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Another aspect of tuning (and the piano in particular) that is interesting is the duality of intervals. A fifth is dual to a fourth (a fourth is a fifth going

Another aspect of tuning (and the piano in particular) that is interesting is the duality of intervals. A fifth is dual to a fourth (a fourth is a fifth going downwards and taken the octave). You can the apply duality also on chords: a major is dual to a minor. If you would like this to be more rigorous, there are a few first steps. For example, you can look at the cyclic group of 12 elements. Lets call them {0,1,...,11}. Then, 7 has inverse 5. And upon translation of this back to the piano you'll see that 7 is the fifth and 5 is the fourth. (7 halftones and 5 halftones.) To see that a major chord is dual to a minor chord you similarly use the cyclic group of 8 elements.

mhh__ 8y ago
What does dual mean in the context of a chord? If it means parralel then why a minor triad (Presumably implied)
- mikorym 8y ago
  To answer that, you have to first answer what the dual means in the first place. And the notion that you get to is dual with respect to the octave. A usual chord is with respect to a fifth. Then a major third is a minor third down. Starting position should be at the tonic. Otherwise, you get at the fifth that when you go a major third down it is a minor third from the tonic.
mikorym 8y ago
This elaboration is late, and maybe will be read only in 10 years time by a unassuming bot scraping the web, but here goes... The musical scale (and a piano) is a logarithmic device. A logarithm by definition exhibits an interplay between addition and multiplication. Addition is repeated increment by one, whereas multiplication is repeated increment by some number. Raising to a power is repeated multiplication by a number. Repetition, or increment by one, is the basis for addition, leading to multiplication and then to the next step of raising to a power. (You can go beyond that again, but I forget what it is called.) In the form of an equation, you can express this as lnx + lny = ln(x*y). As it is a logarithmic device, in a musical scale you go up the scale in integer increments. What I mean by logarithmic is that, x +' 12 in the chromatic scale is equal in the actual frequency of the airwaves to 2 * x. I use +' here to denote addition along the scale. However, as we are working in the cyclic group of order 12 we have in fact that x +' 12 is congruent modulo 12 to x itself; i.e., the octave is the same note as the tonic. We are forgetting for a moment that some people would call the tonic for instance A and the octave A2 in favour of a simpler system for our current discussion. To recap, we already have a connection (by definition) between multiplication and addition in the structure that we call a logarithm and in music we can model the sound wave frequencies as a logarithm. Now, the next step is a more interesting step perhaps for a human that likes to listen to music. We already know as mentioned above that 7 chromatic steps up is a fifth and 5 chromatic steps down is a fifth as well. This is embodied by the cyclic nature of our model (a mathematical group structure). In the same way, in an analogue clock display, 12 + 7 hours is equal to 7 o'clock. 12 - 5 hours is also equal to 7 o'clock. However, time (if you experience it like me) moves forward linearly and indiscriminately. Even though they are inverses in a group, between the hours of 5 o'clock and 7 o'clock themselves, there is not a special physical connection. They are inverses simply due to the way we decided to partition the seconds in our days. (Incidentally, the human hearth rate is usually close to 60 bpm, but that phenomenon is perhaps for discussion another day.) So back to the scale: Within music there is a very special connection between the 7 step chromatic interval and the 5 step chromatic interval. The 7 step interval is a fifth and the 5 step interval is a fourth. There have been countless musical pieces that comprise only the tonic chord, the fourth and the fifth. You can apply duality on chords as well: using the cyclic group of order 7 we have that the major third is dual to the minor third (with respect to the fifth = 7 steps chromatic) giving that a major chord is dual to minor chord. One can continue in this fashion, using cyclic groups, and elements and their inverses will have special significance for music. Of particular note is that we can hear some of these connections, perhaps not rigorously or even explicitly, but indeed we know 1-4-5 is an important chord progression. I am tempted to say that this embodies a physical human interpretation of multiplication and addition, but one can equally argument that this is a physical human interpretation of duality, of group theory, or a very many other mathematical phenomena. We can speculate, but we do know for a fact that songs and progressions that we regard as aesthetically significant indeed turn out in the process of modelling to exhibit mathematical significance.

mhh__ 8y ago

What does dual mean in the context of a chord? If it means parralel then why a minor triad (Presumably implied)

mikorym 8y ago
To answer that, you have to first answer what the dual means in the first place. And the notion that you get to is dual with respect to the octave. A usual chord is with respect to a fifth. Then a major third is a minor third down. Starting position should be at the tonic. Otherwise, you get at the fifth that when you go a major third down it is a minor third from the tonic.

mikorym 8y ago

This elaboration is late, and maybe will be read only in 10 years time by a unassuming bot scraping the web, but here goes... The musical scale (and a piano) is a logarithmic device. A logarithm by definition exhibits an interplay between addition and multiplication. Addition is repeated increment by one, whereas multiplication is repeated increment by some number. Raising to a power is repeated multiplication by a number. Repetition, or increment by one, is the basis for addition, leading to multiplication and then to the next step of raising to a power. (You can go beyond that again, but I forget what it is called.) In the form of an equation, you can express this as lnx + lny = ln(x*y). As it is a logarithmic device, in a musical scale you go up the scale in integer increments. What I mean by logarithmic is that, x +' 12 in the chromatic scale is equal in the actual frequency of the airwaves to 2 * x. I use +' here to denote addition along the scale. However, as we are working in the cyclic group of order 12 we have in fact that x +' 12 is congruent modulo 12 to x itself; i.e., the octave is the same note as the tonic. We are forgetting for a moment that some people would call the tonic for instance A and the octave A2 in favour of a simpler system for our current discussion. To recap, we already have a connection (by definition) between multiplication and addition in the structure that we call a logarithm and in music we can model the sound wave frequencies as a logarithm. Now, the next step is a more interesting step perhaps for a human that likes to listen to music. We already know as mentioned above that 7 chromatic steps up is a fifth and 5 chromatic steps down is a fifth as well. This is embodied by the cyclic nature of our model (a mathematical group structure). In the same way, in an analogue clock display, 12 + 7 hours is equal to 7 o'clock. 12 - 5 hours is also equal to 7 o'clock. However, time (if you experience it like me) moves forward linearly and indiscriminately. Even though they are inverses in a group, between the hours of 5 o'clock and 7 o'clock themselves, there is not a special physical connection. They are inverses simply due to the way we decided to partition the seconds in our days. (Incidentally, the human hearth rate is usually close to 60 bpm, but that phenomenon is perhaps for discussion another day.) So back to the scale: Within music there is a very special connection between the 7 step chromatic interval and the 5 step chromatic interval. The 7 step interval is a fifth and the 5 step interval is a fourth. There have been countless musical pieces that comprise only the tonic chord, the fourth and the fifth. You can apply duality on chords as well: using the cyclic group of order 7 we have that the major third is dual to the minor third (with respect to the fifth = 7 steps chromatic) giving that a major chord is dual to minor chord. One can continue in this fashion, using cyclic groups, and elements and their inverses will have special significance for music. Of particular note is that we can hear some of these connections, perhaps not rigorously or even explicitly, but indeed we know 1-4-5 is an important chord progression. I am tempted to say that this embodies a physical human interpretation of multiplication and addition, but one can equally argument that this is a physical human interpretation of duality, of group theory, or a very many other mathematical phenomena. We can speculate, but we do know for a fact that songs and progressions that we regard as aesthetically significant indeed turn out in the process of modelling to exhibit mathematical significance.