Perfect intervals (8vas, 5ths, 4ths, and their compounds) were termed "perfect" (in contrast to "imperfect") because they sounded perfect to the people who named them "perfect"--going back well before Christ and many hundreds of years before "key" was dreamed of.

I think modern man is less apt to describe the sound of 8vas, 5ths, 4ths, and their compounds as "perfect" (in contrast to "imperfect") than as "pure" or "simple" (in contrast to "rich").

There is a point-blank physical reason for all this. With the limitations of email and just prose, the effect of intervals traditional called "perfect," "imperfect," or "dissonant" can most easily be described to you in terms of the harmonic series, the open notes on your trpt, their transpositions, and, ABOVE ALL, your playing the suggested problems.

Let's start with the Bb trpt's open notes. Think of them in their usual transposition from small c (bs clef) on up. They are small c, C1, G1, C2, E2, G2, Bb2, C3, D3, E3, F#3, G3, etc. Also be think of them as harmonics 1 through 12 else you won't follow the "demonstration" below. Realize that notes 1, 2, 4, 8, 16, etc in the C series are all C's. Realize that notes 3, 6, 12, 24, etc in the series are all G's. Realize that notes 5, 10, 20, etc in the C series are all E's. ETC. Arithmetic progression. Get it?

You should try all of the following with a friend, but the experiments will be hardly audible until you transpose them up about half an 8va or more. In the C series, player #1 hold 4th harmonic (C2) while player #2 holds 3rd harmonic (G1). Whether we like it or not, those notes and their own overtones combine in our ears in such a way as to make us hear harmonic 1 (small c) and harmonic 2 (C1). IF YOU'RE NEW TO THIS, you probably won't hear those buzzing low notes in your ear unless you raise the whole experiment an 8va. Raised an 8va, G2 and C3 held together cause you to hear the lower C's. One of the lower C's will probably mask the other, and you'll probably notice only the lower note, the fundamental of the series (C1) whose 3rd harmonic is G2 and whose 4th harmonic is C3.

If either player is out of tune at all, the primary difference tone (buzzing low note in your ear) will move out of alignment with the G & C generators. Fool around with the intonation. Fool around with dynamics. Transpose the whole experiment. Try it muted.

The math of this experiment is simple:
harmonic #4 minus harmonic #3 equals harmonic #1. Harmonic #1 then interacts with harmonic #4 to produce a weaker #3 (4 minus 1 = 3, thus doubling the one already being played). Harmonic #1 also interacts with harmonic #3 to produce a weaker #2 (3 minus 1 = 2).

For reasons we haven't touched on, in-tune perfect 5ths must be about 2 cents (2% of a half step) larger than the piano's perfect 5th. Conversely, in-tune perfect 4ths must about 2 cents smaller than the piano. If you listen you'll move those 2 cents without even knowing it, but keep in mind that if you happen to be on one your instruments "out-of-tune" notes, you may have to move a lot more than 2 cents to play in tune.

In short, a "perfect" interval played beatlessly in tune produces difference tones that are doublings or 8va doublings of the tones being played. The higher the generating tones, the more obvious will be the difference tones and sense of alignment (or lack thereof).

To experience "imperfect consonance"--a far more interesting experience, sustain E2 and G2. Those notes are harmonics 5 and 6 in the C series. E and G played together in proper alignment will cause you to hear a low C; 6 minus 5 = 1. NOTE that the primary difference tone is a DIFFERENT MEMBER OF THE ALPHABET. Going on with it: 6 - 5 = 1. #1 interacts: 6 -1 = 5, 5 -1 = 4, and all those interact ad infinitum until you hear a complete harmonic series from #5 (the highest generator) on down to the fundamental.

Experiment as above with both major and minor 3rds and 6ths. High and rather loud is the easiest to hear. Major 3rds and their inversions (minor 6ths) must be 14 cents smaller than equal temperament. Minor 3rds and major 6ths must be 16 cents larger than equal temperament. These are rather large descrepancies between just tuning (beatlessly in tune) and equal temperament. Music's main tuning problems are with 3rds & 6ths. Lord help you if you happen to be on a note whose tuning on your instrument happens to be at odds with the intonation demands of music/nature. Good luck!!

NOTE: Players often get the mistaken idea that the top note of a major 3rd, for example, must be lower than in equal temperament. The context dictates whether the top note must be lower, the bottom note must be higher, or the players meet each other.

To the acoustician, a perfect interval is one whose difference tones introduce no new members of the musical alphabet, and an imperfect interval is one whose difference tones introduce one or two new members of the musical alphabet. Dissonance introduces still more difference tones. The tritone seems to be a special case.

It is easy to mix apples and oranges in this subject. Just intonation means tuning justly (absolutely beatless) as you go, regardless of what turns up. Any music that is complicated enough to use a ii chord (supertonic) has tuning requirements that make it either (1) impossible to end on the starting note or (2) impossible to tune every interval as we go--a subject for another post.

Most keyboard instruments must have fixed pitch for the duration of a piece, cannot adjust to musical tuning problems as the piece progresses, and, therefore, cannot be beatlessly in tune throughout. Unless the music at hand is I-IV-V-ish, Just Intonation is not appropriate for keyboards. Keyboards must compromise; hence "equally compromised" or "equal temperament." Other well-known compromises have been called "mean-tone" and "well-tempered." "Stretch tuning," as I understand it, is a variation on "equal tempeament" which compensates for our hearing loss in the upper range of keybd instruments.

Those of us who play WITH a fixed-pitch instrument must continually "do the best we can." The piano, for example, will not come to meet us as did your friend in the experiments above.

IMPORTANT: No matter how little you got from this post, practice playing/aligning high major and minor 3rds & 6ths (imperfect consonance) with a friend. Also practice playing/aligning rather high 4ths and 5ths (perfect consonance).

Believe it or not, this complicated post was written rather quickly. I hope it doesn't show too much. No more time just now. On my way out the door to the NY Brass Conference. Say hello to me there: 70 yrs old, tall, bald, gray fringes, gray beard, tri-focals, blue jacket.

Roger McDuffie