Scala References

Tuning Theory

The following selective list of references to the tuning literature covers material relevant to Scala:

• Balzano, Gerald J. "The group theoretic description of 12-fold and microtonal pitch systems", Computer Music Journal vol. 4 no. 4, 1980, pp. 66-84.

• Bannerman, Aricca; James Emington and Anil Venkatesh. "Optimal Tuning of Two-Dimensional Keyboards", arXiv 1711.05260, WWW, Nov 2017, 14 pages.

• Barlow, Clarence. "Two Essays On Theory", Computer Music Journal vol. 11 no. 1, 1987, pp. 44-60.

• Barlow, Clarence. Bus journey to Parametron, Feedback Papers 21-23, Feedback-Studio-Verlag, Köln, 1980.

• Blackwood, Easley. "Modes and Chord Progressions in Equal Tunings", Perspectives of New Music vol. 29 no. 2, summer 1991, pp. 166-201.

• Breed, Graham. "Prime-Based Error and Complexity Measures", WWW, 2007, 29 pages.

• Breed, Graham. "Tripod Notation. A Magic Cauldron of Musical Possibilities", WWW, 2009, 22 pages.

• Bohlen, Heinz. "13 Tonstufen in der Duodezime", Acustica vol. 39 no. 2, S. Hirzel Verlag, Stuttgart, 1978, pp. 76-86.

• Brun, Viggo. "Euclidean algorithms and music theory", L'enseignement mathématique revue internationale tome X, fasc. 1-2, 1964, pp. 125-137.

• Burt, Warren. "Adventures in Scale Generation Along a Wilsonian Path, part 1", 1/1 vol. 9 no. 3, summer 1996.

• Burt, Warren. "Adventures in Scale Generation Along a Wilsonian Path, part 2", 1/1 vol. 10 no. 1, winter 1998, pp. 4-10.

• Busch, Hermann R. Leonhard Eulers Beitrag zur Musiktheorie, Kölner Beiträge zur Musikforschung 58, Gustav Bosse Verlag, Regensburg, 1970.

• Canright, David. "Superparticular Pentatonics", 1/1 vol. 9 no. 1, 1995, p. 10.

• Carey, Norman and David Clampitt. "Aspects of Well-Formed Scales", Music Theory Spectrum vol. 11 no. 2, fall 1989, pp. 187-206.

• Carey, Norman and David Clampitt. "Self Similar Pitch Structures, Their Duals, and Rhythmic Analogs", Perspectives of New Music vol. 34 no. 2, summer 1996, pp. 62-87.

• Chalmers, John H. Jr. Divisions of the Tetrachord. Frog Peak Music, Hanover NH, 1993. 234 pages.

• Clough, John, Nora Engebretsen and Jonathan Kochavi. "Scales, Sets, and Interval Cycles: A Taxonomy", Music Theory Spectrum vol. 21 no. 1, spring 1999, pp. 74-104.

• Day, C.R. The music and musical instruments of Southern India and the Deccan, 1891. Reprint, Low Price Publications, Delhi, 1990, 181 pages.

• Dudon, Jacques. "Differential Coherence: Experimenting with New Areas of Consonance", 1/1 vol. 11 no. 2, winter 2003, pp. 1, 6-13.

• Erlich, Paul. "A Middle Path. Between Just Intonation and the Equal Temperaments, part 1", Xenharmonikôn vol. 18, 2006, pp. 159-199.

• Fokker, A.D. Rekenkundige bespiegeling der muziek, Noorduijn, Gorinchem, 1945. 228 pages.

• Fokker, A.D. "Les mathématiques et la musique. Trois conférences", Archives du Musée Teyler vol. 10, Martinus Nijhoff, Den Haag, 1947.

• Fokker, A.D. "Unison vectors and periodicity blocks in the three-dimensional (3-5-7-) harmonic lattice of notes", Proceedings of the KNAW, Series B, vol. 72, Amsterdam, 1969, pp. 153-168.

• Gough, Martin. On the Bimodular Approximation and Equal Temperaments. WWW, 2014, 38 pages.

• Helmholtz, Hermann L. F. von. Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik, 6th edition, Friedrig Vieweg & Sohn, Braunschweig, 1913. 668 pages. Olms, Hildesheim, 1983.
  On the Sensations of Tone as a Psychological basis for the Theory of Music, 2nd English edition translated by Alexander John Ellis, based on the 4th German edition of 1877 with extensive notes, foreword and afterword: 1885. Reprint by Dover Publications, 1954.

• Honingh, Aline K. A geometrical approach to find the preferred intonation of chords. Technical Report Series TR/2008/DOC/02, City University London, 2008, 16 pages.

• Kaegi, Werner and Stan Tempelaars. "VOSIM - A New Sound Synthesis System", Journal of the Audio Engineering Society vol. 26 no. 6, June 1978, pp. 419-425.

• Milne, Andrew J.; David W. Bulger; Steffen A. Herff and William A. Sethares. "Perfect Balance: A Novel Principle for the Construction of Musical Scales and Meters", International Conference on Mathematics and Computation in Music, London, June 2015, pp. 97-108.

• Muzzulini, Daniel. "Isaac Newton's Microtonal Approach to Just Intonation", Empirical Musicology Review vol. 15 no. 3-4, 2020, pp. 223-248.

• Partch, Harry. Genesis of a Music: An Account of a Creative Work, its Roots and its Fulfillment, 2nd enl. edition, Da Capo Press, New York, 1974. 517 pages. Reprint, 1979.

• Pearce, Markus. "The Group-theoretic Description of Musical Pitch Systems", 2016, 31 pages.

• Rapoport, Paul. "About 31-tone equal temperament and six songs written in it", 1987.

• Rapoport, Paul. "Notes Toward Quasi-Triskaidekaphilia", Xenharmonikôn vol. 13, spring 1991, pp. 45-53.

• Rapoport, Paul. "The Structural Relationships of Fifths and Thirds in Equal Temperaments", Journal of Music Theory vol. 37 no. 2, pp. 351-389, fall 1993.

• Rapoport, Paul. "The Notation of Equal Temperaments", Xenharmonikôn vol. 16, 1995, pp. 61-84.

• Rasch, Rudolf (ed.) Adriaan Daniël Fokker - Selected Compositions, with introduction by Rudolf Rasch, Corpus Microtonale CM 1, Diapason Press, Utrecht, 1987. 217 pages.

• Roberts, Catherine A. "Stråhle's Guitar Construction", course handout, Mathematics and Computer Science, College of the Holy Cross, Worcester MA, 2011, 5 pages.

• Rothenberg, David. "A Model for Pattern Perception with Musical Applications. Part I: Pitch Structures as Order-Preserving Maps", Mathematical Systems Theory vol. 11, 1978, pp. 199-234.

• Sábat, Eduardo. Principios de la Gama Dinámica, Arca, Montevideo, 1994. 238 pages.

• Sauveur, Joseph. Collected writings on musical acoustics (Paris 1700-1713), R.A. Rasch (ed.), Diapason Press, Utrecht, 1984.

• Schüffler, Karlheinz. "Algebra und Analysis in der Arithmetik musikalischer Intervallsysteme", Vorträge auf der 52. Tagung für Didaktik der Mathematik - Jahrestagung der Gesellschaft für Didaktik der Mathematik vom 05 bis 09 Mar 2018, Paderborn, 2018, pp. 1639-1642.

• Secor, George and David Keenan. "The Miracle Temperament, Decimal Keyboard, and Sagittal Notation", Xenharmonikôn vol. 18, 2004.

• Sethares, William A. "Adaptive Tunings for Musical Scales", Journal of the Acoustical Society of America vol. 96 no. 1, July 1994, pp. 10-19.

• various authors. Xenharmonic Wiki (highly recommended), WWW.

• Wille, Rudolf. "Mathematik und Musiktheorie", in Musik und Zahl, Verlag für systematische Musikwissenschaft, Bonn-Bad Godesberg, 1976, pp. 233-264.

• Wilson, Ervin. "D'Alessandro, Like a Hurricane", Xenharmonikôn vol. 12, spring 1989, pp. 1-39.

See also the Tuning and Temperament bibliography.

Algorithms

Some algorithms used in Scala can be found in the following references:

• Khinchin, A. Ya. Continued fractions. 1935. Translated from Russian. University of Chicago Press, 1964. Dover Publications, New York, 1997, 95 pages.

• Lehr, André. "Change ringing of wisselluiden", Asten, WWW, 6 pages.

• Reingold, Edward M., Jurg Nievergelt and Narsingh Deo. Combinatorial Algorithms. Theory and Practice, Prentice Hall, Englewood Cliffs NJ, 1977, 433 pages.

Modes

Many of the modes in the list of modes come from the following sources:

• Ader, Lidia Olegovna. "Microtonal Storm and Stress: Georgy Rimsky-Korsakov and Quarter-Tone Music in 1920s Soviet Russia", Tempo vol. 63 no. 250, Cambridge University Press, Cambridge, Oct 2009, pp. 27-44.

• Chalmers, John H. Jr. Divisions of the Tetrachord. Frog Peak Music, Hanover NH, 1993, 234 pages.

• Farhat, Hormoz. The Dastgah Concept in Persian Music. PhD thesis, University of California, Los Angeles, 1965. Cambridge University Press, Cambridge, 1990, 2004, 213 pages.

• Fritsch, Ingrid. Analysen von Solo-Honkyoku der Tozan-Schule, Studien zur traditionellen Musik Japans, Band 4, Bärenreiter Verlag, Kassel, 1979, 241 pages.

• Helmholtz, Hermann L.F. von. On the Sensations of Tone as a Psychological basis for the Theory of Music, 2nd English edition translated by Alexander John Ellis, 1885.

• Howlett, George. "Megalist of Ragas (365+)", WWW, 2022.

• Jedrzejewski, Franck. "Algebraic Combinatorics on Modes", WWW, Oct 2017, pp. 1-12.

• Karadeniz, M. Ekrem. Türk Mûsikîsinin Nazariye ve Esaslari. Türkiye IS Bankasi Kültür Yayinlari, Publ. no. 237/238, Ankara, 1965, 1983.

• Oesch, Hans. Außereuropäische Musik. 2 vols. Neues Handbuch der Musikwissenschaft, Band 8 & 9, Carl Dahlhaus (ed.), Laaber Verlag, Laaber, 1987, 367 & 513 pages.

• Pesch, Ludwig. Raga Dhana. An alpha-numerical directory of ragas. 2nd edition, Natana Kairali, Kerala, 1993, 226 pages.

• Pesch, Ludwig. The Illustrated Companion to South Indian Classical Music. 2nd edition, Oxford University Press, New Delhi, 2009, 517 pages.

• Raagtime - From Dawn till Dusk.

• Rasch, Rudolf (ed.) Corpus Microtonale, vol. 1: Adriaan Daniël Fokker - Selected Compositions, with introduction by Rudolf Rasch, Diapason Press, Utrecht, 1987. 217 pages.

• Swets, Wouter. Traditionele Turkse kunstmuziek. KRO, Hilversum, 1983, 179 pages.

• Varadarangan, Krishna Iyengar. Srutis and Srutibheda. Lalitha Prakashana, Bangalore, 2002, 161 pages.

• various authors. Xenharmonic Wiki: EDO

• Wiersma-te Nijenhuis, Emmie. Dattilam. A compendium of ancient Indian music, Phd thesis, Musicology, Universiteit Utrecht. E.J. Brill, Leiden, 1970, 477 pages.

• Wikipedia: Modes and Musical scales

• Zanten, Wim van. Tembang Sunda. An ethnomusicological study of the Cianjuran music in West Java. PhD thesis, University of Leiden, 1987, 320 pages.

• Zonis, Ella. Classical Persian Music: An Introduction. Harvard University Press, Cambridge MA, 1973.

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