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It's the overtone/harmonic series. Some intervals are Pythagorean, some are sharp, some are flat. But giving every interval a technical name is akin to blinding with science, IMO ("subminor third" is nothing more than gobbledegook for "flat minor third"). An alpine horn just uses the top octave, and your embouchure attempts to iron out the poor intonation. It doesn't really amount to a lot more than trying not to make the fourth too sharp. Most horn players wouldn't go above the fifth.
Andrew

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Well you should watch that!
Actually I've oversimplified it. I'm going to bed now. While I'm asleep you can work out the maths of the overtone series. You'll find that it gives "whole tones" of 9/8 and 10/9 (from memory). 9/8 is known as the major whole-tone and 10/9 is known as the minor whole tone, with similar for other intervals. Have fun!
Andrew

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@Gordon Shumway -
Thanks for sending me down another rabbit hole! 🤨
My almost 6 year-old Grandson enjoys Math MUCH more than I do, but I did take a peek at interval info on your favorite site (Wikipedia 🤣).
The Pythagorean Interval Page - has a GREAT chart (with audio!) for comparing the sound of the common, PLUS diminished & augmented intervals!
The List of Pitch Intervals Page - also provided audio for comparison - but way too much crazy stuff, like how are we supposed to tell the difference between the Sixty-seventh Harmonic and the Hard Semitone/two-fifth tone small semitone? 🥴
@Fiddlerman -
Thank you!
I was pretty sure I didn't need to know all these! 🤗
I've learned a few tunes with a perfect 4th double stop, otherwise I probably wouldn't have much reason to be aware of it - and I know I'm playing diminished & augmented intervals in some tunes, so I hope I can learn to identify them properly... down the road. I think listening to the Pythagorean Interval chart is VERY helpful😊
...my intervals might be a Pythagorean Fuzzy Set. 😱
- Emily
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