This piece was originally used as the musical part of a graphical-musical artwork of the same name, displayed at the Olohuone 306.4 km2 city art festival in Turku, Finland, in June 2019. The title of the piece (which is Latin for “hidden proportions”) associates to the main idea of illustrating an analogy between proportions appearing in musical intervals and geometry: all musical intervals are, fundamentally, ratios of frequencies of pitches, just as dimensions of geometric shapes can have certain ratios.

In this piece, ratios of proportions are divided into five separate and spatially distributed groups. The degree of complexity of ratios varies between groups. Here, “degree of complexity” refers to ratios of smaller vs bigger numbers. The simplest possible ratios are integer ratios of small numbers such as 1:1, 2:1, 3:2 etc. (which in musical harmony represent highly consonant intervals such as unisons, octaves and fifths). Ratios involving progressively bigger integers tend to represent less consonant intervals, e.g. 16:15 and 45:32 (minor second and tritone, respectively).

Since pure integer ratios are used in this piece, this also means avoiding the compromise of equal temperament, which is familiar from most traditional keyboard instruments such as the piano. This piece uses instead a so-called asymmetric 5-limit just intonation (with one of its two alternative tritone intervals removed). Also higher harmonic overtones (up to above the 30th) are occasionally used on their own.
In the original 2019 artwork on location, the above-mentioned groups of ratios were located at five separate positions. Each position had one graphic print and stereo loudspeakers, which allowed listeners to walk around and listen to individual positions (groups of ratios) separately, or to the combined harmonies that they create at a distance. This spatial distribution is reproduced also in the stereo image of this album track: going from left to right in the stereo image, ratios are increasing in complexity, i.e. musical intervals used are becoming less and less consonant.

Each of the five positions reproduces only a given subset of ratios (i.e. ratios of a given complexity), and it should be noted that this is determined entirely by the ratios themselves and not by any voice leading rules etc. Part I, Rationes tonorum, simply distributes given notes in a 12-tone scale among the positions based on their corresponding ratios to the tonic. Part II, Rationes intervallorum, does the same with musical intervals between pairs of tones, and part III, Rationes melodiarum, goes further on to short melodies built on these intervals. A sixth position that is independent of the others appears as a kind of background (also sonically, as it is heard further behind the main groups in the stereo image). This sixth “background” position is quite free and does not restrict itself to any given set of ratios (intervals) as the others do.

Some of the original graphics and photographs can be viewed together with the music in a separate video that is also part of this album. All five original graphic prints appear side by side close to the end of the video.

The graphics furthermore illustrates differences between tuning systems in music: just intonation ratios and equal temperament ratios are shown together so that the former use thinner lines than the latter. Since, moreover, gradual appearance of harmonic ratios of increasing complexity has been a kind of megatrend in music during several centuries and millennia, this is also reflected in that the backgrounds used for the graphics stylistically represent different times in history (from antique to modern times).

Instrumentation used in this piece: Haken Continuum synthesizers; electric razor; Native Instruments Reaktor; two of the eight bells of Turku Cathedral, Finland.