Dyadic chord: Difference between revisions

An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit, hence a chord that will not "ring". This is broader than the essentially tempered chord because it includes the possibility that the chord is not tempered at all, and contains a [[wolf interval]]. For example, the [[Augmented_triad|augmented tetrad]] in 5-limit JI is an innate comma chord, because it is impossible to tune all three major thirds to 5/4 while retaining a pure octave: the innate comma here is 128/125 = 41¢. In practice, it might be sung or played justly but with a large odd limit (and hence a wolf interval), as 1/1 – 5/4 – 8/5 – 2/1 or 1/1 – 5/4 – 25/16 – 2/1 (or even 1/1 – 5/4 – 25/16 – 125/64). Or it might be tempered, e.g. in 12edo as 0¢ – 400¢ – 800¢ – 1200¢. In 7-limit JI, one of the major thirds can be tuned to 9/7, reducing the innate comma to 225/224 (only 8¢). This comma can be distributed among the three thirds, tempering each by only a few cents, which is usually close enough to be acceptable. In 11-limit JI, the augmented chord is not an innate comma chord, because it can be tuned justly as 7:9:11:14, a low enough odd limit to "ring". (However, it is debatable whether this chord qualifies as an augmented tetrad, because the middle [[11/9]] interval is a neutral third rather than a major third.)