Peperplanet, what you are referring to is "stretch" and that's used when tuning an instrument to compensate the fact that string thicknesses is altering the strings internal harmonics slightly, and thus making fore example your piano sound better. By this the lower tones get piches slightly lower, an the higher piches get piches slightly higher. An octave is always the ratio 2:1 in its essence, however, an octave on a piano that's tuned with "stretch" is slightly longer than its ideal 2:1 ratio. en.wikipedia.org/wiki/Stretched_tuning waldorfmathematics, a great idea applying music in math classes. I'm a music teacher student trying to do the opposite, explaining the math behind music in my music classes. And it's so true what you're saying, very few musicians know about the mathematical beauty at the core of music and sadly enough music notation is a big obstacle in the sense that is, in many ways, mathematically contradictory. Calling a 5th a 5th when is hasn't even anything remotely to do with 5. Here 7th (7 half tones) or 3:2 (the ratio) would be a better name. The complexity of the ratio has to do with the dissonance/consonance quality of the interval, and a great way it's illustrated is in the "harmonic entrophy" graph here: en.wikipedia.org/wiki/Consonance_and_dissonance#Physiological_basis_of_dissonance As you may notice this subject really engages me, especially because I haven't seen any great books out there about the subject. Musicians have the right to know this. Not just mathematicians, and musician nerds, like me. So maybe I should write a book about it :)
Yes, I wish this was more talked about or written about.The stretching of the octaves is just part of well temperament tuning. In equal temperament tuning there is no stretching, no sharpening of the octaves. I think it sounds better because we are used to hearing this sharpening. From 1450 to about 1800, equal temperament was favored.
It's a common misconception that octaves are just doubling or halving a frequency; 220, 440, 880... Because of the physical properties of strings (no, they're not just numbers in the ether), 'octave stretching' occurs; this means octaves are constantly getting more sharp/flat the farther high/low you go; 219.6, 440, 880.4... It's really noticeable on small pianos-the highest C on in a spinet might be 30 cents sharp to sound like a proper octave! Get a book on piano tuning or just google it!
For an in depth explanation of math and music, try The Great Teaching Courses "How Music and Mathematics Relate". www.thegreatcourses.com/courses/how-music-and-mathematics-relate.html
This dude says for Pythagoras the Perfect Fifth was represented as a ratio 3:2. But he is covered up the truth! In fact this is from Archtyas and Plato saying that Arithmetic Mean (3/2) x Harmonic Mean (4/3) = Geometric Mean Squared (2). And this is not Orthodox Pythagorean philosophy. Orthodox Pythagorean PHilosophy uses noncommutative phase as math professor Alain Connes points out, 2, 3, infinity is noncommutative phase as quantum nonlocality - the real truth of Pythagorean philosophy. So if 3/2 is C to G as the Perfect Fifth then 2/3 is C to F as the Perfect Fifth subharmonic. So if C is 1 and the Octave C is 2 then G = 3= F as noncommutative phase, being in two places at the same time. This is the real truth of Pythagorean harmonics that got covered up by Archtyas and Plato, creating a "pre-established deep disharmony" to quote math professor Luigi Borzaccchini who states that the cover-up of the music origin of Western math is "really astonishing" and "shocking" (2007) - see his article on incommensurability, the continuum and music.
+Vishal Menon A fifth above A is E. C# is a (major) third above A. Evidence of this is the Am triad which contains a minor, or flatted third ( A - C - E ).
+AsNightAsMyWitness i just looked it up and learned that fifths are seven semitones apart - therefore the perfect 5th of A is E. i was under the impression that 5th was referring to the difference in semitones. however, the way it is counted in this video was confusing. why are sharps avoided? he goes 5th from A is A B C D "E" :S
+Vishal Menon It seems that you're thinking musical intervals are based on semitones, and that every time we move a semitone we move a second, then a third, etc... Although this is an intuitive, and sensible, way of counting intervals, it is incorrect. Music intervals are not based not on semitone distances, but on the musical staff. Consider this: if we are at A, and move up one semitone (A#), you might be tempted to say we traveled a 2nd. This might make sense in our heads, but if we did it on a musical staff, we would not have traveled anywhere at all. We would still be on the same "A" line or space that we were before (There would now simply be an accidental [#] symbol placed next to the A.) Now, if we move two semitones from A to B, we would actually have to change our vertical position on the staff to the next line or space above us. It is actually the smallest amount of distance we can move by on the staff. Not in music, but on the staff. This is a second. If we moved up the staff two positions, from the line or space our A was on, to the next line or space above (or B), and then further to the next line or space above that, we will be at C. Now, this is where it gets tricky. We might say we moved three STs from A, putting as at C, and giving us an interval of a minor third. Or, we could say we traveled four STs from A, putting us at C# and giving us an interval of a Major third. In either case, we are at the same position on the staff and that in itself is what allows us to say we traveled a third, be it minor or major. This same idea applies, going up the scale, all the way to the octave. And all those little notes in between, the sharps in this case, are not being avoided; they are simply given their own name based on their nearest natural note relative. C to D is a Major second. C to Db is not quite there. So it's called a minor 2nd instead. It's still a second because our staff positioned changed. C to C# is the same as Db. The difference is our position on the staff didn't change (due to nomenclature alone) and so we cannot call this a 2nd. We have to give it a different name because we're choosing to call it a C# and not a Db, even though they are the same note. (This 'name' is called an "augmented unison.") I hope this helps in some way.
cheers man! i looked it up a bit and sort of understood how these were different mathematical points ; even when counting semi-tones and intervals: 12 semi-tones and only seven intervals :S I couldnt understand the intuition but you made it more clear. it makes more sense you bring the staff into reference. i had to read that second half a couple times but thats a really good way of explaining it. nailed in the concepts! thanks :)
Peperplanet, what you are referring to is "stretch" and that's used when tuning an instrument to compensate the fact that string thicknesses is altering the strings internal harmonics slightly, and thus making fore example your piano sound better. By this the lower tones get piches slightly lower, an the higher piches get piches slightly higher.
An octave is always the ratio 2:1 in its essence, however, an octave on a piano that's tuned with "stretch" is slightly longer than its ideal 2:1 ratio.
en.wikipedia.org/wiki/Stretched_tuning
waldorfmathematics, a great idea applying music in math classes. I'm a music teacher student trying to do the opposite, explaining the math behind music in my music classes. And it's so true what you're saying, very few musicians know about the mathematical beauty at the core of music and sadly enough music notation is a big obstacle in the sense that is, in many ways, mathematically contradictory. Calling a 5th a 5th when is hasn't even anything remotely to do with 5. Here 7th (7 half tones) or 3:2 (the ratio) would be a better name.
The complexity of the ratio has to do with the dissonance/consonance quality of the interval, and a great way it's illustrated is in the "harmonic entrophy" graph here:
en.wikipedia.org/wiki/Consonance_and_dissonance#Physiological_basis_of_dissonance
As you may notice this subject really engages me, especially because I haven't seen any great books out there about the subject. Musicians have the right to know this. Not just mathematicians, and musician nerds, like me. So maybe I should write a book about it :)
Yes, I wish this was more talked about or written about.The stretching of the octaves is just part of well temperament tuning. In equal temperament tuning there is no stretching, no sharpening of the octaves. I think it sounds better because we are used to hearing this sharpening. From 1450 to about 1800, equal temperament was favored.
what are the ratios if you talk about a minor 3rd, and the minor 6th and 7th
It's a common misconception that octaves are just doubling or halving a frequency; 220, 440, 880... Because of the physical properties of strings (no, they're not just numbers in the ether), 'octave stretching' occurs; this means octaves are constantly getting more sharp/flat the farther high/low you go; 219.6, 440, 880.4... It's really noticeable on small pianos-the highest C on in a spinet might be 30 cents sharp to sound like a proper octave! Get a book on piano tuning or just google it!
Peperplanet,
Thank you for this clarification. Quite interesting! I will make note of this in my books.
~Jamie
For an in depth explanation of math and music, try The Great Teaching Courses "How Music and Mathematics Relate".
www.thegreatcourses.com/courses/how-music-and-mathematics-relate.html
5th above F is A not C.. the sharps are being forgotten in the way the intervals are being calculated here...
+Vishal Menon A fifth above F is C. A third above F is A. 'A' is the third note, and the first stacked third, of an F major triad ( F - A - C ).
+AsNightAsMyWitness thanks for that. i stand corrected...
It is difficult stuff and confusing.
1/2 step Jaws theme
1 step. Happy birthday song
1 1/2. What child is this
2steps. Oh when the saints
2 1/2 steps Amazing grace
3steps Marias theme WSS
3 1/2 step Star Wars theme
4 steps
This dude says for Pythagoras the Perfect Fifth was represented as a ratio 3:2. But he is covered up the truth! In fact this is from Archtyas and Plato saying that Arithmetic Mean (3/2) x Harmonic Mean (4/3) = Geometric Mean Squared (2). And this is not Orthodox Pythagorean philosophy. Orthodox Pythagorean PHilosophy uses noncommutative phase as math professor Alain Connes points out, 2, 3, infinity is noncommutative phase as quantum nonlocality - the real truth of Pythagorean philosophy. So if 3/2 is C to G as the Perfect Fifth then 2/3 is C to F as the Perfect Fifth subharmonic. So if C is 1 and the Octave C is 2 then G = 3= F as noncommutative phase, being in two places at the same time. This is the real truth of Pythagorean harmonics that got covered up by Archtyas and Plato, creating a "pre-established deep disharmony" to quote math professor Luigi Borzaccchini who states that the cover-up of the music origin of Western math is "really astonishing" and "shocking" (2007) - see his article on incommensurability, the continuum and music.
My brane hurtz.
5th above an A is Csharp not E...
+Vishal Menon A fifth above A is E. C# is a (major) third above A. Evidence of this is the Am triad which contains a minor, or flatted third ( A - C - E ).
+AsNightAsMyWitness i just looked it up and learned that fifths are seven semitones apart - therefore the perfect 5th of A is E. i was under the impression that 5th was referring to the difference in semitones. however, the way it is counted in this video was confusing. why are sharps avoided? he goes 5th from A is A B C D "E" :S
+Vishal Menon It seems that you're thinking musical intervals are based on semitones, and that every time we move a semitone we move a second, then a third, etc... Although this is an intuitive, and sensible, way of counting intervals, it is incorrect. Music intervals are not based not on semitone distances, but on the musical staff. Consider this: if we are at A, and move up one semitone (A#), you might be tempted to say we traveled a 2nd. This might make sense in our heads, but if we did it on a musical staff, we would not have traveled anywhere at all. We would still be on the same "A" line or space that we were before (There would now simply be an accidental [#] symbol placed next to the A.)
Now, if we move two semitones from A to B, we would actually have to change our vertical position on the staff to the next line or space above us. It is actually the smallest amount of distance we can move by on the staff. Not in music, but on the staff. This is a second. If we moved up the staff two positions, from the line or space our A was on, to the next line or space above (or B), and then further to the next line or space above that, we will be at C. Now, this is where it gets tricky. We might say we moved three STs from A, putting as at C, and giving us an interval of a minor third. Or, we could say we traveled four STs from A, putting us at C# and giving us an interval of a Major third. In either case, we are at the same position on the staff and that in itself is what allows us to say we traveled a third, be it minor or major. This same idea applies, going up the scale, all the way to the octave. And all those little notes in between, the sharps in this case, are not being avoided; they are simply given their own name based on their nearest natural note relative. C to D is a Major second. C to Db is not quite there. So it's called a minor 2nd instead. It's still a second because our staff positioned changed. C to C# is the same as Db. The difference is our position on the staff didn't change (due to nomenclature alone) and so we cannot call this a 2nd. We have to give it a different name because we're choosing to call it a C# and not a Db, even though they are the same note. (This 'name' is called an "augmented unison.")
I hope this helps in some way.
cheers man! i looked it up a bit and sort of understood how these were different mathematical points ; even when counting semi-tones and intervals: 12 semi-tones and only seven intervals :S I couldnt understand the intuition but you made it more clear. it makes more sense you bring the staff into reference.
i had to read that second half a couple times but thats a really good way of explaining it. nailed in the concepts! thanks :)
yeah really, thank you my friend
The ancient Greeks were polytheise.