We have presented a theory of tuning that is precise, concise, and provides insight into the structure of tunings of Western music. Included in the theory are various tuning models which may be applied to a wide variety of musics and instruments. The model best suited to Western tunings was the transposable register-zero tuning. This theory also introduced the fifth-register vector (FRV), a powerful mathematical formalization of diatonic pitch and interval. The goals of just P5 and M3 were explained and used as a basis for judging the efficacy of tunings. Transposable register-zero tunings operating on FRV were studied, illuminating some of the problems and compromises associated with diatonic tunings. The third-fifth-register (TFRV) and triadic tunings were presented along with a discussion of P11 slips and other factors limiting their feasibility.
Although this theory is similar to the work of others, including Helmholtz, Regener, Lindley and Turner-Smith, and Blackwood, it offers certain advantages over this previous work. One advantage it offers compared to existing theories is its thorough treatment of register. Other theorists have correctly identified pitch class as the fundamental element on which a tuning operates, but they have failed to present a mechanism by which a tuning of pitch classes can be extended to a real tuning of all pitches. This may seem like a simple process, and indeed this is probably what these theorists assumed, leading them to ignore it, but in fact it involves some subtlety. Regener is the only other theorist who has treated register in a satisfactory manner, although his work is restricted to regular tunings. One of the main theoretical contributions of this thesis is the synthesis of Regener's ``register-smart'' representation of pitch with Lindley and Turner-Smith's treatment of irregular tunings.
Another advantage of the tuning theory presented in this work is the notion of triadic tuning. Other theorists have presented just intonation as a series of direct modifications to Pythagorean diatonic tuning. Thus it seems more like ``raw'' (ad hoc) intonation than a form of tuning. In contrast, this work arrives at just intonation as a specific tuning of triadic pitch classes. The assignment of triadic pitch class (subscripting) becomes an endeavor in the theoretical domain of harmonic analysis, not a tweaking of raw frequency ratios. This is methodologically important for a number of reasons. One, it allows for triadic tunings other than just. Two, it maintains the theoretical framework of tuning as a map from pitches to frequency ratios. Three, it allows the problems of just intonation to be framed in terms of enharmony, a concept already present in diatonic theory.
Helm, the dynamic intonation software developed as part of this research provides a flexible platform on which to investigate tuning and intonation. It is the only such platform that exists. It is based upon tuning-annotated score files written in a simple score description language. These score files can be converted into MIDI files. They can also be used by a score follower program to achieve dynamic intonation for keyboard performance, allowing one key to produce many frequencies at different points in the score.
Helm was used to conduct various experiments in just triadic tuning. The tuning theory presented in this work suggests that just triadic tuning is of limited feasibility, but it was investigated nonetheless, for a variety of reasons. One, it is important to actually hear the phenomena that the theory predicts will create problems. Two, that it is the most extreme form of tuning since it actually creates beatless, not just slowly beating chords, and therefore provides an opportunity to assess the desirability of beatlessness in musical context. Three, much debate, but little experimentation, has surrounded the feasibility of just intonation for the realization of traditional repertoire.
The musical materials used for this experimentation were as follows:
Not surprisingly, it was difficult to assign triadic functions (subscripts) to notes. In order to avoid slippage, occasional non-just chords were created. The job was made somewhat easier for the first two excerpts because the music was taken from Blackwood [3], who had already given his interpretation of the subscripting.
The results were in no way scientifically studied; they are merely the observations of the author and the impressions he was given by the colleagues for whom he played the examples. 12TET versions were created for comparison. Many of these experiments can be listened to on the web page http://theremin.media.mit.edu/bdenckla/thesis/main.html.
The beatlessness of the chords was noticeable, but only with certain timbres, and only for fairly long chords. It came as no surprise that beatlessness wasn't too important for chords that even in 12TET would only beat a few times during their duration. The importance of timbre was quite surprising, though.
For many timbres, beatless and 12TET chords sounded almost identical. One reason for this is that many timbres have beating (detuning) already built in. This is usually accomplished by playing two or more slightly detuned versions of the same note. This beating within tones makes it hard to hear beating (or the lack of it) between tones. Also, many timbres have vibrato built in, which makes beating hard to perceive. Still, many timbres just didn't seem to make beating show up, but there was no easy explanation for this such as the presence of detuning or vibrato. The timbre that worked best was a simple organ sound with voice doubling at the octave. This voice doubling probably helped because it emphasized higher harmonics, albeit only even ones.
``Mistakes'' in subscripting, leading to chords very far from just and/or different versions of the same note within a short time span, were very noticeable. In the examples in question, these situations were avoidable by redoing the subscripting, but there could easily be situations for which no good solution exists.
The theory of this work and the results of tests using its software
indicate that just triadic tuning has many difficulties and few
benefits. This should not in any way be interpreted as a vindication
of 12TET. Many other diatonic and triadic tunings remain to be
investigated by listening to a variety of pieces rendered in them. In
particular, the meantone tunings seem quite promising because of their
small 
and 
, as well as the historical
prominence of their truncated versions. The software developed in this
work currently only supports just triadic tuning and 12TET, but adding
new tunings is a trivial programming task since the underlying
structure of the software is completely general.
In conclusion, this work has resulted in a new theory of intonation and powerful new software for intonation research.