So in Lesson 4, let's turn to the question of why the number of intervals in scales are so limited. Why are we limited to five note scales, seven note scales, that are in use today and have been for centuries. What's the reason for that? So to answer that question, let's go back to a slide I showed you before which is really the theme of the whole module today and of the previous module on consonants. And that is that, scales today are consequence of harmonic overlap. And that's what I just finished telling you, that the scales that we like are the ones that have high harmonic overlap and therefore a strong correspondence. A strong association with human vocalizations. So let's look at the 12 intervals, 13 notes of the chromatic scale, and I showed you this diagram before. This is a harmonic series of the fundamental indicated by these white symbols here. So let's say this was a musical tone with a fundamental of a hundred hertz, then this harmonic would be 300, 200 hertz, 400 hertz and so on. And the issue here in a harmonic overlap that we talked about before and I'm talking about again now is, can the small number of notes that are present in scales be defined or explain in terms of their degree of harmonic overlap? Here is a blow up in this of just a single octave in this harmonic series and all of these other colored symbols that I told you before indicate the other intervals of the chromatic scale. So you can see from this that the playing of unison is complete harmonic correspondence. Every time you play two notes together, every single harmonic of the two notes is in correspondence. When you play an octave, and this would be the white symbols and the red symbols, then it's the case that every other harmonic in the two notes is in correspondence when you play a major fifth, that's the orange symbol here, every third harmonic in the two notes is overlapping. And so on down the list of the notes that are relatively more or less constant in the chromatic scale series. So what does this mean with respect to the question of why it is that we have a relatively small number of tones and the scales that are popularly used, five intervals, seven intervals, and so on. So the reason is that as you introduce other intervals, you begin to introduce harmonics that don't overlap with any of the harmonics in the tonic. So let's look at this. So here the brown symbols are indicating no combinations where there is overlap with the tonic harmonics. And the yellow symbols indicate the harmonics of the notes that don't correspond to any of the harmonics of the tonic. And you can see that as you generate more intervals in a scale necessarily, you introduce more and more harmonics the yellow symbols. That don't correspond with any of the harmonics in the tonic, these harmonics that are diagrammed here. So there's a tradeoff between the introduction of notes that correspond, that are consonant, that have high correspondence with the tonic. Even though they begin to introduce intervals that don't have any correspondence with the harmonics of the tonic, and that tradeoff begins to lose vocal similarity. So the more notes that you introduce into a scale, the more you reduce the vocal similarity, and the more the scale begins to loose it's ability to signify human vocalization. So there's no limit in principal to the number of notes you can introduce into a scale. Different composers have tried to introduce large numbers of notes in scales. There is a 31 note scale, there is an 81 note scale that people have tried to use. But of course, they're not popular or widely used, and the reason is that very quickly after you've introduced seven intervals, you quickly begin to lose the yellow symbols here in this diagram begin to outweigh the bounds, and those are the correspondences and vocal similarity as you go beyond seven intervals in the scale or eight notes in a heptatonic scale, diatonic scale, you very rapidly lose vocal similarity and losing the vocal similarity is what in a biological explanation, this is really all about.