Baroque and minimalist music are two of my favorite genres. In particular, I like the repetitive note sequences with variations. J.-S. Bach's Goldberg Variations and Phillip Glass's Naqoyqatsi are good examples of this.
What similarities do these genres share in a way that might impact how they are performed? Do they have any common foundations in particular aspects of music theory (besides the trivial)? Are they historically related?
Couple of ideas...
Harmony:
Baroque music was the period when harmonic progression (instead of just counter point) started to become a fundamental part of composition (See Functional Tonality). Dissonance is resolved to consonance throughout a phase.
Minimalism tends to focus more on a stable harmony, but in both case the Harmony tends to be a very exposed feature of the music.
Timbre:
Although Baroque was regarded as heavily ornamented music at the time, I would say both Baroque and Minamlist music tend to have a more uniform and stable sound.
How might this effect how they are played?
If you feel that these characteristics I gave as examples are important attributes, you can try to emphasize these or other attribute when you play.
With the Harmony, having accurate pitch and tone might help you expose the harmonies better. With timbre, you might try to keep that timbre stable until there is an intentional change of contrast in the music.
Identifying the contrast between these two styles also might help you understand the genres better. With Baroque you want to strongly identify the functional tonality so you can feel the flow of the harmony, and that generally will naturally translate to your playing. With the stable harmonic nature of Minimalism, you might want to emphasize that by having stable steady sections that coincide with those harmonies. In orchestral auditions, awareness of the contrast between styles and periods of music can be heard and is a major factor.
Keep in mind, these are very sweeping generalizations, music is full of exceptions.
You are absolutely right in saying that both have functional harmony/resolve dissonances but it is in fact more useful in drawing comparisons to compare both of these styles with everything in between them. For example, the shift over the Classical and Romantic periods tended to shift, harmonically, towards countering listeners' expectations over a very brief scales of time (there are countless examples, take any symphony by Brahms and listen to the rapidly evolving harmony) whereas Baroque and Minimalist compositional practices adhered to quasi-Schenkerian 'structural dissonances' whereby the music very broadly moves its tonal center containing smaller harmonic intricacies. Take, for example, Steve Reich's Music for 18 Musicians, which moves broadly from stacked 5ths chords build over a D, shifting down to C sharp with much 'darker' harmonies above, then back to D, creating a huge paradigmic harmonic arc over the whole work or Terry Riley's 'In C'. Similarly Bach's 5th Brandenburg concerto which, despite containing normal harmonically functional dissonances, moves from D major through relative minors but then a very extended highly chromatic harpsichord cadenza, then back to the major. This broad scale of harmonic change would not be found in Classical/Romantic/trad 20th century music, which might adhere to other things such as sonata form or cleverly manipulated passages of modulation of differing harmonic rhythm.
Secondly, there is much similarity to be found in the textures that are used - Baroque devices such as a ground bass/passacaglia (see Pachelbel's canon/the passacaglias of the French school such as those of Charpentier) has a repetitivity about it that surely has influenced the devices of repetition, modification and imitation of the minimalists. The Baroque saw some incredibly complex polyphony such as any traditional form of the time (Fugue etc, or even larger forms such as the Masses) as well as constant ornamentation. If you listen to Minimalist music, you will notice a preoccupation with texture such as the interweaving lines of yet more Steve Reich (Drumming, Nagoya Marimbas etc), or the elaborate orchestral effects of John Adams' Two Fanfares for Orchestra or Philip Glass's Violin Concerto.
Having performed both styles of music extensively, I can say that there are certainly similarities in the way you play them, especially compared with Romantic music. The trend of authentic Baroque performance fits with that of minimalism - in neither would you use excessive Rubato/vibrato or other aspects of 'over-expression'. In both there seems to be a trend away from over-interpretation of the composers' writing and a general feeling of letting the music flow completely from start to finish with no interruptions - indeed this is one aspect that strongly characterises the two styles in a similar way. In terms of beat, both Baroque and Minimalism seem to be characterised by a constant unchanging pulse (look at the relentless semiquavers of Vivaldi violin concertos or the constant pulse of Adams' 'Short Ride in a Fast Machine'). Also both seem to draw inspiration from dance patterns and forms, look at how Bach's cello suites reflect now-archaic dance forms such as the minuet, sarabande or gigue, and similarly how minimalist pieces such as 'The Chairman Dances' or 'Music for Pieces of Wood' and how they so clearly contain the aspects of pulse and rhythm found in more recent forms of dance music.
