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. Det Springende Punkt presents Keys to the Harmonic Universe

The first four octaves (windings) of the octave spiral with vowel sounds, circle numbers and solfège inserted

From the perspective of the harmonic series music is the language of natural numbers - Its atoms and molecules are primes and compounds of primes One winding of the octave spiral corresponds to the ratio 1:2, which is one octave: We return to the same quality on a new pitch level: 1-2-4-8-16-... are octaves of the fundamental. 3-6-12-... are octaves of the perfect fifth. 5-10-... are octaves of the just major third. Etc. 1:2 First octave: 2:4 Second octave: 4:8 Third octave: 8:16 Fourth octave:

1:2 octave. 2:3 perfect fifth; 3:4 perfect fourth. 4:5 just major third; 5:6 just minor third; 6:7 septimal third; 7:8 septimal second. 8:9 major whole-tone; 9:10 minor whole-tone;... 15:16 major semitone. Etc.

The correspondences between tone, body resonance and phonetics should not be interpreted too rigidly! In this example the harmonic series springs forth from a G in the deep bass range from where harmonics #2 and #3 actually cannot be accentuated (the human voice can produce overtones within the range 330-2.640 Hz ~ E3-E6, approximately), but naturally any note in any pitch will carry the harmonic series with it. What is invariable is the sequence of intervals between the elements. The body is a very complex resonance system, so you may also find resonances for deep tones high up in the body and vice versa. But generally: Deep notes : 'down' and high notes : 'up', as indicated intuitively in the language. The vowel sounds can be generated synthetically by a compound of two or three sine wave frequencies from tone generators but by overtone singing you may accentuate elements from the harmonic series by shaping the vocal tract to positions similar to vowel pronunciation. Please notice again that the correlation is not absolute: You may be able to accentuate the harmonics with different vowel qualities depending on voice type, pitch etc.

Generally the lower elements of the harmonic series are accentuated at the same regions of the vocal tract as the rounded back vowels, whereas the higher elements correspond with the front vowels. To accentuate distinct overtones demands mastery of the tongue as a precision instrument. The integers indicate multiples of the fundamental frequency of the vocal folds. Please bear in mind that overtone singing technique is not limited to the vocal tract. A complex interaction between abdomen, chest and throat musculature, breathing and body posture is essential.

It is a big advantage to become aware of the three areas of the vocal tract where you may fundamentally widen or narrow the passage of the sound waves emitted by the vocal folds: - Between the lips: m - Between the tip of the tongue and hard palate: n - Between the back of the tongue and soft palate: ŋ By consciously opening from these humming sounds – the primary nasal consonants – into open vowel sounds, you may accentuate the elements from the harmonic series very distinctly. The overtones – like the flageolet tones of a string instrument – are created at the shorter part of the divided sound wave (string). In case of the human voice that is between the points where the passage has been narrowed by the tongue and the lips respectively.

A vibrating string - as well as the vocal folds - gives rise to a range of simultaneous frequency conditions. Imagining using a finger against a fretboard to divide the string in smaller fractions, the 1/1, 1/2, 1/3, 1/4, 1/5 and 1/6 (bold) markers indicate the points of natural harmonics of the string - the points where you may find the flageolet tones. Furthermore you may find other fundamental musical functions. Here is only shown a part of the full picture as we have set the limit to 6 out of a potentially endless number of elements.

The bowed lines of these illustrations do not indicate a vibrating string but interval relations (italic) between elements of the harmonic series (upright figures). The length proportions refer to the size of the interval. Here the harmonics #7 and #11 have been omitted because the interval names refer to tuning systems (Pythagorean, just intonation, equal temperament etc.) where these harmonics do not have a place but actually sound out of tune in the context.

The just intervals between elements 1-12: 1:2 octave 2:3 perfect fifth 3:4 perfect fourth 4:5 just major third 5:6 just minor third 8:9 major whole-tone 9:10 minor whole-tone 3:5 just major sixth 5:8 just minor sixth 5:9 just minor seventh

The vowel mandala is a map for singers giving an organic structure to the various sounds which can be used - in the rhythmic language (primarily the plosives/ stop consonants); - for humming (the nasals) - and for producing sound colour (the vowels). The lastly mentioned are here represented by initials of Danish girls names. The three fundamental organic sequences can be practised in either direction with long sustained tones. Going from the open [É‘] in the center they all converge towards the nasals.

The vowel mandala was not introduced in an attempt to replace the standard IPA phonetic chart, which is here for reference.

Another diagram elucidating the close correlations between music and speech. It shows the formation of the Swedish vowel sounds as a result of two dominant frequencies. (from Johan Sundberg: RÜstlära) The formants are not only resonance frequencies but are also linked to the degree of opening or narrowing particular areas of the vocal tract - incarnating music!

English is missing the term klang which in German and Scandinavian languages indicates the quality aspect of any sound - which may be conceived as the sum of its overtones. 'Timbre' doesn't quite cover the meaning. Overtone singing forms a unity of three very fundamental fields of expression and knowledge essential to consciousness as such: - Music, here represented by solfège names of primary intervals - Language, here represented by the vowels which may accentuate the corresponding harmonics - Numbers, here represented by the relevant ratios. ... A universal and deeply rooted language!

Tone is braiding time (frequency) and space (wavelength). According to the formula for the correlation between the two aspects: f x 位 = c, they are inversely proportional. Here f is the frequency of the tone, 位 is its wawelength and c is the constant of speed of sound wawes in air at about 20oC ~ 343 m/s So for example the wavelength of the note c at 256Hz: 位 = 343 m/s : 256 s-1 = 1,34 m

Albrecht D端rer (1471-1528) who was active in Nuremberg made this proportion study of the human anatomy. Do notice the measure at the far left - the proportions of the harmonic series: 1/1- 1/2- 1/3- 1/4- 1/5-1/6- 1/7- 1/8- 1/9- 1/10. D端rer's masterly hand is also behind the sketch pasted into the note system at the fourth slide of this presentation.

Links to further information og listening experiences:

Keys to the Harmonic Universe  

Plates and diagrams elucidating correspondences of music, language and number from the persepective of harmonics and overtone singing.

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