Chord progressions
Let the cycle write the harmony. Each step n moves the root of the chord by its interval: a fifth goes up a fifth, a fourth up a fourth, the unison and the octave stay, and silence rests. With the Pentametric cycle this gives only three chords, I, IV and V, and the root comes home after every period. Play it, then download it as MIDI with smooth voice leading, a bass line and, in flamenco style, palmas.
How the chords are made
The cycle gives intervals, not chords, so a rule turns one into the other. Here the rule moves the root: H(n) = 3 (the fifth, 3:2) moves it up a fifth, H(n) = 4 (the fourth, 4:3) up a fourth, H(n) = 1 and 2 (unison and octave) leave it where it is, and H(n) = 0 is a rest. On each root the page builds the chord of the chosen scale, and the voicing moves each voice to the nearest note of the next chord.
Starting from the tonic, fifths and fourths alone can only reach the fourth and fifth degrees, so the Pentametric cycle plays I, IV and V and nothing else: the harmony of the blues, folk and early rock, produced by the formula without being chosen. Each period of 25 holds four fifths and four fourths, and a fifth plus a fourth is an octave, so the root is back home after every period (the musical face of “the consonances multiply to 2⁸”).
Flamenco. The box switches to the Spanish minor (in A by default): i and iv from the natural minor and a major V, so the same rule gives Am, Dm and E (Am7, Dm7 and E7 with a minor ninth when sevenths are on). That is the backbone of a lot of rumba flamenca. Choose rumba for a 4/4 strum with palmas on 2 and 4, or compás de 12 for one step per beat with palmas accenting 3, 6, 8, 10 and 12, as in soleá and bulerías. A period has 24 transitions, exactly two compases of 12; this is a numerical coincidence, recorded as such. Between the blues and flamenco there is a structural resemblance: a 12-bar blues is three 4-bar phrases, and heard one bar to a beat it becomes a 12-beat cycle like the compás, with its phrases grouped differently (4 + 4 + 4 against 3 + 3 + 2 + 2 + 2), like the same language spoken with two accents. Choose 12 steps with 4 beats per chord for a 12-bar form, then switch flamenco on to hear it as compás. The classic flamenco cadence Am–G–F–E steps downwards and does not come from the formula.
With the silence as 5 (Lluis's reading, exploratory). The 9 silences of a period become major thirds 5:4, and a third moves the root up 4 semitones. The harmony is then no longer only I, IV and V: chains of thirds reach nine roots (in C: C, E, G♯/A♭, C♯/D♭, F, A, D♯/E♭, G and B), many outside the scale, the chromatic mediants of Romantic and film music. The root still comes home after every period: 4 fifths, 4 fourths and 9 thirds make 4·7 + 4·5 + 9·4 = 84 semitones, exactly 7 octaves. proved The root walk counts semitones, so here it closes exactly. With pure intervals it would not: nine pure thirds 5:4 fall 123 cents short of three octaves (each is 14 cents narrower than the equal-tempered third), so a walk in pure thirds would sink by more than a semitone every period. computed
13-cycle (exploratory). Uses ⌊n²/13⌋ mod 13 with the same rule H : (H − 1), so the root can move by every ratio from 2:1 to 12:11, each rounded to the nearest semitone (5:4 → 4, 6:5 and 7:6 → 3, 8:7 to 12:11 → 2). It visits all twelve roots and also comes home after each period of 169. Roots outside the scale get a major chord (a dominant seventh with sevenths on). Chords sound in equal temperament, so they can be played on any instrument.
Tuning (playback and MIDI file: the file gives each sounding note its own channel with a pitch bend, range ±2 semitones, so any General MIDI synthesizer plays the tuning). Pentametric fifths tunes every note by the cycle's own pure fifth 3:2 stacked up from the tonic, Pentametric fourths by its pure fourth 4:3 (as on The chord wheel); both are Pythagorean tunings that go only one way round the circle, so the comma falls inside these progressions: with fifths the IV chord is reached by eleven fifths up (its root is E♯ in C, 21.5¢ high, and its fifth is a wolf), with fourths the V chord likewise (its root is A𝄫, 21.5¢ low). Listen for it. Pythagorean goes both ways from the tonic, a fifth up and a fourth up, as the cycle's own root walk does, so I, IV and V all keep pure fifths and pure roots (five fifths below the tonic, six above). Kirnberger fifth narrows each pure fifth by one schisma, which is exactly (√10/3 ÷ 256/243)², the gap between the √10/3 curiosity and the limma; it sounds like equal. Meantone, Werckmeister III, Kirnberger III and Vallotti are the historical temperaments, measured from the tonic.