Chapter VII: Conclusions

We have studied the various elements of sound and harmony that have inspired different cultures to make the most profound connection to a higher reality through music.  In the West, the Gregorian songs employ only the purest of intervals, which can be validated using mathematics, the language of nature. In the East cultures have consistently used certain pitches derived apparently intuitively, that can be mathematically related to cosmic ratios. Other philosophers and mystics have made their own proposals, and all of this data fits nicely into a single musical scale.

Based on the analysis of the most appropriate tuning system and the optimal number of notes in a scale presented in Chapter V, in combination with the coincidental findings described in Chapter VI, we recommend an optimal scale for modal music with an uplifting, transcending intention using the Pythagorean tuning system centered at A = 432 Hz and replacing the C# and G by the corresponding tones of the Earth’s year and Earth’s day, resulting in the following data:

 

 Note 

Frequency (Hz)

Ratio

Comment

A

432

1:1 

Tonic, sacred number, digit sum = 9

A#

455.1

 256:243

Minor Second Pythagorean interval, symmetric to Major Seventh 

B

486

 9:8

Major Second Pythagorean interval, harmonic interval, digit sum = 9 

 C

512

 32:27

 Minor Third Pythagorean interval, power of 2, symmetric to Major Sixth, Rudolf Steiner's recommendation

C#

544.4

 712:565

Mediant, tone of the Earth's year, tone of OM

D

576

 4:3

Perfect Fourth Pythagorean interval, harmonic interval, symmetric to Fifth, digit sum = 9

D#

606.8

 1024:729

Diminished Fifth Pythagorean interval

E

648

 3:2

Dominant, Pythagorean interval, harmonic interval, symmetric to Fourth, digit sum = 9

F

682.7

 128:81

Minor Sixth Pythagorean interval

 F#

729

 27:16

Major Sixth Pythagorean interval, harmonic interval, symmetric to Minor Third, digit sum = 9

G

776.7

 989:550

Minor Seventh Pythagorean interval, tone of the Earth's day

G#

820.1

 243:128

Major Seventh Pythagorean interval, symmetric to Minor Second

Table 4: TranscendentalTone's recommended modal scale

 

By understanding and combining these features one can design sounds that aim to influence our state of mind for the better. Sounds we can deeply relax to, tones that bring us well-being and joy, and perhaps even make us feel closer to our inherent Divinity.

We have designed our Tones of the Earth and our 432 Hz meditations in this way; we hope you enjoy our carefully created, conscientious compositions.

 

As far as a tempered scale is concerned (one that can be used to play polymodal music, as explained in Chapter V), we recommend the use of Maria Renold's 1962 tempered tuning, which is a variation of the 12-tone Pythagorean scale. Renold's brilliant contribution results in both interval temperament and mathematically sound interval ratios simultaneously. She bases her resulting tones on strict Pythagorean frequencies for the diatonic scale (the white keys on the piano), but then forms the remaining 5 tones using the geometric middle between two of these Pythagorean notes. Maria Renold, a German musician and anthroposophist, documented her discovery in great detail on her book Intervals, Scales, Tones and the Concert Pitch 128 Hz.

Renold's scale could simply replace the current standard of equal tempered tuning completely, resulting in true intervals that work consistently for every harmonic combination in any style of music. Unfortunately, the logistic endeavour to accomplish a new global tuning standard would be monumental, not to mention the lack of interest of the majority of the public on the subtle, yet transcendental impact of employing archetypal tones in music. However, you can tune your own instrument to this wonderful scale and benefit from its outstanding qualities.

You can listen to a composition recorded using Renold's scale directly from our site by clicking on the title "Concierto Improvisado para Arpa Mágica y Ranas Sinfónicas", on the player at the bottom of our Home Page

Audio Samples: Recommended Modal Scale

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