LCM trajectory · prototype

BWV 38.6 · Aus tiefer Not schrei ich zu dir

Phrygian-flavored harmonization of the Luther hymn. Home tonic A (Krumhansl 0.93). The piece closes on V (E major), the characteristic Phrygian half-cadence ending.

E majorlocal key · 0.73I⁴⁄₂m1 · b1.0161215203040721201442401̂5̂2̂6̂3̂7̂♯4̂♭2̂♭6̂♭3̂♭7̂4̂1̂home: AB4SE4AG#3TD3B
Phrase 1 of 5 · mm. 1–3
Tempo60 bpm (1.00×)
Position
m1 · b1.0
event 1 / 57 · dur 2♩
Roman numeral
I⁴⁄₂
Local key
E major (0.73)
Pitch LCM (max / mean)
144 / 42.8
Chord LCM
root 1 · internal 420
Region LCM (vs A)
6
Voices
S: B4A: E4T: G#3B: D3
Tendency tones
none
jump:compare:

How to read this

Center is the local tonic (changes across phrases as Krumhansl re-estimates the key). Radial distance is each sounding pitch's LCM measure against the local tonic. The dashed rings mark canonical LCMs — closer in is more consonant.

Angle follows the circle of fifths: 12 o'clock is the tonic, 1 o'clock is the dominant, and so on clockwise. So pitches that are tonally close also sit visually close.

Color coding: cream = ordinary pitch; gold = chord root (the lowest-LCM pitch in the texture, approximately); red = leading tone; amber = Phrygian pull (♭6̂ or ♭2̂); purple = chordal seventh; green = secondary- function tone (temporary leading tone / chordal seventh).

Green dashed circle appears when the local key differs from home (A) — it shows where the local tonic itself sits on the home-key gradient.

What to look for

Use the cadence-jump buttons. Compare a leading-tone HC (m5, m7, m11, m13 — the G♯ pitch sits on the LCM 120 ring) against a Phrygian-pull moment (m1 b3, m2 b1 — the F-natural sits on the LCM 40 ring). The Phrygian half-step from above is acoustically 3× closer to home than the leading-tone half-step from below — exactly what Claim 6 predicts.