Das Clavier oder der Flügel läßt sich, wie Musikverständigen bekannt ist, nach dem Quintencirkel stimmen, in folgender Ordnung der Töne: c g d1 d a e1 e [etc]

As those with an understanding of music know, keyboard instruments are tuned according to a circle of fifths in the following order of the tones: c g d1 d a e1 e
- Johann Gustav Karsten, Anfangsgründe der Naturlehre 1790

“Sharps are low and flats are high.” How many times have we heard HIP music students recite this maxim? And in a certain sense, it is true; a sharp should be a bit less than half a tone above the natural form of the generating note, and a flat a bit less than half a tone below its respective natural. Unfortunately, this idea is all too often misinterpreted as meaning that sharps and flats are respectively lower and higher than their equivalent positions in the larger scheme of 12-tone Equal Temperament. While this may be true in certain limited cases, this is largely an illusion created by the ahistorical use of the wrong reference pitch.

These two graphics illustrate the situation. Both show the deviation from Equal Temperament for 38 historical temperaments, from various flavors of strict meantone, through a number of “mollified meantones” to the circulating temperaments of the 18th century. The only difference between the two is the reference pitch, the first showing the situation when A is taken as the unvarying pitch and the second taking C as the reference point.

deviation from ET with constant A

deviation from ET with constant C

The first thing which becomes obvious is that when A is taken as a reference, the traditionally-constant notes of C or F are both sharp, at times by as much as 15 cents! This explains why tuners are so often asked to tune a bit lower, at 438 instead of 440, or 428 instead of 430, or 414, 412 or even 410 instead of 415. Every professional tuner has had this happen time and time again. The cause is that the wind musicians are just responding to the fact that with unequal temperaments, the natural notes come out too high when A is used as a reference, and they must fight to push them up so high. Therefore they almost always ask that A be set a little bit lower, precisely as it would be if everybody – musicians and tuner alike – were using historical reference pitch practice! As Homer Simpson would say, D’oh!

The other problem when A is taken as a reference is the false idea it gives about accidental inflection relative to the standard by which most modern musicians think: Equal Temperament. Using A, it would indeed appear as though the ubiquitous maxim where true; sharps are almost always lower than in Equal, and flats are usually higher, with a handful of exceptions. However, when we acknowledge the fact that the practice of taking A as the reference pitch was never followed during the times when these unequal temperaments were in use, we must admit that this is a complete misunderstanding of the historical reality. When a proper C reference is used, almost all notes – not just the sharps – are lower than their relative positions in Equal, and taking an F reference would push them even lower yet. Only in the case of strict meantone (and the most conservative meantone mollifications) does the ubiquitous modern maxim hold true.

Suppose, then, that you were a gigging oboist in a large German city during the early 18th century, for example, playing around in various church, theater and court orchestras. How would you have thought about intonation? To the degree that there was any consistency of pitch level at all, it would have resembled the second situation. In the church, you most likely would have been confronted with an organ tuned in pure 1/4 comma meantone. Since the organ was completely inflexible, you would have made your reeds and staples so as to function well at this pitch level (assuming, of course, that the church did not have it’s own set of instrument specifically made to work well with the organ, as was sometimes the case). In orchestral settings, however, the situation was completely turned around. You, as the primary wind instrument, would have been the arbiter of the center of a flexible tonal universe by providing the reference note – most often C – to the harpsichordist in each and every situation. He could have then continued to temper his instrument using anything from meantone to a mollified meantone to a rational circulating temperament, even Equal! Thus you and all the other instrumentalists, to the degree that you would have mimicked the intonation of the keyboard at all, would have had to primarily lower ALL notes compared to Equal.

Of course, in those times, musicians were not cursed with the affliction of having been indoctrinated through a lifetime of hearing Equal Temperament, and the idea that each note had some theoretical neutral position within a virtual matrix of intonational reference would have seemed completely foreign to them. Their internalized point of reference would have been the sounds of absolutely pure fifths, fourths, thirds and sixths, taken in reference to another note which itself also would have had a flexible identity. Each note could have a multitude of possible inflections, dictated by the harmonic function of the moment, depending on the particular interval it formed, its relationship to other notes (i.e. root, third, fifth, or seventh of the harmony, including inversions), its direction of relationship to the generating note (up or down), and whether or not it was to be realized in its pure form or using some degree of tempering. Thus the task of adapting the intonation of each note was one of fitting into the harmonic reality of each and every unique moment, using pure intonation whenever possible, and going along with the “imperfect tuning” of the keyboard to a greater or lessor degree whenever not. The further one ventured from the center of this vast universe of possibilities – C – the greater the possible variation up or down could be for any given note in any given situation. Yet the general drift would have been that indicated by the second of the two graphics shown here above. In no case would the mere application of some simplistic formula such as “sharps are low and flats are high” have been sufficient. In fact, they most likely would have thought completely the opposite, since their reference was the pure version of each interval; therefore, sharps would occasionally have to be performed too high and flats too low in order to coincide with the tempered version on the keyboard. Thus, coming from the viewpoint of pure thirds and sixths, their equivalent maxim would have been “sharps can sometimes be too high and flats can sometimes be too low.”

Therefore, while we may well argue that any comparison whatsoever to an imaginary neutral reference of an Equal Tempered chromatic scale represents a fundamental misconception of the historical reality, the basic point remains valid; if we are ever to hope to regain a truly historically-based sense of intonation, in other words, if we want to reacquire an 18th century sense of “musical understanding”, we must unlearn the misconceptions of the earlier generations of the HIPP movement. Learning to take your bearings from the proper point of reference is a critical first step on this far-reaching journey.