At the beginning of the chapter on Just Intonation, you say "Within the broad conglomeration of tuning systems (in Just Intonation) we will concentrate on elaborating what we call Functional Tuning, limited to the ratios 2/1 (the octave), 3/2 (the perfect fifth), and 5/4 (the major third), which will seem the most natural and satisfying to the ear, and which will apply to music of all styles and periods." Do you mean by this that the music of all styles and periods will be tuned in the same fashion?

Not at all. What we meant was that this system of Functional Tuning is capable of tuning music of all styles and periods, each in its own fashion. Thanks for bringing this to our attention. A correction will be made shortly.

Thanks for answering my previous email. In the Just Intonation section you also say: "Functional Tuning is music itself, and cannot be studied without constantly referring to the structure of Melody and especially Harmony (for they are not tuned the same way)." By this do you mean that Just Intonation is more than a tuning procedure and that it plays a role in the very structure of music?

Yes, by all means. This is where music theory should start. The simplest mathematical proportions (ratios) are what give the octave, the fifth and the major third their preferential status in the structure of music (both Melody and Harmony). Harmony, in particular, is traditionally described as either (a) superimposed thirds on the different degrees of the scale, or (b) coming directly from the Natural Harmonics. We claim that neither of these theories will produce reliable results, and that the series of fifths is the beginning of music with the thirds of chords (which we call the medians) placed within the "frame" of each fifth. Once this musical generation is set in motion, it is of secondary importance that it be tuned according to its original generation or tuned according to equal temperament, which all agree is out of tune but which is very useful for fixed tuning (like keyboards). I hope that this answers your question. Do not hesitate to write back.

I've recently been getting into the functional tuning section on the website, and I have a couple of questions on this very interesting subject. First of all, are there any fixed-pitch melody instruments out there with Pythagorean/Trunk tuning? Also, are all the clips from the website in functional tuning? If so, how is this done with instruments such as the piano?

     In answer to the first question, there are instruments that could be tuned Pythagorean, such as piano, various types of harps, fretted instruments with moveable frets (guitars, lutes, etc.). These aren't strictly "melody instruments" since they are capable of playing chords, but they could be used as melody instruments. String players (violin, etc.) usually tune their open strings in perfect fifths, which is Pythagorean - but of course these are not "fixed-pitch." There may be others that we are not aware of at this time. Perhaps that would be an interesting research project. Let us know if you find any more info !
     As for the clips on the web site, they are all in equal temperament, using standard midi note values. However, we have done experiments with functional tuning using midi note-bend messages. You will find examples of MIDI Functional Tuning in our analysis of the Planck Enigma in the Just Intonation Performance page.
     The acoustic piano is not capable of playing in functional tuning in any practical sense. (Theoretically, it would be possible to tune a piano to be able to play in functional tuning using a single swing progression, or possibly two swings if some motrices are left out, but this would be very limiting). Using electronic instruments with programmable pitch-bend would make it possible for keyboard instruments to play in functional tuning.

As a cello player I wonder about the convenience of tuning my instrument in temperated fifths (instead of perfect fifths) specially when I play with a piano in order to fit its tuning. Aternatively it could be interesting to tune the piano with the "Cordier" temperament (that preserves perfect fifths thus stretching octaves). The standard practice (cello in perfect fifths and piano in temperated fifths) doesn't seem to be the ideal. Playing alone (or in a string ensemble) perfect fifths seem better but what a mess if I should change to temperated fifths when I play with a piano. Or not? Thanks for your advice. PS: incredibly interesting your MusicNovatory. I feel there are some coincidences with Mathieu's book Harmonic Experience in the explanation of Just Intonation. Are there?

1. We certainly would not recommend that you tune your cello in tempered fifths unless very exposed use of open strings clashed with the piano. It seems far preferable to keep the cello in tune with itself. However, tuning with the A of the piano might not be the best solution. The difference between the perfect fifth and the tempered fifth is almost 2 cents (1.955 to be more precise). Tuning with the A of the piano makes your low C almost 6 cents flat to the piano. However, tuning with the D of the piano would make your A 2 cents sharp but the C only 4 cents flat, already an improvement. We are very flattered by the fact that you ask us for advice but would it not seem more appropriate to consult other cellists? How do they tune when they play with piano? We are not very familiar with the "Cordier temperament" but stretching the octave seems very difficult to accept. Along with the additionnal stretch on the thirds, it seems like a high price to pay for the fifths. You can probably find out more on Cordier tuning from some of its proponents and users, including this reference from the Nydana Notation web site FAQ answer to question #6 (see paragraph 6)
2. Concerning Mathieu's HARMONIC EXPERIENCE, we were also quite astonished to see how much his theories coincided with that of Music Novatory's. At a certain point, they diverge. We could talk more about that in the future.

Is temperament inherent to atonal or post-tonal music? Is atonal or post-tonal music an epiphenomenon of temperament? Should we strictly avoid just intonation in atonal or post-tonal music? Thanks.

     There may not be simple and definitive answers to these questions, but they may nonetheless be worth exploring. Before proceeding, it will be necessary to lay some groundwork so that we have a common understanding of what is meant by certain terms.
     First, by "temperament" it is assumed that you mean twelve tone equal temperament (there are other temperaments). It is very important that we recognize what twelve tone equal temperament (TTET) is, and what it is not. Though its gradual acceptance occurred during the 18th and 19th centuries, TTET has been advocated by some theorists since as early as the 16th century, hundreds of years before what is commonly called atonality came onto the scene. It would behoove us to remember that it was developed as a PERFORMANCE tool to allow fixed-pitch instruments the flexibility to play in a greater variety of keys and harmonic situations than would have ever been possible with fixed unequal temperaments. It is a compromise which allows the same pitch to stand for and approximate a few different pitches that are close enough to it that the musical function of the approximated pitch will, in context, be evident. Everything, other than the octave, is slightly out of tune, but this has been accepted by many as a desirable trade-off for the increased flexibility. The twelve-tone equal tempered chromatic scale is not, therefore, in our view, a fundamental building block of the musical language. It belongs to the realm of performance, not to the realm of generative structure.
     If we agree on this, we can proceed to the next step and explore the question: "Is temperament inherent to atonal or post-tonal music?"
     In answering this, it may help to realize that music can be atonal and yet still functional. There is a section of the Music Novatory web site where we present four forms of harmony, two of which (free harmony and rhythmic harmony) are atonal. These types of harmony, though functional, are defined as atonal because they have no tonal center and no diatonic restraints. Equal temperament is NOT inherent to this type of harmony, although this type of harmony could certainly be performed on an instrument tuned in equal temperament. The best method of tuning, however, would be Functional Tuning, wherein each pitch would be tuned according to its "birth" within the generative structure. Of course, this would not be possible on a fixed-pitch instrument; in this case TTET seems to be the most feasible alternative for any music that has more than severely restricted harmonic movement.
     Based on the above, it would appear that the answer to the first question is: no, TTET is not inherent to atonal music, because atonal music can exist without TTET and TTET can exist without atonality.
     However, we are aware that the music you are asking about may not fit into our description of functional atonal harmony. The terms "atonal" and "post-tonal" may have different meanings to different people. You may be referring to a particular school of composers, i.e., pre-serial Schoenberg and his followers. It is doubtful that one would be able to decipher much operative functionality (as we are using that word) in the works of this genre, at least not, in our opinion, enough to provide a basis for tuning decisions.
     Some schools of composition, such as dodecaphony (twelve-tone serial composition), and perhaps others in the "atonal" category, take the twelve notes of the equally tempered chromatic scale to be basic building blocks. On a theoretical level, this seems rather strange: to take a tuning device meant to be an approximation and ascribe to it the level of primary structure, and in so doing to disregard the primary generative structure that it was developed to approximate. (This is simply a theoretical observation, and should not be taken as a value judgment of any kind.) In any case, in this restrictive sense, it does appear that equal temperament could be thought of as an essential element of dodecaphony, and perhaps other specific forms of atonality.
     This might be a good segue into the next question, which was: "is atonal or post-tonal music an epiphenomenon of temperament?"
     If atonality is defined as absence of tonal center and absence of distinction between diatonic and chromatic, then atonality certainly could have evolved (or been discovered) without equal temperament. However, if there was a belief in the twelve equally tempered tones as being some sort of basic musical structural entity, other than simply an approximated tuning device, then it seems reasonable that those who held such a belief would attempt to create a musical system based on that belief.
     The last part of the question was "should we strictly avoid just intonation in atonal or post-tonal music?"
     That is a question which must be answered by the performer. Music Novatory is mainly concerned with the generative structure of music. Our contribution to performers is to present a sound theory, as well as a means of developing musical intuition, both of which they may choose to rely upon when making performance decisions. Our specialty is dealing with a theory of generative principles that we see as inherent to the musical language. With those qualifications in mind, we can offer this advice: If you are performing music that is functional, i.e., based upon the naturally occurring generative structure of music, whether tonal or atonal, then Functional Tuning would be the best choice. If you are performing music that you are having difficulty analyzing as functional, we suggest that you simply use your ear and intuition, and that you continue to develop both your musical intuition and theoretical understanding. If you are performing music that you know to be based on some artificial intellectual contrivance, then we aren't really the best people to ask about how it should be tuned - that is beyond our expertise.

In the new major 7 tuning section it says that the chromatic line C, B, Bb, A from tonic to counter is unadvisable. However, this goes against what has been common practice. This line is a staple in love songs and sounds great. Some of the many examples include Something, You Are So Beautiful, and Can't Take My Eyes Off of You. They all have a progression that goes I,Imaj7,I7,IV in the beginning.

You have no idea how precious these comments are to us ! We will be examining these 3 songs carefully and get back to the drawing board.