*In which we explore harmony through the image of a background web* # Three Ways to Create a Chord Progression Three ways to create a chord progression: - stack on a root line - grow through efficient voice leading - derive from a melody ## Stack on a root line This is a fast and unfussy way to write a chord progression and get reasonably good spacing and voice-leading, starting with a root line. ![[fast-4-part.png]] The only things you need to know about harmony: - Use the notes of one scale. (A key signature helps here). - Avoid the 7th note of the scale in your root line (produces a diminished triad). **Write a chord progression this way.** ### The Rules If You Want Them For maximally efficient diatonic voice leading: If your root is moving by 4th or 5th: - One upper voice holds - Two move by step If your root is moving by 3rd or 6th - Two voices hold - One moves by step If your root is moving by 2nd or 7th - All voices move by step - Move them in the opposite direction from the bass. Make the top line a good melody. - Usually one of the top 3 lines will be better than the others --- fewer repeated notes, a nice shape, a single peak. Move that one to the top. ## Follow voice leading A more protean, organic, open way to get a chord progression is to start with a chord and voice-lead yourself into interesting situations. ![[efficient-voice-leading-ex.png]] Even though there are odd chords in here, and these chords don't 'belong together' in a normal way, every move is smooth and makes sense because: **Some notes hold while others move** **Moving notes usually don't move by much** (half steps are the smoothest) In other words the moves are *efficient;* we are moving from chords to nearby chords by the shortest paths possible. This subverts our 'normal' rules of harmony. Notice how C major relates to E major and Ab major. These are distantly related keys on the circle-of-5ths but actually as close as can be. Third relations - C E Ab - are always next-door-neighbors in this space. I think of them like wormholes across the circle of fifths. (In 7th chords, the next door neighbors are minor thirds apart.) ![[third-relations.png]] This approach is very rewarding but it takes some trial and error. Sit at a piano and follow your ears. Figuring out a bass line from here also can take some trial and error. When chords are ambiguous (like my 5th one) just try each of its notes as the bass and choose your favorite. In academia, this way of thinking about chords is called \"Neo-Riemannian theory.\" It relates chords to each other, not to an external tonic or key, and builds on the work of Hugo Riemann (1849-1919). Write a chord progression by Taking a voice leading adventure. ## Derive from a line When we write chords to fit a line, want the notes of the chord to reinforce the important notes in a line. What makes a note important? Lines project harmony through their **leaps**. That's the key to understanding how to fit lines and chords together. When you leap to or from a note, the ear hears it as a harmonic tone. So if you have a melody, and want to find chords that fit it, look for the notes that get leapt from and to. Other things that make notes important: - starting/ending notes, of a phrase, or of the bar - long-held notes - notes beginning/ending stepwise runs (or top or bottom of a curve) - notes that happen often Notes to probably disregard: - passing tones - notes in the middle of stepwise runs - neighbor tones - notes that might be suspensions (because they resolve downward) Compare these two lines. The first only leaps to and from notes of a D minor chord. The second uses the same notes but leaps from/to D, G, E, C and G, projecting a C chord with a D bass. ![[deriving-a-chord-from-a-line.png]] Example of the thought process used in harmonizing a line: ![[chords-from-line.png]] 1. First note of a bar and a phrase. Likely harmonic tone C. 2. Lower neighbor, ignore. 3. Leaps to 4. Potential harmonic tone C. 4. Leapt to. Likely harmonic tone G. 5. Strong beat. Potential harmonic tone A, or upper neighbor (disregard). 6. Passing tone, ignore. 7. Passing tone, ignore. 8. Passing tone, ignore. Bar one candidates: C, G, A. Triads in this key that include C and G: C major. Triads that include C and A: A minor, F major. Possibilities for harmony: C major, A minor, F major. You choose! 1. First note of a bar. Nadir of a curve. Likely harmonic tone D. 2. Passing tone, ignore. 3. Held note, leapt from. Likely harmonic tone F. 4. Leapt to. Likely harmonic tone D. 5. Passing tone, ignore. Bar two candidates: D, F. Triads in this key that include D and F: D minor, Bb major. You choose! 6. First note of a bar. Likely harmonic tone F. 7. Lower neighbor, ignore. 8. Leapt from. Likely harmonic tone F. 9. Leapt to. Likely harmonic tone A. 10. Leapt to. Likely harmonic tone C. 11. Passing tone, ignore. 12. Passing tone, ignore. 13. Leapt from, possible harmonic tone G. Bar three candidates: F, A, C, G. F A and C make an F major triad. Solid solution. Ignore G. 14. Leapt to, start of a bar. Likely harmonic tone E. 15. Passing tone, ignore. 16. Ending note, held note, likely harmonic tone G. Bar four candidates: E, G. Triads that contain E and G in this key: C major, E minor. You choose! **Take a melody you have written and derive the possible chords for each bar. Try them all and choose your favorites.** # Three Webs ## First Web: The Chord Tone Mesh Imagine the notes of a chord progression as a stack of independent threads. This grid stretches and contracts to adjust to new chord shapes. Notice the threads don't cross or tangle, they just each move to the nearest chord tone in the next chord: ![[Music as a Lattice.png]] And now imagine that structure repeating upwards and downwards in all octaves: ![[Music as a Lattice-1.png]] This is the background harmonic space of a chord progression. With this image in mind, writing becomes a game of connect-the-dots, with these rules: 17. Leap from thread to thread. (Chord tones) 18. Connect threads with stepwise lines. (Passing tones) 19. Decorate threads with the steps above and below. (Neighbor tones) 20. ☆ Bonus points for **showing where the threads change.** (Not required, but always a nice choice.) ![[Music as a Lattice-2.png]] This line on a staff (written to demonstrate types of motion, not to be a great line!) ![[Music as a Lattice-3.png]] **Take the chord progression you just wrote, And, following these rules, write a line through it. Write 5 More.** ## Second Web: The Tonnetz This is a segment of the **Tonnetz** (*tone net*, Leonhard Euler, 1739). Pitches are arranged horizontally in fifths, and vertically in thirds, infinitely. Triangles are triads. Triangles that share points are harmonically \'close.\' Functional root motion tends to move to the left (classical and jazz), and retrofunctional root motion (rock) to the right. ![[Tonnetz-Tymoczko.png]] In this transformation of the Tonnetz[^1], letters represent chords: capital for major, lower case for minor. Chords are arranged such that each step along the path gives you voice leading where only one voice has to move by one step (half or whole). **Using B, write out (in notation) a chord progression that walks along the path, using efficient voice leading.** ## Third Web: Pure Harmony A very brief introduction to tuning. We start with a note - C. The natural resonance of any note contains the whole harmonic series. To make a second note fit into the first, we tune it to one of these harmonics. ![[full-harmonic-series.svg]] The first five harmonics are the strongest, we'll limit ourselves to them for now. And harmonics 2 and 4 just give you C, so they aren't useful for tuning a second note. That leaves us two harmonics, 3 and 5, which give you the 5th (G) and major 3rd (E). (A source of confusion or useful mnemonic: the third harmonic gives you the 5th, the fifth harmonic gives you the 3rd). ![[harmonics-1-3-5-1.svg]] We\'ll start with the C and draw the fifth to the right and the third above: ![[three-chords.png]] We can also tune in the opposite direction: here\'s the F a fifth below C, and the Eb a third below G. ![[Music as a Web-1.png]] Remember: we're always connect fifths on the horizontal axis and thirds on the vertical axis, same as the Tonnetz. Now let\'s tune an A to that F. ![[Music as a Web-2.png]] And let\'s tune a few more fifths: G D A. ![[Music as a Web-3.png]] Now we have something really interesting: two As on the chart, arrived at by two different paths. These are not the same A. How are they different? Let\'s go back to the harmonic series. The +2 over 3 means that the harmonic is \~2 cents sharper than equal temperament. So for every fifth, we have to add 2 cents. ![[harmonics-1-3-5-1.svg]] The -14 over the 5th harmonic means it is \~14 cents flatter than equal temperament. So for every third, we have to subtract 14 cents. ![[Music as a Web-4.png]] It turns out that A (the third of F) and the A (the fifth of D) are \~22 cents apart! (To be precise, the ratio 81:80.) This is called the **Syntonic comma**. Syn (together) - tonic (tone). The point is this: there aren\'t just 12 notes. There\'s an infinite lattice of relationships for us to traverse and explore. And this is just a two dimensional lattice, because we\'re only using two harmonics, 3 and 5. Add another and it becomes a cube\... We\'ll look at this more in depth, with more math and more practical application, in our Near Stasis class. # Chord Reductionisms It\'s interesting to think about how different analytical tools give different answers to the question of what chords \"belong together.\" Take these three chords: ![[three-chords.png]] ### Roman numeral analysis Chords are related by key, tonal function, and the circle of fifths. \(a\) C: I7 \(b\) Dm: ii7 \(c\) ??? Advantage: describes important features of Western tonal music. Drawback: falls apart in non-functional and atonal music. Ignores spacing. Practice: generate diatonic families of chords, secondary dominants, modulations, etc. ### Neo-Riemannian analysis Chords are related by voice-leading. (a \> b) \[+2, 0, 0, -1\] (C goes up to D, E holds, G, holds, B down to Bb) (b \> c) \[-4, +1, -1, -7\] (voices are compressed to an octave to find efficient motion) Advantage: maps the geometry of voice-leading space. Drawback: doesn\'t represent qualities of a chord\'s sonority. Ignores spacing. Practice: generate sequences of chords by efficient motion. ### Set class Chords are related by pitch class content, transposition and inversion. C is 0, inversions are equivalent. See also Tone Clock Theory (McLeod). Pitch classes / Prime Form (most compressed to 0) \(a\) is \[0,4,7,11\] \[0,1,5,8\] \(b\) is \[2,4,7,9,10\] \[0,2,5,8\] \(c\) is \[7,8,9,10\] \[0,1,2,3\] Advantage: can define any chord structure irrespective of transposition and inversion. Drawback: conflates significant differences (major and minor triads are \'the same\') Practice: generate related chords by transposition, inversion, complementarity. ### Interval vector Chords are related by interval content. Inversions are equivalent. format: \<m2/M7, M2/m7, m3/M6, M3/m6, P4/P5, TT\> \(a\) is \<1, 0, 1, 2, 1, 0\> (read: 1 m2/M7, 0 M2/m7\... \(b\) is \<0, 1, 3, 0, 1, 0\> \(c\) is \<4, 0, 0, 0, 0, 0\> Advantage: captures something about the way the chord *sounds.* Drawback: conflates significant differences and ignores spacing. Practice: The Chord Trick # \"The Chord Trick\" Louis Andriesson taught this to Quinn Collins who taught it to me. It\'s a way to generate a \'family of chords\' that are related because they share the same interval vector and top note. These families can span wildly different keys, but the chords share an essential quality. The common tone connection further links them, an axis around which the intervals pivot. Here\'s how you do it: Write a chord. Arrange the chord in its first inversion (bottom note to top, not mirror image). Transpose the chord down such that the top note matches the original chord\'s top note. Repeat. ![[chord-trick.png]] a. original chord b. first inversion c. transpose down so that the top note stays A. Repeat until\... eventually you will reach the chord you started with (5 note chord = 5 forms) # Spacing Matters All of these analytic tools ignore spacing. One model for thinking about spacing is **resonance.** Resonance has a natural spacing, the harmonic series, in which there is wider spacing at the bottom and narrower spacing at the top. ![[full-harmonic-series.svg]] So we get our chords into their optimally resonant arrangement when we mimic this shape -- wide spacing at the bottom, narrower spacing at the top. For example, take the amazing chord from Dutilleux\' *Ainsi la Nuit.* Compress it to a cluster, and it has little resonance or discernible character: ![[compressed-ainsi.png]] But spread it out as Dutilleux does and it becomes an extremely powerful chord. Notice the three interlocking fifths, the overtonal C\# major triad at the bottom, and the wide spacing at the bottom and narrow spacing at the top. Spacing matters. ![[ainsi-chord.png]] # Register Matters Feldman often does the reverse, putting wide spacing at the top. Here are four chords from his *Piano and String Quartet*. ![[feldman.png]] I think the beauty of Feldman\'s chords here depends on their register. An octave down and they would be muddy, but here they are crystalline. In this way register is asymmetrical -- lower notes exert more harmonic effect; higher notes less. Often chords are organized around this, where the \'in\' notes are towards the bottom and the \'out\' notes towards the top (the \'upper structures\'). # Some famous chords ![[famous-chords.png]] # Some chords I love ![[chords-i-love.png]] # Macroharmony This idea is from Dmitri Tymozcko: an important part of our perception of music is the experience of **macroharmony**. If a harmony is a chord, the macroharmony is the harmonic impression we get over a medium-length of time based on **what pitches occur** and **how often**. This is a weighted collection: pitches that happen more often have more weight. It evolves over time in our mind as new pitches appear and old ones are forgotten. It's this that gives us an impression of our harmonic space, especially in complex textures without an explicit chordal basis. Here are some examples: This chart shows what notes occur and how often. The Y values are 'relative probability:' 10 doesn't mean anything in itself, but the fact that all notes have the same value means they're all equally likely to occur. This is us a truly random distribution of pitches (white noise). ![[macro1-equal-chromatic.mp3]] ![[chart1.png]] (What you are hearing is purely random note choice, within 3 octaves, random short or long notes, random volumes, random sustain. Even this is quite fun to listen to right away!) Now all 12 pitches can occur, but C and C\# are much more likely. ![[macro2-chromatic-weighted-01.mp3]] ![[chart2.png]] Here, all 12 pitches can occur, but the notes of a C major pentatonic scale are much more likely: ![[macro3-chromatic-weighted-major-pent.mp3]] ![[chart3.png]] Let's move to a scale. Here is equal distribution in C major: ![[chart4.png]] Here is the same major scale but where the notes D F A (the ii chord) are more likely: ![[macro5-diatonic-ii-chord.mp3]] ![[chart5.png]] Here is the same major scale but where the notes C E G (the I chord) are more likely. Same notes, very different sound. ![[macro6-diatonic-I-chord.mp3]] ![[chart6.png]] # Macroharmony Trajectories Now imagine those probabilities changing over time. Over 45 seconds let's transition gradually through 3 states: C only, Gb major, and then F and Ab only. ![[trajectory-charts.png]] Let's also transition our volume and sustain parameters --- they'll still be random, but the range of possible values will go from small to large. ![[macro7-transition.mp3]] When you approach music this way --- starting with randomness, and then shaping what's allowed to happen over time --- you end up with a very different 'theory of music.' Instead of a metaphor of constructing interlocking objects, you start to imagine music as a flow. Instead of sweating the details, step back and focus on shaping the big-picture trajectories. I\'ve created a Max for Live device that helps you explore and compose with this idea. ![[macro-harmonizer.jpeg]] [^1]: From Tymoczko, Dmitri. *The Geometry of Music.*