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Djuro Zivkovic/Difference Tones


Djuro Zivkovic often works with chord sequences created from combinations of different tones. We implemented his approach in various ways and compared the results with Zivkovic’s handwritten solutions.


  • Base Tone
  • Interval Vector
  • Pitch Class
  • Folk Music
  • Chord Sequence

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  1. 1.

    Which comes from “nous” (\(\nu \!\acute{o}\! o\!\varsigma \))—intellect.

  2. 2.

    For example, as strings vibrate in certain frequencies where the result of it has a specific effect on the listener, rhythms in a certain number of beats or accents gives a result which can be perceived as correct or incorrect rhythm according to lyrics, dance or emotional conditions of the listener.

  3. 3.

    Here I have in mind not only the origin of a human, but also the whole set consisting of the human pre-existence and as well both its ontology and eschatology, where music serves as an ultimate medium to re-establish the true, divine source of human nature and the whole environment.

  4. 4.

    narikac̆e (Serbian).

  5. 5.

    By the spiritual knowledge (\(\gamma \nu \tilde{\omega }\sigma \iota \varsigma \)) is referred to the knowledge of the intellect as distinct from that of reason. The intellect acts here as the highest faculty of man, through which he knows the inner essences or principles of created things by means of direct apprehension or spiritual perception. Unlike the reason, the intellect does not function by formulating abstract concepts and then arguing on this basis to a conclusion reached through reasoning, but it understands the divine truth by means of immediate experience, intuition or “simple cognition”—term used by St. Isaac the Syrian [4, Glossary].

  6. 6.

    I could compare the “Whatever-Music” with the freedom of choice. Whatever we choose, we are still “obligated” to respect architecture of the universe, and still we have an infinite freedom of choice. We can choose “whatever” but still in the frame of the natural law. The “Whatever-Music” is also a free choice inside of the frame of the spiritual and esthetical law.

  7. 7.

    This is the case, for instance, in the total serialism, or any kind of musical structures, which are manipulated only by numbers which represent the musical matter like pitch, rhythm, form, instrumentation, etc.

  8. 8.

    It is not possible to rationalise it even through a metaphysical understanding. Here, the spiritual journey has a very strong apophatic character.

  9. 9.

    Guiseppe Tartini (1692–1770), Italian violinist, composer and music theorist.

  10. 10.

    Used by Carl Friedrich Gauß (1777–1855), a German mathematician and physical scientist, at the age of 9 on the occasion of having to sum the numbers from 1 to 100 at school. However the formula has been known since ancient times.

  11. 11.

    By enumeration we found that there is one single case in which even this doesn’t break the tie, it comes from splitting the integer 12 into the tuple (1 3 1 3 3 1). Then for a chord with interval vector (1 3 1 3 3), e.g. (C Db E F Ab B) there exist two equidistant maximally compressed chords with the interval vectors (1 3 3 1 1) and (3 1 1 3 1), here (E F Ab B C Db) and (Ab B C Db E F). In that very special case a choice would have to decide with which chord to proceed.

  12. 12.

    Zivkovic also calls it mirroring scale since the interval vectors 1211 and 1121 frame a base tone (not to be confused with a tonic). E.g. Ab is the base tone of D# E F# G Ab A Bb C C# (D#).

  13. 13.

    It should be mentioned that for a long historical period triads without thirds were not considered valid in 4-part voice leading. However, as we are not developing a measure for music in a certain tradition itself, but rather for the association to a certain tradition, we include the third-less seventh chord for its strong association to a fundamental.


  1. Cariani PA, Delgutte B (1996) Neural correlates of the pitch of complex tones. I. Pitch and pitch salience. J Neurophysiol 76(3):1698–1716

    Google Scholar 

  2. Kemp DT (1978) Stimulated acoustic emissions from within the human auditory system. J Acoust Soc Am 64(5):1386–1391

    CrossRef  MathSciNet  Google Scholar 

  3. Oster G (1973) Auditory beats in the brain. Sci Am 229(4):94–102

    CrossRef  Google Scholar 

  4. Palmer G, Sherrard P, Ware K (1995) The Philokalia, vol 4. Faber and Faber, London

    Google Scholar 

  5. Plato (1997) Philebus. In: Cooper JM, Hutchinson DS et al (eds) Plato: complete works (trans: Frede D). Hackett Publishing, Indianapolis, pp 398–456

    Google Scholar 

  6. Tartini G (1754) Trattato di musica secondo la vera scienza dell’armonia. G. Manfré, Padua

    Google Scholar 

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Correspondence to Gerhard Nierhaus .

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Zivkovic, D., Mayer, D., Nierhaus, G. (2015). Djuro Zivkovic/Difference Tones. In: Nierhaus, G. (eds) Patterns of Intuition. Springer, Dordrecht.

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