Mapping Periodicity
I’m collecting examples of regular two-dimensional mappings of linear, periodic data. If you know of any good examples, please let me know.
Here are the best examples I know about.
The earth is a three-dimensional object, but maps are two-dimensional. They have to discard an entire dimension in order to present the curved surface of the Earth in a conveniently-flat manner, and there are lots of projections than organize this discarding in a systematic manner.
But what if you want to go the other way ‘round? What if you have low-dimension data, and you want to display it at a higher dimensionality?
Chemistry provides a famous example. Each of its elements has an atomic number, corresponding to the number of protons in its nucleus. Atomic numbers form a one-dimensional continuum from 1 (Hydrogen) to at least 118 (Ununoctium). The continuum of atomic numbers has no inherent periodicity, but the physical properties of the atom impose a periodic structure on this otherwise-undifferentiated continuum of atomic numbers, as seen in the Periodic Table of the Elements.
Another example is the calendar. One can think of time as a one-dimensional count of days, such as the Julian Day Number, which has no inherent periodicity. Divide it up into 7-day weeks, however, and you get a calendar which is seven days wide and infinitely tall (usually clipped to display a single month’s days).
Music provides the third example (and the point of this article). The human ear has a hearing range that runs from about 20 Hz up to about 20,000 Hz. This one-dimensional range of frequencies has no inherent periodicity, but if two tones from a harmonic source are sounded together within that range, the coincidence of their harmonics makes some inter-frequency ratios sound more consonant than others, giving rise to the musical intervals recognized as octaves, perfect fifths, and so on. Using only the octave and tempered perfect fifth to generate all other tonal intervals, one can map linear frequency into a periodic set of intervallic relationships, as seen in isomorphic keyboards.
If linear data contains periodicity, then that periodicity can supply the information needed to provide an extra dimension.
This seems to me to be rather magical: An entire dimension of data conjured up out of thin air!
If you know of any other examples, please let me know! :-)
Here are the best examples I know about.
The earth is a three-dimensional object, but maps are two-dimensional. They have to discard an entire dimension in order to present the curved surface of the Earth in a conveniently-flat manner, and there are lots of projections than organize this discarding in a systematic manner.
But what if you want to go the other way ‘round? What if you have low-dimension data, and you want to display it at a higher dimensionality?
Chemistry provides a famous example. Each of its elements has an atomic number, corresponding to the number of protons in its nucleus. Atomic numbers form a one-dimensional continuum from 1 (Hydrogen) to at least 118 (Ununoctium). The continuum of atomic numbers has no inherent periodicity, but the physical properties of the atom impose a periodic structure on this otherwise-undifferentiated continuum of atomic numbers, as seen in the Periodic Table of the Elements.
Another example is the calendar. One can think of time as a one-dimensional count of days, such as the Julian Day Number, which has no inherent periodicity. Divide it up into 7-day weeks, however, and you get a calendar which is seven days wide and infinitely tall (usually clipped to display a single month’s days).
Music provides the third example (and the point of this article). The human ear has a hearing range that runs from about 20 Hz up to about 20,000 Hz. This one-dimensional range of frequencies has no inherent periodicity, but if two tones from a harmonic source are sounded together within that range, the coincidence of their harmonics makes some inter-frequency ratios sound more consonant than others, giving rise to the musical intervals recognized as octaves, perfect fifths, and so on. Using only the octave and tempered perfect fifth to generate all other tonal intervals, one can map linear frequency into a periodic set of intervallic relationships, as seen in isomorphic keyboards.
If linear data contains periodicity, then that periodicity can supply the information needed to provide an extra dimension.
This seems to me to be rather magical: An entire dimension of data conjured up out of thin air!
If you know of any other examples, please let me know! :-)
Labels: calendar, isomorphism, periodic table
posted by ThumMeister | 8:47 PM
1 Comments:
Here's my depiction of the pattern of musical harmonics, and why it is matches the Thummer layout - the combination makes quite a synergy.
The basis of harmony:
http://musicscienceguy.vox.com/library/post/the-simple-secret-of-harmony.html
Ken.
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