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Research

Appendix I

18-Tone microtonal division of the octave derived from:

Ratios of Just Intonation & degrees of the overtone series

 

Diagram 1: 18 Pitches Defined by Frequency Value and Ratio to the Fundamental C 

(+) or (-) indicates a microtonal alteration of the scale degree up or down

Diagram 2: Pitch Pyramid Displaying the Hierarchy of Ratios in the 18-Tone Scale

3:2

4:3       5:3

5:4                   7:4

6:5       7:5       8:5       9:5

7:6                                           11:6

33:32   17:16   9:8      11:8     25:16   31:16

1:1                                                                        2:1

 

 

Diagram 3: Twelve Tone Divisions of the Octave Beginning on A

 

Degrees of the scale as derived From the Cycle of Fifths (C. 5th) in relation to degrees of the scale as derived from the Harmonic Series (H. S.) and Just Intonation (J. In.)

 

C. 5th  Degree & Value in Hertz      Ratio to Previous 5th                    Value as Fraction

H. S.    Degree & Value in Hertz      Ratio to Related Tonic                   Value as Fraction

J. In.     Degree & Value in Hertz      Ratio to Related Tonic                  Value as Fraction

                       

I           C.5th    A = 55 hz                    Ratio to Prev. 5th = 1:1           1

            H.S.      A = 55 hz                     Ratio to Rel. Tonic = 1:1         1

            J.In.      A = 55 hz                     Ratio to Rel. Tonic = 1:1         1

V          C.5th    E = 82.5 hz                Ratio to Prev. 5th = 3:2            1.5

            H.S.       E = 82.5 hz                 Ratio to Rel. Tonic = 3:2         1.5

            J.In.       E = 82.5 hz                  Ratio to Rel. Tonic = 3:2         1.5

II         C.5th    B = 123.75 hz            Ratio to Prev. 5th = 3:2            1.125

            H.S.     B = 123.75 hz             Ratio to Rel. Tonic = 9:8         1.125

            J.In.      B = 123.75 hz              Ratio to Rel. Tonic = 9:8         1.125

 

VI        C.5th    F# = 185.625 hz        Ratio to Prev. 5th = 3:2            1.6875

            H.S.      F# = 185.625 hz         Ratio to Rel. Tonic = 27:16     1.6875

            J.In.     F# = 183.333 hz          Ratio to Rel. Tonic = 5:3         1.6667

 

III        C.5th    C# = 278.439 hz        Ratio to Prev. 5th = 3:2            1.2656

            H.S.      C# = 275.000 hz         Ratio to Rel. Tonic = 5:4         1.25

            J. In.     C# = 275.000 hz          Ratio to Rel. Tonic = 5:4         1.25

 

VII       C.5th    G# = 417.658 hz        Ratio to Prev. 5th = 3:2            1.8984

            H.S.      G# = 412.500 hz         Ratio to Rel. Tonic = 15:8       1.875  

            J.In.      G# = 408.571 hz          Ratio to Rel. Tonic = 11:6       1.8333

            J.In.      G# = 412.500 hz          Ratio to Rel. Tonic = 15:8       1.875

 

#IV      C.5th    D# = 626.487 hz        Ratio to Prev. 5th = 3:2            1.4238

            H.S.      D# = 632.500 hz         Ratio to Rel. Tonic = 23:16     1.4375

            J.In.      D# = 616.000 hz          Ratio to Rel. Tonic = 7:5         1.4

 

bII        C.5th    Bb = 939.731 hz       Ratio to Prev. 5th = 3:2             1.0678

            H.S.       Bb = 935.000 hz         Ratio to Rel. Tonic = 17:16     1.0625

            J.In.       Bb = 938.666 hz          Ratio to Rel. Tonic = 16:15     1.0667

bVI       C.5th    F = 1409.597 hz       Ratio to Prev. 5th = 3:2            1.6018

            H.S.       F = 1402.500 hz         Ratio to Rel. Tonic = 13:8       1.625

            J.In.       F = 1408.000 hz          Ratio to Rel. Tonic = 8:5         1.6

 

bIII      C.5th    C = 2114.396 hz        Ratio to Prev. 5th = 3:2            1.2013

            H.S.      C = 2090.000 hz         Ratio to Rel. Tonic = 19:16     1.1875

            H.S.      C = 2145.000 hz         Ratio to Rel. Tonic = 39:32     1.2187

            J.In.      C = 2112.000 hz          Ratio to Rel. Tonic = 6:5         1.2

 

bVII     C.5th    G = 3171.594 hz        Ratio to Prev. 5th = 3:2            1.802

            H.S.      G = 3190.000 hz         Ratio to Rel. Tonic = 29:16     1.8125

            J.In.      G = 3168.000 hz          Ratio to Rel. Tonic = 9:5         1.8

 

IV        C.5th    D = 4757.391 hz        Ratio to Prev. 5th = 3:2            1.3515

            H.S.      D = 4620.000 hz         Ratio to Rel. Tonic = 21:16     1.3125

            H.S.      D = 4730.000 hz         Ratio to Rel. Tonic = 43:32     1.3437

            J.In.      D = 4693.333 hz          Ratio to Rel. Tonic = 4:3         1.3334

 

I           C.5th    A = 7136.048 hz        Ratio to Prev. 5th = 3:2           1.0136432

            H.S.      A = 7040.000 hz         Ratio to Rel. Tonic = 2:1         1

            H.S.      A = 7150.000 hz         Ratio to Rel. Tonic = 65:64     1.015625

            J.In.      A = 7040.000 hz          Ratio to Rel. Tonic = 2:1         1

 

 

Diagram 4: Microtonal Divisions of the Octave Beginning on ‘A’

 

H. S.    Degree & Value in Hertz     Ratio to Related Tonic        Value as Fractio

J. I.      Degree & Value in Hertz      Ratio to Related Tonic        Value as Fraction

           

#I         H.S.      A ¼# = 907.500 hz      Ratio to Rel. Tonic = 33:32     1.0312

            J.In.     A ¼# = 916.667 hz      Ratio to Rel. Tonic = 25:24     1.0416

 

#V        H.S.      E ¼# = 1375.000 hz    Ratio to Rel. Tonic = 25:16     1.5625

            J.In.     E ¼# = 1364.000 hz    Ratio to Rel. Tonic = 31:20     1.55

 

#II       H.S.      B ¼# = 1430.000 hz    Ratio to Rel. Tonic = 37:32     1.1562

            J.In.     B¼# = 1425.600 hz     Ratio to Rel. Tonic = 7:6         1.1667

 

#VII     H.S.      G ¾# = 426.250 hz      Ratio to Rel. Tonic = 31:16     1.9375

            J.In.     G ¾# = 422.400 hz      Ratio to Rel. Tonic = 48:25     1.92

 

#IV      H.S.      D ¼# = 605.000 hz      Ratio to Rel. Tonic = 11:8       1.375

            J.In.     D ¼# = 606.222 hz      Ratio to Rel. Tonic = 62:45     1.3778

 

VII       H.S.      G = 3080.000 hz          Ratio to Rel. Tonic = 7:4         1.75

            J.In.     G = 3080.000 hz          Ratio to Rel. Tonic = 7:4         1.75

 

 

 

A Notational Method for the 18-Tone Division of the Octave

 

The music notation will reflect a strong horizontal trajectory. Extended suspensions in the accompanying voices will be used to impart modal colour to melodic passages; any vertical harmony will be considered incidental, the result of overlapping polyphonic structures. Because of the harmonic transparency of the electroacoustic soundtrack, and its protracted rate of harmonic change, it will be possible for the choral voices to articulate a microtonal melodic division of the octave.

 

Excerpt 1: 12-Tone Chromatic Scale Beginning on ‘C’

Upon this structure, a further 6 ‘colour-tones’ will be imposed. These tones are derived primarily from the ratios of Just Intonation and the first 33 degrees of the Harmonic Series. They appear here, in the way in which they will be described and employed in the score.

 

Excerpt 2: 6 Additional Microtones for a Scale Beginning on ‘C’

Two new symbols are introduced to describe their microtonal relationship to the fundamental ‘C’. The single-barred sharp indicates the note is to be raised slightly more than a quartertone above the natural: (¼ +). The inverted flat indicates the note is to be lowered slightly less than a quartertone beyond the traditional flat: (¾ -). Here are all the tones of the scale, as they will appear in the score in ascending order.

Excerpt 3: 18-Tone Microtonal Scale Beginning on ‘C’

Initially, the appearance of microtones may be intimidating to some members of the choir; their natural response will be to over state them. It will help vocalists to think of these altered tones as momentary dissonances, which resolve to more consonant intervals. Understanding the trajectory of the melody, and identifying points of harmonic repose within it will help in its execution. Choristers may be reminded that in the context of this piece, the intended purpose of these colour-tones is the evocation of subtle emotion. If they identify with them as an emotive tool, (and not an intellectual one) they will enjoy more success. The technical accuracy of colour-tones may be refined as the choir comes to understand and internalize the material.

 

Excerpt 4: Extract of Music Incorporating this Notational Method