Meantone Temperament - Extended Meantones

Extended Meantones

All meantone tunings fall into the valid tuning range of the syntonic temperament, so all meantone tunings are syntonic tunings. All syntonic tunings, including the meantones, have an infinite number of notes in each octave, that is, seven natural notes, seven sharp notes (F♯ to B♯), seven flat notes (B♭ to F♭), double sharp notes, double flat notes, triple sharps and flats, and so on. In reality, double sharps/flats are uncommon, but still needed; triple sharps/flats are never seen. In any syntonic tuning that happens to divide the octave into a small number of equally wide smallest intervals (such as 12, 19, or 31), this infinity of notes still exists, although some notes will be enharmonic. For example, in 19-ET, E♯ and F♭ are the same pitch.

Many musical instruments are capable of very fine distinctions of pitch, such as the human voice, the trombone, unfretted strings such as the violin, and lutes with tied frets. These instruments are well-suited to the use of meantone tunings.

On the other hand, the piano keyboard has only 12 physical note-controlling devices per octave, making it poorly suited to any tunings other than 12-ET. Almost all of the historic problems with the meantone temperament are caused by the attempt to map meantone's infinite number of notes per octave to a finite number of piano keys. This is, for example, the source of the "wolf fifth" discussed above. When choosing which notes to map to the piano's black keys, it is convenient to choose those notes that are common to a small number of closely related keys, but this will only work up to the edge of the octave; when wrapping around to the next octave, one must use a "wolf fifth" that is not as wide as the others, as discussed above.

The existence of the "wolf fifth" is one of the reasons why, before the introduction of well temperament, instrumental music generally stayed in a number of "safe" tonalities that did not involve the "wolf fifth" (which was generally put between G♯/A♭ and D♯/E♭).

Throughout the Renaissance and Enlightenment, theorists as varied as Nicola Vicentino, Francisco de Salinas, Fabio Colonna, Marin Mersenne, Constantijn Huygens, and Isaac Newton advocated the use of meantone tunings that were extended beyond the keyboard's twelve notes, and hence have come to be called "extended" meantone tunings. These efforts required a concomitant extension of keyboard instruments to offer means of contolling more than 12 notes per octave, including Vincento's Archicembalo (shown in Figure 3), Mersenne's 19-ET harpsichord, Colonna's 31-ET sambuca, and Huygens' 31-ET harpsichord. Other instruments extended the keyboard by only a few notes. Some period harpsichords and organs have split D♯/E♭ keys, such that both E major/C♯ minor (4 sharps) and E♭ major/C minor (3 flats) can be played without wolf fifths. Many of those instruments also have split G♯/A♭ keys, and a few have all the 5 accidental keys split.

All of these alternative instruments were "complicated" and "cumbersome" (Isacoff, 2003), due to (a) not being isomorphic, and (b) not having the ability to transpose electronically, which can significantly reduce the number of note-controlling buttons needed on an isomorphic keyboard (Plamondon, 2009). Both of these criticisms could be addressed by electronic isomorphic keyboard instruments (such as the open source Thummer, shown in Figure 4), which could be simpler, less cumbersome, and more expressive than existing keyboard instruments.