What's the difference between “modal music” and “tonal music”?

Answer

The difference between modal and tonal are in the harmonic languages surrounding the tonal center. Tonality implies the system of common-practice harmony well-established by the eighteenth century that uses major and minor keys. The tonal center of a tonal work is the first note of the major or minor scale in use as the pitch collection. The harmonic implications of tonality are more than just the use of major and minor scales, as functional harmony is also a feature of tonal music. The progression from the dominant sonority (a major triad with or without a minor seventh from the triad root based on the fifth note of the major or minor scale in use, or a similar-sounding substitute such as a fully-diminished seventh chord based on the leading tone) to the tonic triad to end a work is just one characteristic of functional harmony. This characteristic is so important that, if the dominant sonority is instead a minor chord (thereby lacking the leading tone), the work no longer sounds tonal. This means that even in a minor key, the seventh note of the scale is very often raised so that it becomes the leading tone.
Modal music uses diatonic scales that are not necessarily major or minor and does not use functional harmony as we understand it within tonality. The term modal is most often associated with the eight church modes. The tonal center of these modes is called its "final." All the church modes use a pattern of half and whole steps that could be played on the white keys of a piano. You may notice that there are only four different patterns among the church modes; the difference between e.g. "dorian" and "hypodorian" is whether the final occurs at or near the bottom of the melodic range or whether the final occurs in the middle of the melodic range. The term "modal" has expanded in more modern music to encompass any non-tonal music that uses a diatonic pitch collection and has a tonal center.
There are many types of music other than modal and tonal. Some examples include:
  1. chromatic music, which uses all twelve of the standard Western pitch classes instead of the diatonic pitch collection, and which may or may not have a tonal center;
  2. serial music, sometimes called "dodecaphonic," which is chromatic music that intentionally avoids a tonal center, often by avoiding repetition of a pitch class until all twelve pitch classes have been used;
  3. bitonal or polytonal music, which uses multiple diatonic pitch collections and multiple tonal centers simultaneously;
  4. microtonal music, which uses pitches with frequencies between those of the standard twelve Western pitch classes;
  5. whole-tone music, which uses a six-note scale comprised entirely of whole steps; and
  6. non-Western music, which uses a pitch collection outside the twelve Western pitches (this is not a good classification, as there are many cultures with many different kinds of music that are very different from one another in pitch collection).
I did not even touch on music that does not use pitches at all; for example, an unpitched percussion work would clearly not be modal or tonal.
There are entire books on functional harmony, modes, etc., but I hope this has been a reasonable summary to answer your question.

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