Tuning Bach
is the first of several planned case studies looking at musical applications of rational intonation. It includes my first public doctoral presentation, which took the form of a lecture-performance. Violinists Sara Cubarsi and Xenia Gogu joined me on 19. October 2024 in Organo, Musikkitalo, at the Sibelius Academy, University of the Arts Helsinki, Finland. You can view the presentation on youtube, read the scores, and look at the slides as a pdf document. If you have a little more time and would like to help with my study, please fill out this webropol survey.
The Tree
is a project I am currently working on. It is an ongoing cycle of pieces for open instrumentation inspired by the concept of harmonic radius, which is a way of comparing the relative harmonicities of chords sounding in rational or just intonation (JI).
All notes in “The Tree” are part of a collection of 81 tones tuned in JI as harmonic partials (overtones) of written C. The partials have been selected to minimise pitch-class harmonic radius of intervals and chords. The largest odd partial pitch class included is 255°; only those partials that combine to form an harmonic chord (i.e., a collection of 3 or more pitches following an harmonic series) are included. The set resulting from these two conditions has odd limit 255 and prime limit 47 (no partials with prime factors larger than 47 are included).
The music is written, for the most part, at sounding pitch. Any reference frequency suitable to the instruments playing may be chosen freely. The microtonal JI tuning is notated using accidentals of the Helmholtz-Ellis JI Pitch Notation (HEJI), devised in collaboration by Marc Sabat and Wolfgang von Schweinitz. In addition, cents deviations from the nearest 12edo notes are given so that electronic tuners may be used to determine fingerings and verify accuracy. In live performance, many vertical intervals or chords may be tuneable by ear by listening for smooth beatless consonance or a clear, steady periodic purring. While precise and stable tuning is central, phrasing and flow are equally important.
The textures include various contrapuntal lines and “continuo” materials (chords with some specified voice leading) that may be realised by tuning instruments in an appropriate scordatura to access the pitches needed. Scordatura tunings for Lumatone and chromatic pedal harp are provided in the score. Harp is notated in conventional notation (flat, natural, sharp), indicating the playing position of notes. Every octave of the harp is tuned differently, so transposing these pitches into different registers is not possible. Lumatone is notated at sounding pitch, using a layout that maps the partials and their constituent prime factors onto associated colour combinations.
In other instruments, octave transpositions may be made as needed, if the results sound harmonically clear and musically interesting. Players may choose to stay on a given line or to move between lines, shaping the melodies to their instruments as sounds best and sharing the parts appropriately amongst available players. The flow of time may be treated with great flexibility, fermatas applied where helpful, as long as vertical coordination and the sequence of tones is observed. Ornamental tones, glides, and very occasional expressive vibrati may be added where beautiful and as long as they aid and support accurate tuning. Dynamics are free.
Additionally, this piece may be thought of as continually evolving, welcoming arrangement, extension and ornamentation by the musicians in any manner, as long as the underlying collection of pitches, and a focus on hearing pitch relations and interactions, perceiving harmonicity, and playing in rational intonation (JI) are maintained. A performance of “The Tree” can include any of the scored pieces, played in any instrumentation and in any order, for any duration.