Chord Loops

I'll get back to that last point later. Right now, I have something else on my mind. Let's take an excursion into chord loops.

Chord loops are repeated sequences of several chords—usually 2–4 (but often 4) chords (or, perhaps 4+4 with a variation in the repeat). In classical or jazz traditions, we are used to more variety, less repetition, and longer sequences. Chord loops have arisen in popular music mainly as a result of unskilled guitar labor, but they have certain advantages for the listener as well. They amount to short harmonic motives, harmonic phrases that needn't align with the melodic phrases. In unfamiliar pieces they are the sort of device that quickly teaches listeners how to listen to the piece, especially in short pieces where they won't wear out their welcome. Repeated listenings aren't necessary because the repetition is self-contained, but they are desirable because strong expectations are set up. They are a good way to get noticed on the radio.

Now, I think that the best way to compose a harmony, whether repetitive or not, is to define a set of sonority-flavor hoops that you want it to jump through and use my algorithm to find the sonority sequence that best satisfies your criteria. I introduced all of this in AMFB §1.3.3. Still, chord loops are an intriguing idea, and we can explore them without resorting to flavor harmony.

My recent interest was brought on by two things: Philip Tagg's Everyday Tonality, and this video by 12tone: My Brand-New Take On Four-Chord Loops.  Both of them are steeped in tonal harmony and talk about chord loops with the assumption that untrained listeners experience chords as places in an abstract space. I've explained why this is a fallacy elsewhere, but tonal harmony can still be an interesting game to play (especially if you play it with my hypertonal, polychrome temperaments), so let's play a little.

Tagg and 12tone use some confusing locational analogies in the tonal tradition. They are worth listening to, but I think they miss the essential point: chord loops create short harmonic phrases demarcated by the strength of progressions. The transition from one sonority to the next can be more or less strong, and the best way to measure it is by using my fuzzy-logic algorithm that will give you a value for how "progressive" it is on a scale of [0,1]. Today, though, let's be much more approximate and simply put chord transitions into two categories: weak and strong.

Harmonic phrases end with a strong transition (it punctuates them) followed by a weak one (another strong one would seem to continue the phrase). A loop of all strong transitions is possible (e.g., the Andalusian cadence: iv→III→II→I→ in Phrygian, or i→VII→VI→V→ in minor), but is less stable—it challenges the listener to orient himself to the beginning of the loop based on smaller differences in transition strength or on the melodic phrase structure.

So, chord loops come in these varieties (for later reference, I'll name them with capital letters):

2-chord loops:
A: ws,
B: ss.
(sw would be perceived as ws with the beginning shifted; ww would be perceived as wandering.)

3-chord loops:
C: wws,
D: wss,
E: sss.

4-chord loops:
F: wwws,
G: wwss,
H: wsws,
I: wsss,
J: ssss.

Notice how the different types have different characters. F wanders a long time before deciding on a cadence. I has a clear beginning with a new thought which it takes to a logical conclusion. H has an alternative point of stability in its middle—an ambiguity which might be played with by the melodic phrasing. I'll leave you to consider how these harmonic motions might be used to depict some mental or emotional trains of thought.

The world of sonorities and transitions is vast. Let us restrict ourselves to the useful diatonic modes (excluding Locrian) and not bother about the tuning—we are being approximate. Let's also use the modal notation {1,...,7} for the root of chords, so that, in Ionian on C, 1=C major, 2=D minor, etc.; and in Dorian on D, 1=D minor, 2=E minor, etc. Generally, within a diatonic mode, motion by seconds, fourths and their inverses is strong; motion by thirds, sixths, and tritones is weak. In other words, if the motions are described as addition under mod 7, then {1, 3, 4, 6} are strong and {2, 5} are weak, along with some exceptions to be made for tritones later.

This will be clearer if we look at a little table that rank-orders the strength of transitions,
in general:

For example, if we are on 5, motion 5→1 (like an authentic cadence) is very strong; 5→2 (like a plagal cadence) is a little weaker; and so on down to 5→7 and 5→3 which are very weak.

There is room for disagreement here. The last two rows might be switched. The line between weak and strong might be placed differently—it may be reasonable to regard root motion by a descending second as weak. Let's use this table though.

The diatonic modes contain a tritone (e.g., b–f) that causes some exceptions. Transitions between the following pairs are weak:
in Ionian: 4↔7,
in Dorian: 3↔6,
in Phrygian: 2↔5,
in Lydian: 1↔4,
in Mixolydian: 3↔7,
in Aeolian: 2↔6,
in Locrian: 1↔5.

So that, in Ionian, a corrected table would read:

For reference, I'll rewrite the six tables with the "from" line on top and color coded: red is strong, black is weak; higher rows are stronger than lower ones:

So, for example, if you are playing a 3 in Lydian on F (an A minor), your strongest progression is to 6 (D minor), your weakest is to 1 (F major), and to list them from weakest to strongest, you simply read up the 3 column: {1, 5, 2, 4, 7, 6}.

Notice that nothing up to this point depends on the traditional functions—we are talking about motion from any chord to any chord without defining any as central or otherwise. But, to apply these tables without swamping ourselves with data, we can make a tonal choice and say that "1" is the tonic, the center, the destination. Elsewhere, I've described "hypertonal" well temperaments—tunings that distribute proportionally more dissonance to remote chords and keys (as defined by the tonal tradition), thus increasing the tonal pressure to resolve to the tonic in the home key. Choosing "1" as the tonic and using a hypertonal temperament will make any progression especially piquant.

Also, I want to follow an interesting convention of rock music. Normally, we are used to concluding melodic and harmonic phrases together on the tonic (or a substitute). In rock though, it's usual to begin phrases with the tonic chord and end them with a dominant, so we are relentlessly pulled into the next phrase by the harmony. I like this idea. Let's define all our loops to begin on 1.

Now I'll list all the groovy loops for each diatonic mode, by type (from weakest to strongest; beginnings more significant than endings). I'll reject some loops:

1. loops that end on 1 (because they are equivalent to a loop one size smaller, e.g., 161=16);
2. four-chord loops that are just a repeated two-chord loop (e.g., 1717); and
3. redundant loops of type H and J that are simply rotations (e.g., 1714=1417);
   but, I'll include loops that repeat chords otherwise (e.g., 1714 (I remove the weaker rotation), 1525, etc.).

For the sake of curiosity, I'll count the number in each group.

In Ionian:

A (ws):   none!
B (ss):   17, 12, 15, 14
C (wws):  164, 135
D (wss):  165, 167, 162, 132, 134, 137
E (sss):  176, 125, 154, 152, 145
F (wwws): 1647, 1642, 1357
G (wwss): 1645, 1617, 1612, 1615, 1614, 1317, 1312, 1315, 1314, 1354, 1352
H (wsws): 1657, 1674, 1675, 1672, 1635, 1627, 1624, 1327, 1324, 1347, 1342,
          1374, 1375, 1372, 1364
I (wsss): 1654, 1652, 1632, 1634, 1637, 1625, 1325, 1345, 1365, 1367, 1362,
J (ssss): 1765, 1767, 1762, 1732, 1734, 1737, 1217, 1232, 1234, 1237, 1265,
          1267, 1262, 1254, 1252, 1545, 1565, 1567, 1562, 1525, 1517, 1512,
          1432, 1434, 1437, 1454, 1452, 1417, 1412, 1415
(A=0, B=4, C=2, D=6, E=5, F=3, G=11, H=15, I=11, J=30; total=87)

In Dorian:

A (ws):   none!
B (ss):   17, 12, 15, 14
C (wws):  164, 135
D (wss):  165, 167, 162, 132, 134, 137
E (sss):  174, 125, 154, 152, 145, 147
F (wwws): 1635, 1642, 1364, 1357
G (wwss): 1632, 1634, 1637, 1645, 1647, 1617, 1612, 1615, 1614, 1365,
          1367, 1362, 1317, 1312, 1315, 1314, 1354, 1352
H (wsws): 1657, 1675, 1672, 1627, 1624, 1327, 1324, 1342, 1375, 1372
I (wsss): 1654, 1652, 1674, 1625, 1325, 1345, 1347, 1374
J (ssss): 1765, 1767, 1762, 1745, 1747, 1732, 1734, 1737, 1217, 1232,
          1234, 1237, 1265, 1267, 1262, 1254, 1252, 1545, 1547, 1565,
          1567, 1562, 1525, 1517, 1512, 1432, 1434, 1437, 1454, 1452,
          1417, 1412, 1415, 1474
(A=0, B=4, C=2, D=6, E=6, F=4, G=18, H=10, I=8, J=34; total=92)

In Phrygian:

A (ws):   none!
B (ss):   17, 12, 15, 14
C (wws):  164, 135
D (wss):  165, 167, 162, 132, 134, 137
E (sss):  174, 154, 145, 147
F (wwws): 1642, 1642, 1352, 1357
G (wwss): 1645, 1647, 1612, 1617, 1615, 1614, 1312, 1317, 1315, 1314, 1354
H (wsws): 1652, 1657, 1675, 1672, 1635, 1625, 1627, 1624, 1325, 1327, 1324,
          1342, 1375, 1372, 1364
I (wsss): 1654, 1674, 1632, 1634, 1637, 1345, 1347, 1374, 1365, 1367, 1362
J (ssss): 1765, 1767, 1762, 1745, 1747, 1732, 1734, 1737, 1217, 1232, 1234,
          1237, 1265, 1267, 1262, 1545, 1547, 1565, 1567, 1562, 1517, 1512,
          1432, 1434, 1437, 1454, 1417, 1412, 1415, 1474
(A=0, B=4, C=2, D=6, E=4, F=4, G=11, H=15, I=11, J=30; total=87)

In Lydian:

A (ws):   none!
B (ss):   17, 12, 15
C (wws):  142, 135
D (wss):  145, 147, 165, 167, 162, 132, 137
E (sss):  125, 152
F (wwws): 1427, 1642, 1357
G (wwss): 1417, 1412, 1415, 1425, 1465, 1467, 1462, 1645, 1647, 1617, 1612,
          1615, 1317, 1312, 1315, 1352
H (wsws): 1435, 1457, 1475, 1472, 1657, 1675, 1672, 1635, 1627, 1327, 1342,
          1375, 1372
I (wsss): 1432, 1437, 1452, 1652, 1632, 1637, 1625, 1325, 1345, 1347, 1365,
          1367, 1362
J (ssss): 1765, 1767, 1762, 1745, 1747, 1732, 1737, 1217, 1232, 1237, 1265,
          1267, 1262, 1252, 1545, 1547, 1565, 1567, 1562, 1525, 1517, 1512
(A=0, B=3, C=2, D=7, E=2, F=3, G=16, H=13, I=13, J=22; total=81)

In Mixolydian:

A (ws):   none!
B (ss):   17, 12, 15, 14
C (wws):  164, 137, 135
D (wss):  165, 167, 162, 132, 134
E (sss):  174, 125, 154, 152, 145, 147
F (wwws): 1642, 1375, 1372, 1357
G (wwss): 1645, 1647, 1617, 1612, 1615, 1614, 1374, 1317, 1312, 1315,
          1314, 1354, 1352
H (wsws): 1657, 1675, 1672, 1637, 1635, 1627, 1624, 1327, 1324, 1342,
          1364, 1361
I (wsss): 1654, 1652, 1674, 1632, 1634, 1625, 1325, 1345, 1347, 1365,
          1367, 1362
J (ssss): 1765, 1767, 1762, 1745, 1747, 1217, 1215, 1232, 1234, 1265,
          1267, 1262, 1254, 1252, 1545, 1547, 1565, 1567, 1562, 1525,
          1517, 1512, 1432, 1434, 1454, 1452, 1417, 1412, 1415, 1474

(A=0, B=4, C=3, D=5, E=6, F=4, G=13, H=12, I=12, J=30; total=89)

In Aeolian:

A (ws):   none!
B (ss):   17, 12, 15, 14
C (wws):  162, 164, 135
D (wss):  165, 167, 132, 134, 137
E (sss):  174, 125, 154, 152, 145, 147
F (wwws): 1627, 1624, 1642, 1357
G (wwss): 1625, 1645, 1647, 1617, 1612, 1615, 1614, 1317, 1312, 1315, 
          1314, 1354, 1352
H (wsws): 1657, 1675, 1672, 1635, 1327, 1342, 1375, 1372, 1362, 1364
I (wsss): 1654, 1652, 1674, 1632, 1634, 1637, 1325, 1345, 1347, 1374,
          1365, 1367
J (ssss): 1765, 1767, 1745, 1747, 1732, 1734, 1737, 1217, 1232, 1234,
          1237, 1254, 1252, 1545, 1547, 1565, 1567, 1525, 1517, 1512,
          1432, 1434, 1437, 1454, 1452, 1417, 1412, 1415, 1474
(A=0, B=4, C=3, D=5, E=6, F=4, G=13, H=10, I=12, J=29; total=86)

Grand totals:
(A=0, B=23, C=14, D=35, E=29, F=22, G=82, H=75, I=67, J=175;
2-chord loops = 23, 3-chord loops = 78, 4-chord loops = 421;
total=522)

There are no cases of type A. That's because our categories of strong and weak are symmetrical. If we consider, for example, transitions by a second strong when ascending and weak when descending, 17 would fall into type A rather than type B.

Notice that, in contrast with flavor harmony, we have done nothing to analyse dissonance, tension, modality (majorness vs. minorness), richness, or counterpoint. All we've done is taken a ham-handed approximation of progression, oversimplified it into two categories, and found the interesting loops on that basis. That's tonal harmony for you.

So, to use these, you simply choose a mode, a loop of whatever type, learn the chords, and strum away. Say, Lydian on F, type H loop 1675 (F→d→e→C→). Then you might improvise melodies and bass parts above and below that, in the same key (or maybe even in another diatonic mode).  Remember me when you hit the bigtime.