life when objective evidence supports a subjective claim as in (after David Lewin) "I hear x and y in such and such a piece, and I think you can too. Other approaches to analysis include: zooming in, zooming out, following marked elements through a work, connecting basic materials (latent) to the piece (manifest), and tracing transformations of note-chord-key across a piece.

      • music analysis: marking

        Music Analysis: Implication-Realization and Marking.

        Listen to Beethoven, Symphony no. 4 in B-flat major, I: pdf for evidence of musical marking in terms of the pitch-class G-flat. From a relatively simple flat-6 to 5 motion in the initial measures, the pitch-class undergoes transformations at measures 17-18, 22, and 304-305.

      • music analysis: the structural gap

        Music Analysis: Implication-Realization and the Structural Gap. In the mid 1950s American theorist Leonard Meyer wrote Emotion and Meaning in Music. In that book, Meyer applies to music analysis some of the principles of Gestalt Psychology. Gestalt psychologists claim that the human brain cannot tolerate incomplete patterns; given an incomplete pattern, the brain will do anything possible to complete that pattern. Think about how hard it is to proofread a paper; proofreading is not hard because it's exhausting and boring; we have (according to Gestalt psychologists) an auto-correct hard-wired in our brains. We fix mistakes in our brain without even realizing it! Meyer applies this idea to music. For Meyer, composers establish patterns and then break them in structural gaps. At that moment of pattern disruption a surge of emotion occurs in the listener; filling that gap, or even leaving it open, intensify the emotional response in the listener.

        Listen to Chopin, Prelude no. 2 in A minor: pdf For me, Chopin marks his musical material in this work by first establishing a pattern and then breaking it; the moment at which the pattern is broken is marked for memory and produces a structural gap. Chopin establishes a harmonic pattern (sequence) from measure 1 through measure 10; what is it? What do we expect based on mm. 1-10 that the downbeat of measure 11 denies? What sonority is measure 11 precisely, and how does Chopin get from that sonority to tonic A minor?

      • music analysis: motivic parallelism

        Listen to the first movement of Mozart's Symphony no. 40 in G minor: pdf and listen for evidence that a small motivic idea becomes a chord in a larger context and then a key in an even larger context.

      • music analysis: structural levels

        schenkerian voice-leading is predicated on the assumption that tonal pieces of music are constructed in levels--each of which represents a transformation of a deeper, simpler, preceding level. here is an example from mm. 1-12 of chopin's prelude in e minor opus 28.


        Towards the Dissolution of Tonality

        Throughout the 19th Century, composers wrote music that incorporated chromaticism at an ever deepening level in their music; eventually these composers felt themselves at the threshold of the dissolution of the tonic / dominant binary that had supported diatonic music for at least two hundred years. In central European painting of the same period, painters increasingly were more interested in how light hit the eye, how the imagination processed what it saw, what it imagined (no pun intended) and what it felt rather than any claim to realistic representation. This move in painting can be seen in a rather precise form in a series of tree paintings by Piet Mondrian. As you look at Mondrian, Red Tree: jpg and Mondrian, Blue Tree: jpg and Mondrian, Yellow Tree: jpg and Mondrian, Grey Tree: jpg and Mondrian, Red and Black Tree: jpg and Mondrian, White Tree: jpg think about the gradual dissolution of background / foreground relations; think about how expressive the images become; think about how the images evoke various kinds of webs, networks, blades, bones, and organs.

        Composers did with works of music something very similar to painters' dissolution of visual representation. Listen to the tonal and atonal sonorities in Berg, Sonata For Piano Opus 1: pdf. How do these elements relate to one another?

        Turn of the 20th Century: expressionism (painting), abstraction (poetry), and atonality (music)

        The turn of the 20th century in central european culture saw the birth of psychoanalysis and the idea of the individual and collective unconscious (Freud and others). Culture was increasingly marked by large-scale industrialization and the age of the machine. As a way of depicting the increasing fragmentation of modern life, as a way of expressing the unconscious, artists broke through conventional notions of representation (painting), meaning (poetry), and tonality (music). In painting, artists were much more interested in depicting states of emotion rather than a realistic image of how things appeared; for example take a look at Arnold Schoenberg's Red Gaze: pdf--a representation of intensity of vision. See also works of Edvard Munch, George Grosz, Ernst Ludwig Kirchner, and Emil Nolde.

        And think about this extraordinarily cryptic poem (translation mine) by the poet Guillaume Appolinaire: "Chantre" "Et l'unique cordeau des trompettes marines" (And the unique measuring cord of sea trumpets). In this fragment (grammatically speaking), Appolinaire creates a complex and contradictory image of synesthesia (interpreting one sense on terms of another).

        Listen to Webern, Bagatelle for String Quartet Opus 9, no. 5: pdf

      • intervals and interval classes in modernism

        In atonal music there are four ways of talking about intervals: ordered pitch intervals, unordered pitch intervals, ordered pitch class intervals, and unordered pitch class intervals.

        ordered pitch intervals = the number of half steps from one pitch to another; + for ascending / - for descending.

        unordered pitch intervals = the number of half steps between two pitches without indication of direction.

        ordered pitch class intervals. Let the first note be X and the second Y; designate X and Y with pitch class numbers. The ordered pitch class interval between X and Y is (Y-X) mod 12 (to be discussed in class).

        unordered pitch class intervals = (Y-X) mod 12 or (X-Y) mod 12 whichever is smaller.

        In tonal music some vertical intervals are usually (if not always) dissonant; they much be approached, sounded, and resolved in certain ways. Take the tritone:pdf for example. When written as an augmented fourth (say F-natural on the bottom and B-natural on the top) the interval resolves outwards to a sixth; when written as a diminished fifth (say B-natural on the bottom and F-natural on the top) the interval resolves inwards to a third.

        Consider the tritone in Bartok's Microcosmos no. 10 Book I: pdf and Schubert "Der Wegweiser (excerpt): pdf

      • divisions of the octave

        Divisions of the Octave

        Diatonic tonal music is based on the asymmetrical division of the octave at the fifth and fourth (remember that this is present in the overtone series from partials 2, 3, to 4). This asymmetry is like the division of the picture plane in painting very roughly at the third (rarely the middle) in representational works from the Renaissance through the nineteenth century. Notice the asymmetry between the division of the picture plane that results in much more sky than land in Jacob Isaaksz. van Ruisdael: jpg--a 17th century dutch painter.

        At roughly the turn of the 20th century, artists shifted an emphasis on asymmetry to symmetry. In music this resulted from musical materials based on the symmetrical division of the octave: pdf. In painting this resulted in splitting the picture frame roughly in half as in Matisse, Portrait of Lydia Delectorskaya: jpg Composers such as Debussy used interval cycles (symmetrical divisions of the octave in their works such as Voiles: pdf. Listen for evidence of some of these interval cycles: pdf in the work.

      • atonal pitch class set theory: normal order

        Given the artists' interest in expressionism, abstraction, and atonality, it can be difficult to absorb and understand the modern art of central Europe at the turn of the twentieth century. Understanding expressionist painting involves looking at how specific colors, images, evoke emotions (as the intense red-ringed eyes of Schoenberg's "Red Gaze"); understanding poetic abstraction involves enjoying taking apart conventional meanings (as in Appolinaire's "Chantre"); atonal pitch-class set theory is one way of understanding the pitch structure of pieces such as Webern's Bagatelle Opus 9, no. 5 for String Quartet. Pieces written in the central European tradition between roughly 1905 and 1923 are (more or less, in myriad different ways) atonal. Composers avoid tonal centers and they blur or obliterate the binary opposition between consonance and dissonance. If I ask you to listen and understand this music in terms of atonal pitch-class set theory, I'm asking that you imagine that as you listen, your ear and mind group notes as you hear them. Imagine that your attention span is like a cursor on the screen. At any given moment in a piece you are hearing sounds in a present tense--a "now"; as sounds enter that now, some sounds recede into the past and perhaps you anticipate some sounds to come. In atonal pitch-class set theory, we start by marking-off, provisionally, a series of "now"s in which you hear notes. This process is called segmentation. It looks like this: Segmented score of Webern's Bagatelle Opus 9, no. 5: pdf

        Theorists in the 1950s and 1960s developed atonal pitch-class set theory as an attempt to use group theory from logic and mathematics to study relations between and among notes and intervals between and among them. Once you've segmented a work, you find a basic shape of pitch-classes within each group. We call each group a set and they are labelled here with roman numerals. Sets may contain from 3 to 9 pitch-classes. If a pitch-class occurs more than once in a set, consider it as one pitch class. The basic shape of a set is the set in normal order. To find normal order you perform three steps: 1) arrange the pitch classes ascending within an octave, 2) rotate the collection as many times as there are notes in the set, and 3) choose the ordering that has the smallest interval from first to last note. Consider set VII in the Webern Bagatelle Opus 9 no. 5. The first violin plays pitch class A-flat; the second violin plays pitch class G-sharp; the viola plays pitch class A natural; the cello plays pitch class G. Since A-flat and G-sharp are enharmonically equivalent, we count only one of them. Here are the steps of finding the normal order for pitch class set VII of Webern's Bagatelle Opus 9, no.5: pdf

      • atonal pitch class set theory: the set class

        Consider the pitch-class set [235]; if you would like a way to refer to the intervallic structure of that set without regard to pitch classes, you would need a way to see "there's a half step from 2 to 3 and a whole step from 3 to 5." This is thinking of the pitch class set "left to right;" but we could also say "there's a whole step from 5 to 3 and a half step from 3 to 2." This is thinking "right to left." When deciding to count "left to right" or "right to left", choose the direction that starts with the smaller interval--so "left to right" in this case wins. The prime form of pitch class set [235] is (013). You start from the left with 0; 0 in the context of prime forms means "I'm starting to count numbers of half steps in a pitch class set right here." (01...) refers to the half step from 2 to 3; (...13) refers to the whole step from 3 to 5. Here are some examples of pitch class sets in normal order and their prime forms. Most theorists (but not all) use this system--[pitch class sets in normal order] and (prime form of pitch class sets in normal order).

      • serialism: the row

        Serialism (1923 to 1945): the row

        Remember that World War I took place between 1914 and 1918. It was the first international war; it was the first war to use airplanes as weapons; it was the first war to use poison gas; and it was a war that killed hundreds of thousands of soldiers in trenches. In the devastation of the postwar years, the aesthetic of expressionism waned. In its place die neue Sachlichkeit (the new orderliness) emerged. In the visual arts the cleanliness of synthetic cubism emerged with its use of collage and bright colors (after the dark and angular aesthetic of analytic cubism). In music, the 12-tone row or series became the basic of composition. In a nutshell, the row is a particular ordering of the 12 available chromatic pitches. First generation, central European composers of serial music Schoenberg, Berg, and Webern performed four operations upon rows to generate music: prime (the row played in order left to right); retrograde (the row played in order (or backwards) from right to left; inversion (the row played in order upside down) and retrograde inversion (the row played in order upside down and backwards).

        There are 12! 12-tone rows in the universe--5,748,019,200 by my calculation. Now not all of these rows are interesting. Schoenberg, Berg, and Webern liked, in particular, rows with maximum variety--all interval rows. I will illustrate the row and its four permutations using the all-interval row of Berg's Lyric Suite for String Quartet 1925-1927.

        You invert a row by inverting pitch classes around C. Consider D natural--pitch class 2. It is two half steps above C; its inversion is B-flat or pitch class 10--two half steps below C. Notice that 2 + 10 = 12. Take a look at this map of pitch classes as they invert to pitch classes with their sums equaling 12.

      • serialism: the matrix

        Serialism (1923 to 1945): the matrix

        The matrix is a square of numbers and / or pitch-class names that shows the prime form of the row at the top, left to right; the retrograde at the top right to left; the inversion top to bottom; and the retrograde inversion bottom to top. Here is a score to the theme of Webern, Symphony Opus 21, II (theme): pdf in addition to the matrix of Webern's Symphony Opus 21, II (matrix): pdf showing all permutations of the row.

      • writing about music in graduate school

        Unless you are working towards an Artist's Certificate, you will all be writing papers, theses, and / or dissertations in graduate school. Perhaps some of you have written a lot of academic work; perhaps some of you have written very little academic work. I would like students in the former group to benefit from a review of contemporary writing practices and conventions in graduate school; I would like students in the latter group to possess a mastery of the following essential components of good, graduate writing about music: 1) a thesis, 2) an unfolding of that thesis into a paper, chapter, thesis, or dissertation, 3) a ternary model of discourse (introduction / musical example / comment), 4) consistent documentation, and 5) proper formatting of the finished product (page numbers, spacing, margins, fonts, etc). To those ends, here is a list of more detailed features of good, writing about music in graduate school: pdf

        There are many forms of documentation (showing the reader the source of material you have used, cited, or put into your own words); consult your graduate school or advisor for the particular style you are to use. The components of all documentation may include: 1) footnotes or endnotes, and / or 2) internal markers in your writing alerting the reader to a citation, and 3) a list of works cited / bibliography. Each style will deal with these three components quite differently. You must choose the appropriate style of documentation for your writing and be consistent and accurate in terms of all details of font type, font size, punctuation, spacing, abbreviations, and numbering.