Perfect indeed. The cherry on that sundae would have been to further show that the 2nd, 3rd, 4th, 5th harmonics he played together are actually the standard barbershop C major chord: C, G, C, E. Barbershop tags usually end with the 3rd on top since the top part (tenor) sings over the melody (lead).

Harmonics and humoronics lol

Yeah! I wish someone showed me this when I was younger!

Using the 7th harmonic as the 7th of the scale wouldn't produce what is normally referred to as the Mixolydian mode today. Western music evolved from the modal framework of ancient Greek and Roman music, still used to this day across the Eastern Mediterranean region. Within it, the primary consonances were the perfect fourth and the perfect fifth. Usually, the modes are built by stacking two tetrachords, with the difference tone to complete the octave placed either between the two tetrachords (making them disjunct tetrachords), or on top of the second tetrachord (making them conjunct tetrachords which share one note). Ancient Greek and later medieval Islamic music theorists saw the fifth and the fourth as the smallest consonant intervals, meaning that the most consonant modes had the highest number of perfect fourths between the scale degrees. The modern Mixolydian mode, made of two conjunct major tetrachords and a difference tone (CDEF + FGABb + C) fits the description. But this means that the Bb you get here isn't derived from the 7th harmonic, but rather as the perfect fourth from F (a 16/9 interval from C). Using the 7th harmonic would undermine the structural integrity of a scale based on the perfect 4th and the perfect 5th as the most consonant intervallic relationships. Now, how about the tonal system, which later developed in the West? Here, the 5th harmonic is used to arithmetically divide the perfect fifth (3/2) into two consonant intervals of differing sizes: a major + a minor third (5/4 x 6/5). But the old modal consonances still formed the skeleton of the scale. There's the tonic C, along with its perfect fourth (F) and fifth (G). It is from these three individual notes, taken as separate fundamentals, that three separate major chords are played (456) to derive the fundamental scale, thus producing C E G, F A C, and G B D. The middle notes in the chords (E, A, B) will be a syntonic comma lower than in the modal system. There are two different ways to arrive at a minor 7th for Mixolydian now: either as the minor third of G (making it 9/5 from C), or as the perfect fourth of F (like in the modal system), which is a syntonic comma lower. However, if one were to use the 7th harmonic (7/4) from C, this would make the minor third from G to Bb unusually small (7/6 instead of 6/5), even compared to the one in the modal system, while the whole step from Bb to C would be unusually large (8/7 instead of 9/8). Forming new chords with this Bb would now necessitate a number of notes far beyond the standard 12, and the scale structure would be completely obscured. So, the problem with using the 7th harmonic is that it fits into neither of the two frameworks (modal and tonal) that formed the basis of music in the West for hundreds of years. It detracts from the smallest interval considered a consonance in Western music (the minor third 6/5) and introduces a layer of complexity into the scale structure with two new melodic intervals (a smallish minor third 7/6, and a largeish whole tone 8/7).

I found a piano where F5 note is 20c flat, G7 (G3-D4-B4-F5) chord sounds better than the piano in tune.

That's interesting, thanks.

Well, not really. They weren't used much in vocal music either. Just look at medieval songs, and you will notice that the harmonies (on strong beats) are mainly based on 4ths, 5ths and octaves. 3rds and 6ths only became consonant during the renaissance. Now, singers were probably trained to sing in Pythagorean tuning (and Pythagorean tuning is probably one of the contributing factors to why thirds weren't used much), but still, they could have adjusted the notes by ear. It's probably just that the "beauty" of the third wasn't yet "discovered" because people didn't know exactly how to deal with thirds. You need to put a concept into use before people can get familiar with it. And if 3rds had really never been used "as consonances", then people simply weren't used to the sound of it. It's hard to imagine how people before us perceived music. It's easy to think that of course what sounds great to our ears has always sounded great to people's ears. But that's simply not the case. Music has evolved to become more and more dissonant over time, and there's a reason to it - when the previous generation comes up with a new thing, they can teach that to the next generation, and this way new knowledge builds on top of older knowledge. Now we have the tools to make more dissonant things sound good, and those tools are common knowledge.

Thanks Lee, but I do say that you can’t play fractions of wavelengths. And wavelengths and frequencies are highly related as you know! The whole number ratio of frequencies is exactly equivalent to the whole number of waves.

I'd never heard of her, but yes I do!

You are right, and that is why I go on to other length tubes at the end. And the notes are written in the staff in the background...

It is 3 octaves above the lowest possible note.

@@martinlawrence1148 interesting. So that sound is just made up of harmonics. I wonder if I can play that on the guitar somehow.

@@elementsofphysicalreality yes, an elephant's trunk is acoustically the same as my natural horn. And the harmonics on a guitar string are in the same place too. You can hear them if you place your finger lightly on the string in the right place, as I expect you know. Might be difficult to get as much sustain as a horn, unless it's an electric...

You may be right. I don’t know about Akkadian or Sumerian music (I would love to hear some!). But Pythagoras certainly talks about dividing the string up into whole number intervals to get the harmonic series, and knew what we call a fifth was the ratio 3:2 etc. The Greek modes, as I understand it, used scales derived from the harmonic series, and didn’t use what we would call triadic harmony, a much later invention I agree. But you could say that triadic harmony was derived from Pythagorean scales.

@@martinlawrence1148 That's not quite right. Pythagoras (who is a semi-mythological figure, none of whose writings survive) is credited with being the first to mathematically describe the intervals 3:2, 4:3, and 2:1. But he did not describe the harmonic series. That mathematical development didn't happen until the 14th century in Western Europe. "But you could say that triadic harmony was derived from Pythagorean scales." Pythagorean-ism describes a system of tuning where all intervals can be produced by adding or subtracting 3:2 and 2:1. If you stack seven 3:2's and rearrange it so it fits within one octave, you get the scale 1:1 9:8 81:64 4:3 3:2 27:16 243:128. We would hear this as the major scale. The problem is that the triadic harmony derived from this scale is really bad. The major third 81:64 is around 408 cents. That's 8 cents sharper than the equal tempered major third of 400 cents, and 22(!) cents sharper than the "pure" or "harmonic" major third of 5:4 (386 cents). What happened in Europe is that around the 15th century, musicians and composers because increasingly attracted to the sound of "good" thirds and sixths -- the simple intervals 6:5, 5:4, 8:5, and 5:3. But none of these are obtainable in a Pythagorean system. So what started as fudging intervals here and there eventually developed into meantone temperament, where the purity of 3:2 is sacrificed to improve the thirds and sixths. At this point it can no longer be considered Pythagorean. The general contours of the scales being used (i.e. the diatonic modes, the major/minor scale, dorian, phrygian etc. etc.) stayed the same, but the tuning actually moved away from Pythagorean as triadic harmony was being developed.

@@karawethan I stand corrected on Pythagoras. But the pure 3rds and 6ths you refer to ARE derived from the harmonic series, which is the point I am making. I'd be interested to know why Greek scales ended up with 7 notes in them rather than 5 (as is common in many cultures) or indeed any other number. Can you shed any light on that?

@@martinlawrence1148 I personally don't like the language of musical scales/intervals "deriving" from the harmonic series. It makes it sound like people were aware of the harmonic series in some abstract/mathematical sense and then consciously applied it to music. The real reason has to do with what is called phase locking, basically the auditory perception of how the overtones of 2+ sounds line up with one another (or don't). An interval like 27/16 (major 6th) will sound more dissonant than nearby 5/3 because the overtones of that interval do not sync-up quite as well. BUT...27/16 is simply the 27th partial of the harmonic series brought down into a usable octave. Conversely, there is no 5/3 partial. It is instead the relationship between the 5th and 3rd partial. We recognize 5/3, 6/5 (difference between 6th and 5th partial) and 8/5 (octave inversion of 5/4) as highly consonant, even though they don't appear "naturally" in the harmonic series. That, and we favor those over actual low-level partials like 7/4 or 11/8, which have no close approximation in our musical system (but actually sound pretty consonant once you get used to them). The real kicker is that so far, we have only been considering harmonic timbres. There are all sorts of musical sounds with *in*harmonic timbres (mostly idiophones), meaning that their overtone profile looks nothing like the harmonic series. Consequently, intervals like 5/4, 6/5 etc. may sound more dissonant than nearby irrational intervals. Sorry for the overly long explanation. Regarding ancient Greek scales having 7 notes, I am not from ancient Greece so I couldn't really tell you. But there does appear to be a strong preference not only cross-culturally but really throughout time for 5 and 7 tone scales. Some have hypothesized that this relates to the idea of a scale being "well-formed" or "maximally-even." You want a scale where the tones are distributed more or less evenly across the octave (not bunched up at either end or at the middle). Symmetry can be a desirable feature, but totally perfect symmetry is not (that may have something to do with the relatively rare occurrence of 6 and 8 tone scale systems). And, you don't want so many tones that it becomes hard to tell them apart and remember melodies. There may be other reasons, but 5 and 7 seems to be a natural sweet spot.

Ah, that’s because of the Pythagorean comma! If you tune all the fifths to exactly the ratio 3:2 as they should be, after 13 octaves you should get back to the initial note, but you don’t, it’s sharp by a small fraction. Do the maths! So you have to squash each fifth equally so they fit.

The 11th and the 13th harmonics are not the problem, because Western European music neither uses them, nor does it even attempt to approximate them. Western music uses intervals mathematically derived from the relations between the first three prime numbers, their powers and multiples: 2, 3 and 5. These relations produce the most sonically pure consonances. To get octaves, one need only use 2 (doubling), while for perfect fifths and fourths the prime 3 is also needed: the perfect fifth is a 3/2 ratio, and the perfect fourth its inverse at 4/3. Finally, pure thirds and sixths need all the three primes (major third: 5/4; minor sixth: 8/5; minor third: 6/5; major sixth: 5/3). Tuning a scale using simple ratios produces sonically pure consonances, but it has problems. One is that it features discrepancies between similar tones derived in different ways, which are called commas. The two commas of relevance here are the syntonic comma and the Pythagorean comma. The syntonic comma is the discrepancy between intervals derived from the first two primes (octaves and perfect fifths/fourths), and those which also use the third prime (thirds). For example, an E derived as a pure major third from C, a 5/4 interval, and an E arrived at after four consecutive perfect fifths (3/2) from C, which amounts to 81/64 from C after octave reduction, are separated by a syntonic comma, which is about 21.5% of a semitone. The Pythagorean comma, on the other hand, is the discrepancy between the first and the second prime. Multiplying by 3/2 (a perfect fifth) twelve times doesn't in fact equal multiplying by 2 (an octave) seven times, because twelve fifths are slightly larger than seven octaves. The amount by which the resulting interval after 12 consecutive fifths is larger than seven octaves is called the Pythagorean comma, and it's about 23.46% of a semitone. The circle of fifths never closes in just intonation for this reason. Different systems of temperament attack different problems of just intonation. The system used for the piano and all modern fixed temperament instruments is called 12-tone equal temperament. It narrows (tempers) all the fifths very slightly, by exactly the twelfth of a Pythagorean comma, so that after 12 of them the circle of fifths closes. This is in practice the same as logarithmically dividing 2 (an octave) into 12 equal parts (semitones). This makes all the fifths/fourths very slightly out of tune for the sake of making all 12 keys sound the same, but the approximation of thirds/sixths isn't good at all. The latter are quite poor from a harmonic perspective (the major third is 14% of a semitone sharp, the minor third 16% flat). This can be mathematically verified: the 12-tone equal temperament major third is 2^(4/12)=1,2599210..., while the pure major third is 5/4=1.25.

That is a natural horn, made by Andreas Jungwirth of Vienna, a copy of a Bohemian horn from 1790, when Mozart was still alive!

Google “equal temperament”