by Robert Asmussen, PhD
This chorale, "Ach Gott, wie manches Herzeleid", contains identifiable tonal contradictions. For example, in measure 6, beat 4, the chord type is viio/ii, with a passing D in the bass. If the D is tuned according to Rule 24, then A#4 (25:3) is increased by a major third (is multiplied by 5:4) for a value of 125:12. Adjusted downward 2 octaves (divided by 4), the value for D3 in the bass is 125:48, roughly 2.604.
If the same D is tuned according to Rule 29, then C#4 (10:1) is decreased by a major seventh (is divided by 15:8) for a value of 16:3. Adjusted downward 1 octave (divided by 2), the value for D3 in the bass is 8:3, roughly 2.667.
The Csound score created for this chorale provides silence for the duration of each of the three contradictory chords. These chords can be found in Measure 6 Beat 4, Measure 8 Beat 2.5, and Measure 9 Beat 1.5.
The fraction representing the frequency for each note, given as p5/p6 in each line of the Csound score, is multiplied by 55, which is an octave equivalent of A1. For instance, in the first event of the Csound score is the following line:
i1 3 0.5 0 8 1 ; (A4, Rule: 1.a)
For this event, parameter fields are defined as follows:
p1 (instrument number) is 1
p2 (start time) is 3
p3 (duration) is 0.5
p4 (unused) is 0
p5 (frequency numerator) is 8
p6 (frequency denominator) is 1
The frequency for this event is therefore 8/1 * 55, which is 440 Hz.
Both traditionally notated and Csound versions are given.
sr = 44100
kr = 441
ksmps = 100
nchnls = 1
instr 1
a1 pluck 9000, (p5/p6)*55, (p5/p6)*55, 0, 1, 0, 0
out a1
endin
instr 2
a1 pluck 9000, (p5/p6)*55, (p5/p6)*55, 0, 1, 0, 0
out a1
endin
instr 3
a1 pluck 9000, (p5/p6)*55, (p5/p6)*55, 0, 1, 0, 0
out a1
endin
instr 4
a1 pluck 9000, (p5/p6)*55, (p5/p6)*55, 0, 1, 0, 0
out a1
endin
; Chorale File Number: 000306b_.sco
; Riemenschneider #156
; "Ach Gott, wie manches Herzeleid"
; Key: A Major
; 1/1 = A1 (A1 = 55 cps)
t 0 40 ; tempo 40 beats per minute
a 0 0 2.5 ; advance sound by 2.5 beats to avoid silence
; Measure 1 Beat 4 Chord Type: I (-5)
i1 3 0.5 0 8 1 ; (A4, Rule: 1.a)
i2 3 0.5 0 5 1 ; (C#4, Rule: 1.b)
i3 3 0.5 0 4 1 ; (A3, Rule: 1.a)
i4 3 0.5 0 4 1 ; (A3, Rule: 1.a)
; Measure 1 Beat 4.5 Chord Type: I (Embedded Dom Passing)
i1 3.5 0.5 0 8 1 ; (A4, Rule: 1.a)
i2 3.5 0.5 0 16 3 ; (D4, Rule: 22{A->D} 23{A->D})
i3 3.5 0.5 0 9 2 ; (B3, Rule: 27{D->B})
i4 3.5 0.5 0 4 1 ; (A3, Rule: 1.a)
; Measure 2 Beat 1 Chord Type: I
i1 4 0.5 0 8 1 ; (A4, Rule: 1.a)
i2 4 0.5 0 6 1 ; (E4, Rule: 1.c)
i3 4 0.5 0 5 1 ; (C#4, Rule: 1.b)
i4 4 0.5 0 2 1 ; (A2, Rule: 1.a)
; Measure 2 Beat 1.5 Chord Type: I (Embedded Dom Passing)
i1 4.5 0.5 0 8 1 ; (A4, Rule: 1.a)
i2 4.5 0.5 0 6 1 ; (E4, Rule: 1.c)
i3 4.5 0.5 0 16 3 ; (D4, Rule: 22{A->D})
i4 4.5 0.5 0 9 4 ; (B2, Rule: 23{E->B})
; Measure 2 Beat 2 Chord Type: I6
i1 5 1 0 8 1 ; (A4, Rule: 1.a)
i2 5 1 0 4 1 ; (A3, Rule: 1.a)
i3 5 1 0 6 1 ; (E4, Rule: 1.c)
i4 5 1 0 5 2 ; (C#3, Rule: 1.b)
; Measure 2 Beat 3 Chord Type: IV
i1 6 0.5 0 20 3 ; (F#4, Rule: 11.b)
i2 6 0.5 0 16 3 ; (D4, Rule: 11.a)
i3 6 0.5 0 4 1 ; (A3, Rule: 11.c)
i4 6 0.5 0 8 3 ; (D3, Rule: 11.a)
; Measure 2 Beat 3.5 Chord Type: IV (Embedded Dom Passing)
i1 6.5 0.5 0 15 2 ; (G#4, Rule: 24{E->G#})
i2 6.5 0.5 0 16 3 ; (D4, Rule: 11.a)
i3 6.5 0.5 0 4 1 ; (A3, Rule: 11.c)
i4 6.5 0.5 0 3 1 ; (E3, Rule: 23{A->E})
; Measure 2 Beat 4 Chord Type: vi
i1 7 0.5 0 8 1 ; (A4, Rule: 18.b)
i2 7 0.5 0 5 1 ; (C#4, Rule: 18.c)
i3 7 0.5 0 4 1 ; (A3, Rule: 18.b)
i4 7 0.5 0 10 3 ; (F#3, Rule: 18.a)
; Measure 2 Beat 4.5 Chord Type: I4/3
i1 7.5 0.5 0 8 1 ; (A4, Rule: 3.a)
i2 7.5 0.5 0 5 1 ; (C#4, Rule: 3.b)
i3 7.5 0.5 0 15 4 ; (G#3, Rule: 3.d)
i4 7.5 0.5 0 3 1 ; (E3, Rule: 3.c)
; Measure 3 Beat 1 Chord Type: V6/V
i1 8 0.5 0 9 1 ; (B4, Rule: 15.a)
i2 8 0.5 0 9 2 ; (B3, Rule: 15.a)
i3 8 0.5 0 27 8 ; (F#3, Rule: 15.c)
i4 8 0.5 0 45 16 ; (D#3, Rule: 15.b)
; Measure 3 Beat 1.5 Chord Type: V6/5/V
i1 8.5 0.5 0 8 1 ; (A4, Rule: 16.d)
i2 8.5 0.5 0 9 2 ; (B3, Rule: 16.a)
i3 8.5 0.5 0 27 8 ; (F#3, Rule: 16.c)
i4 8.5 0.5 0 45 16 ; (D#3, Rule: 16.b)
; Measure 3 Beat 2 Chord Type: V (-5)
i1 9 0.5 0 15 2 ; (G#4, Rule: 15.b)
i2 9 0.5 0 6 1 ; (E4, Rule: 15.a)
i3 9 0.5 0 15 4 ; (G#3, Rule: 15.b)
i4 9 0.5 0 3 1 ; (E3, Rule: 15.a)
; Measure 3 Beat 2.5 Chord Type: I4/3 (-3)
i1 9.5 0.5 0 15 2 ; (G#4, Rule: 3.d)
i2 9.5 0.5 0 6 1 ; (E4, Rule: 3.c)
i3 9.5 0.5 0 4 1 ; (A3, Rule: 3.a)
i4 9.5 0.5 0 3 1 ; (E3, Rule: 3.c)
; Measure 3 Beat 3 Chord Type: V/V
i1 10 1 0 27 4 ; (F#4, Rule: 15.c)
i2 10 1 0 45 8 ; (D#4, Rule: 15.b)
i3 10 1 0 9 2 ; (B3, Rule: 15.a)
i4 10 1 0 9 4 ; (B2, Rule: 15.a)
; Measure 3 Beat 4 Chord Type: V/V
i1 11 0.5 0 27 4 ; (F#4, Rule: 15.c)
i2 11 0.5 0 45 8 ; (D#4, Rule: 15.b)
i3 11 0.5 0 9 2 ; (B3, Rule: 15.a)
i4 11 0.5 0 9 4 ; (B3, Rule: 15.a)
; Measure 3 Beat 4.5 Chord Type: V4/2/V
i1 11.5 0.5 0 27 4 ; (F#4, Rule: 16.c)
i2 11.5 0.5 0 45 8 ; (D#4, Rule: 16.b)
i3 11.5 0.5 0 9 2 ; (B3, Rule: 16.a)
i4 11.5 0.5 0 4 1 ; (A3, Rule: 16.d)
; Measure 4 Beat 1 Chord Type: V6
i1 12 0.5 0 15 2 ; (G#4, Rule: 15.b)
i2 12 0.5 0 6 1 ; (E4, Rule: 15.a)
i3 12 0.5 0 9 2 ; (B3, Rule: 15.c)
i4 12 0.5 0 15 4 ; (G#3, Rule: 15.b)
; Measure 4 Beat 1.5 Chord Type: V6/4 (+4, 4 in bass)
i1 12.5 0.5 0 15 2 ; (G#4, Rule: 15.b)
i2 12.5 0.5 0 6 1 ; (E4, Rule: 15.a)
i3 12.5 0.5 0 9 2 ; (B3, Rule: 15.c)
i4 12.5 0.5 0 4 1 ; (A3, Rule: 22 (E->A))
; Measure 4 Beat 2 Chord Type: viio/V (+4, 4 in bass)
i1 13 0.5 0 8 1 ; (A4, Rule: 20.c)
i2 13 0.5 0 27 4 ; (F#4, Rule: 20.b)
i3 13 0.5 0 45 8 ; (D#4, Rule: 20.a)
i4 13 0.5 0 15 4 ; (G#3, Rule: 22{D#->G#})
; Measure 4 Beat 2.5 Chord Type: viio6/V
i1 13.5 0.5 0 8 1 ; (A4, Rule: 20.c)
i2 13.5 0.5 0 27 4 ; (F#4, Rule: 20.b)
i3 13.5 0.5 0 45 8 ; (D#4, Rule: 20.a)
i4 13.5 0.5 0 27 8 ; (F#3, Rule: 20.b)
; Measure 4 Beat 3 Chord Type: V (+2)
i1 14 0.5 0 9 1 ; (B4, Rule: 15.c)
i2 14 0.5 0 27 4 ; (F#4, Rule: 23{B->F#})
i3 14 0.5 0 15 4 ; (G#3, Rule: 15.b)
i4 14 0.5 0 3 1 ; (E3, Rule: 15.a)
; Measure 4 Beat 3.5 Chord Type: V4/3/V
i1 14.5 0.5 0 9 1 ; (B4, Rule: 16.a)
i2 14.5 0.5 0 45 8 ; (D#4, Rule: 16.b)
i3 14.5 0.5 0 4 1 ; (A3, Rule: 16.d)
i4 14.5 0.5 0 27 8 ; (F#3, Rule: 16.c)
; Measure 4 Beat 4 Chord Type: V6
i1 15 0.5 0 6 1 ; (E4, Rule: 15.a)
i2 15 0.5 0 6 1 ; (E4, Rule: 15.a)
i3 15 0.5 0 9 2 ; (B3, Rule: 15.c)
i4 15 0.5 0 15 4 ; (G#3, Rule: 15.b)
; Measure 4 Beat 4.5 Chord Type: vi6/5
i1 15.5 0.5 0 20 3 ; (F#4, Rule: 19.a)
i2 15.5 0.5 0 6 1 ; (E4, Rule: 19.d)
i3 15.5 0.5 0 5 1 ; (C#4, Rule: 19.c)
i4 15.5 0.5 0 4 1 ; (A3, Rule: 19.b)
; Measure 5 Beat 1 Chord Type: V6/4
i1 16 0.5 0 15 2 ; (G#4, Rule: 15.b)
i2 16 0.5 0 6 1 ; (E4, Rule: 15.a)
i3 16 0.5 0 9 2 ; (B3, Rule: 15.c)
i4 16 0.5 0 9 2 ; (B3, Rule: 15.c)
; Measure 5 Beat 1.5 Chord Type: V7/V (-3)
i1 16.5 0.5 0 8 1 ; (A4, Rule: 16.d)
i2 16.5 0.5 0 27 4 ; (F#4, Rule: 16.c)
i3 16.5 0.5 0 9 2 ; (B3, Rule: 16.a)
i4 16.5 0.5 0 9 2 ; (B3, Rule: 16.a)
; Measure 5 Beat 2 Chord Type: V/V
i1 17 0.5 0 27 4 ; (F#4, Rule: 15.c)
i2 17 0.5 0 45 8 ; (D#4, Rule: 15.b)
i3 17 0.5 0 9 2 ; (B3, Rule: 15.a)
i4 17 0.5 0 9 4 ; (B2, Rule: 15.a)
; Measure 5 Beat 2.5 Chord Type: V7/V
i1 17.5 0.5 0 27 4 ; (F#4, Rule: 16.c)
i2 17.5 0.5 0 45 8 ; (D#4, Rule: 16.b)
i3 17.5 0.5 0 4 1 ; (A3, Rule: 16.d)
i4 17.5 0.5 0 9 4 ; (B2, Rule: 16.a)
; Measure 5 Beat 3 Chord Type: V
i1 18 1 0 6 1 ; (E4, Rule: 15.a)
i2 18 1 0 9 2 ; (B3, Rule: 15.c)
i3 18 1 0 15 4 ; (G#3, Rule: 15.b)
i4 18 1 0 3 1 ; (E3, Rule: 15.a)
; Measure 5 Beat 4 Chord Type: V
i1 19 1 0 9 1 ; (B4, Rule: 15.c)
i2 19 1 0 15 2 ; (G#4, Rule: 15.b)
i3 19 1 0 6 1 ; (E4, Rule: 15.a)
i4 19 1 0 3 1 ; (E3, Rule: 15.a)
; Measure 6 Beat 1 Chord Type: I
i1 20 0.5 0 10 1 ; (C#5, Rule: 1.b)
i2 20 0.5 0 8 1 ; (A4, Rule: 1.a)
i3 20 0.5 0 6 1 ; (E4, Rule: 1.c)
i4 20 0.5 0 4 1 ; (A3, Rule: 1.a)
; Measure 6 Beat 1.5 Chord Type: vi (+2, 2 in bass)
i1 20.5 0.5 0 10 1 ; (C#5, Rule: 18.c)
i2 20.5 0.5 0 8 1 ; (A4, Rule: 18.b)
i3 20.5 0.5 0 20 3 ; (F#4, Rule: 18.a)
i4 20.5 0.5 0 15 4 ; (G#3, Rule: 23{C#->G#})
; Measure 6 Beat 2 Chord Type: V/vi (susp. 4, 4 in bass)
i1 21 0.5 0 10 1 ; (C#5, Rule: 15.a)
i2 21 0.5 0 5 1 ; (C#4, Rule: 15.a)
i3 21 0.5 0 15 2 ; (G#4, Rule: 15.c)
i4 21 0.5 0 10 3 ; (F#3, Rule: 22{C#->F#})
; Measure 6 Beat 2.5 Chord Type: V6/5/vi
i1 21.5 0.5 0 10 1 ; (C#5, Rule: 15.a)
i2 21.5 0.5 0 5 1 ; (C#4, Rule: 15.a)
i3 21.5 0.5 0 15 2 ; (G#4, Rule: 15.c)
i4 21.5 0.5 0 25 8 ; (F3, Rule: 15.b)
; Measure 6 Beat 3 Chord Type: vi
i1 22 0.5 0 8 1 ; (A4, Rule: 18.b)
i2 22 0.5 0 20 3 ; (F#4, Rule: 18.a)
i3 22 0.5 0 5 1 ; (C#4, Rule: 18.c)
i4 22 0.5 0 10 3 ; (F#3, Rule: 18.a)
; Measure 6 Beat 3.5 Chord Type: V7
i1 22.5 0.5 0 9 1 ; (B4, Rule: 16.c)
i2 22.5 0.5 0 15 2 ; (G#4, Rule: 16.b)
i3 22.5 0.5 0 16 3 ; (D4, Rule: 16.d)
i4 22.5 0.5 0 3 1 ; (E3, Rule: 16.a)
; Measure 6 Beat 4 Chord Type: viio/ii (passing D)
; i1 23 0.5 0 10 1 ; (C#5, Rule: 20.b)
; i2 23 0.5 0 25 3 ; (A#4, Rule: 20.a)
; i3 23 0.5 0 160 27 ; (E4, Rule: 20.c)
; i4 23 0.5
; (D3, CONTRADICTION Rule 24 (A#->D) D=125:48
; violates Rule 29 (C#->D) D=8:3)
; Measure 6 Beat 4.5 Chord Type: viio6/ii
i1 23.5 0.5 0 10 1 ; (C#5, Rule: 20.b)
i2 23.5 0.5 0 25 3 ; (A#4, Rule: 20.a)
i3 23.5 0.5 0 160 27 ; (E4, Rule: 20.c)
i4 23.5 0.5 0 5 2 ; (C#3, Rule: 20.b)
; Measure 7 Beat 1 Chord Type: ii
i1 24 0.5 0 32 3 ; (D5, Rule: 5.b)
i2 24 0.5 0 80 9 ; (B4, Rule: 5.a)
i3 24 0.5 0 20 3 ; (F#4, Rule: 5.c)
i4 24 0.5 0 20 9 ; (B2, Rule: 5.a)
; Measure 7 Beat 1.5 Chord Type: viio6
i1 24.5 0.5 0 32 3 ; (D5, Rule: 20.c)
i2 24.5 0.5 0 9 1 ; (B4, Rule: 20.b)
; (Held Mutable Tone)
i3 24.5 0.5 0 15 2 ; (G#4, Rule: 20.a)
i4 24.5 0.5 0 9 4 ; (B2, Rule: 20.b)
; (Held mutable tone)
; Measure 7 Beat 2 Chord Type: I6
i1 25 0.5 0 10 1 ; (C#5, Rule: 1.b)
i2 25 0.5 0 6 1 ; (E4, Rule: 1.c)
i3 25 0.5 0 8 1 ; (A4, Rule: 1.a)
i4 25 0.5 0 5 2 ; (C#3, Rule: 1.b)
; Measure 7 Beat 2.5 Chord Type: IV7
i1 25.5 0.5 0 10 1 ; (C#5, Rule: 12.d)
i2 25.5 0.5 0 20 3 ; (F#4, Rule: 12.b)
i3 25.5 0.5 0 8 1 ; (A4, Rule: 12.c)
i4 25.5 0.5 0 8 3 ; (D3, Rule: 12.a)
; Measure 7 Beat 3 Chord Type: V
i1 26 1 0 9 1 ; (B4, Rule: 15.c)
i2 26 1 0 15 2 ; (G#4, Rule: 15.b)
i3 26 1 0 6 1 ; (E4, Rule: 15.a)
i4 26 1 0 3 1 ; (E3, Rule: 15.a)
; Measure 7 Beat 4 Chord Type: vi
i1 27 1 0 8 1 ; (A4, Rule: 18.b)
i2 27 1 0 20 3 ; (F#4, Rule: 18.a)
i3 27 1 0 5 1 ; (C#4, Rule: 18.c)
i4 27 1 0 10 3 ; (F#3, Rule: 18.a)
; Measure 8 Beat 1 Chord Type: V6
i1 28 0.5 0 9 1 ; (B4, Rule: 15.c)
i2 28 0.5 0 6 1 ; (E4, Rule: 15.a)
i3 28 0.5 0 9 2 ; (B3, Rule: 15.c)
i4 28 0.5 0 15 4 ; (G#3, Rule: 15.b)
; Measure 8 Beat 1.5 Chord Type: viio
i1 28.5 0.5 0 9 1 ; (B4, Rule: 20.b)
i2 28.5 0.5 0 16 3 ; (D4, Rule: 20.c)
i3 28.5 0.5 0 9 2 ; (B3, Rule: 20.b)
i4 28.5 0.5 0 15 4 ; (G#3, Rule: 20.a)
; Measure 8 Beat 2 Chord Type: I
i1 29 0.5 0 10 1 ; (C#5, Rule: 1.b)
i2 29 0.5 0 5 1 ; (C#4, Rule: 1.b)
i3 29 0.5 0 3 1 ; (E3, Rule: 1.c)
i4 29 0.5 0 4 1 ; (A3, Rule: 1.a)
; Measure 8 Beat 2.5 Chord Type: viio7/V
; (Rule 21.a: CONTRADICTION)
; i1 29.5 0.5 (C#5)
; i2 29.5 0.5 (D#4)
; i3 29.5 0.5 (F#3)
; i4 29.5 0.5 (A3)
; Measure 8 Beat 3 Chord Type: V
i1 30 0.5 0 9 1 ; (B4, Rule: 15.c)
i2 30 0.5 0 6 1 ; (E4, Rule: 15.a)
i3 30 0.5 0 15 4 ; (G#3, Rule: 15.b)
i4 30 0.5 0 3 1 ; (E3, Rule: 15.a)
; Measure 8 Beat 3.5 Chord Type: V7
i1 30.5 0.5 0 9 1 ; (B4, Rule: 16.c)
i2 30.5 0.5 0 16 3 ; (D4, Rule: 16.d)
i3 30.5 0.5 0 15 4 ; (G#3, Rule: 16.b)
i4 30.5 0.5 0 3 1 ; (E3, Rule: 16.a)
; Measure 8 Beat 4 Chord Type: vi
i1 31 0.5 0 8 1 ; (A4, Rule: 18.b)
i2 31 0.5 0 5 1 ; (C#4, Rule: 18.c)
i3 31 0.5 0 4 1 ; (A3, Rule: 18.b)
i4 31 0.5 0 10 3 ; (F#3, Rule: 18.a)
; Measure 8 Beat 4.5 Chord Type: I6/4
i1 31.5 0.5 0 8 1 ; (A4, Rule: 1.a)
i2 31.5 0.5 0 5 1 ; (C#4, Rule: 1.b)
i3 31.5 0.5 0 4 1 ; (A3, Rule: 1.a)
i4 31.5 0.5 0 3 1 ; (E3, Rule: 1.c)
; Measure 9 Beat 1 Chord Type: V6/5/V
i1 32 0.5 0 8 1 ; (A4, Rule: 16.d)
i2 32 0.5 0 27 4 ; (F#4, Rule: 16.c)
i3 32 0.5 0 9 2 ; (B3, Rule: 16.a)
i4 32 0.5 0 45 16 ; (D#3, Rule: 16.b)
; Measure 9 Beat 1.5 Chord Type: viio7/V (Rule 21.a: CONTRADICTION)
; i1 32.5 0.5 (A4)
; i2 32.5 0.5 (F#4)
; i3 32.5 0.5 (C#4)
; i4 32.5 0.5 (D#3)
; Measure 9 Beat 2 Chord Type: V7 (-5)
i1 33 1 0 15 2 ; (G#4, Rule: 16.b)
i2 33 1 0 6 1 ; (E4, Rule: 16.a)
i3 33 1 0 16 3 ; (D4, Rule: 16.d)
i4 33 1 0 3 1 ; (E3, Rule: 16.a)
; Measure 9 Beat 3 Chord Type: I
i1 34 2 0 8 1 ; (A4, Rule: 1.a)
i2 34 2 0 6 1 ; (E4, Rule: 1.c)
i3 34 2 0 5 1 ; (C#4, Rule: 1.b)
i4 34 2 0 2 1 ; (A2, Rule: 1.a)
e
Robert Asmussen
2008
