Harmonotonic tuning: Difference between revisions
Cmloegcmluin (talk | contribs) →Table of monotonic tunings: links |
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* A diatonic tuning is '''not''' monotonic because it goes back and forth between whole and half steps. | * A diatonic tuning is '''not''' monotonic because it goes back and forth between whole and half steps. | ||
* A segment of the | * A segment of the overtone series '''is''' monotonic because its steps always decrease in size (within the interval of repetition). | ||
* An EDO tuning '''is''' monotonic because the steps are all the same size. | * An EDO tuning '''is''' monotonic because the steps are all the same size. | ||
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Here are the three different '''shapes''', according to their pitches sorted in ascending order: | Here are the three different '''shapes''', according to their pitches sorted in ascending order: | ||
# decreasing step size (e.g. | # decreasing step size (e.g. overtone series) | ||
# equal step size (e.g. EDO) | # equal step size (e.g. EDO) | ||
# increasing step size (e.g. | # increasing step size (e.g. undertone series) | ||
And here are the three different '''types''': | And here are the three different '''types''': | ||
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Basic examples of arithmetic tunings: | Basic examples of arithmetic tunings: | ||
# the | # the overtone series has equal steps of frequency (1, 2, 3, 4, etc.; adding 1 each step) | ||
# any EDO has equal steps of pitch (12-EDO goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step) | # any EDO has equal steps of pitch (12-EDO goes 0/12, 1/12, 2/12, 3/12, etc.; adding 1/12 each step) | ||
# the | # the undertone series has equal steps of length (to play the first four steps of the undertone series you would pluck the whole length of a string, then 3/4 the string, then 2/4, then 1/4; adding -1/4 length each step) | ||
=== Sequences === | === Sequences === | ||
Other arithmetic tunings can be found by changing the step size. For example, if you vary the | Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <span><math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math></span>, or in other words, a class iii [[isoharmonic_chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4. | ||
If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ — 1, 1+φ, 1+2φ, 1+3φ, etc. — we could have the AFSφ. | If the new step size is irrational, the tuning is no longer JI, so we use a different acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ — 1, 1+φ, 1+2φ, 1+3φ, etc. — we could have the AFSφ. | ||
OS and AFS are equivalent to taking | OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see Derivation of OS. | ||
The same principles that were just described for frequency are also possible for length. By varying the | The same principles that were just described for frequency are also possible for length. By varying the undertone series step size to some rational number you can produce a utonal sequence (US), and varying it to an irrational number you can produce an arithmetic length sequence (ALS). Analogously, by shifting the undertone series by a constant amount of string length, the step sizes remain equal in terms of length, but alter their relationship in pitch. | ||
=== Divisions === | === Divisions === | ||
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== Non-arithmetic tunings == | == Non-arithmetic tunings == | ||
We've shown that new arithmetic tunings can found by adding (or subtracting) a constant amount of frequency from the | We've shown that new arithmetic tunings can found by adding (or subtracting) a constant amount of frequency from the overtone series. But addition is not the only operation we could try applying to the frequencies of a basic monotonic overtone series. | ||
The next operation above addition is multiplication. This operation is not very interesting, however, because multiplying frequency is equivalent to adding pitch, which does not meaningfully change a tuning; this merely transposes it. The reason multiplying frequency is equivalent to adding pitch is because pitch is found by taking the logarithm of frequency, and taking the logarithm of something effectively gears it down one operation lower on the hierarchy of operations: addition, multiplication, exponentiation, tetration, etc. | The next operation above addition is multiplication. This operation is not very interesting, however, because multiplying frequency is equivalent to adding pitch, which does not meaningfully change a tuning; this merely transposes it. The reason multiplying frequency is equivalent to adding pitch is because pitch is found by taking the logarithm of frequency, and taking the logarithm of something effectively gears it down one operation lower on the hierarchy of operations: addition, multiplication, exponentiation, tetration, etc. | ||
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The next operation above multiplication is exponentiation. Exponentiating frequency is equivalent to multiplying pitch. Multiplying all pitch values does give you meaningfully new tunings. However, it does not preserve the arithmetic quality of a tuning for frequency or for pitch. So, these are now non-arithmetic tunings. | The next operation above multiplication is exponentiation. Exponentiating frequency is equivalent to multiplying pitch. Multiplying all pitch values does give you meaningfully new tunings. However, it does not preserve the arithmetic quality of a tuning for frequency or for pitch. So, these are now non-arithmetic tunings. | ||
For example, we could start with the | For example, we could start with the overtone series, then take the square root of all the frequencies. This results in something like the overtone series, except you don't reach the 2nd harmonic until the 4th step, the 3rd harmonic until the 9th step, or the 4th harmonic until the 16th step, etc. Because the square root is the same as raising to the power of 1/2, this is equivalent to multiplying all pitches by 1/2 (i.e. dividing them by 2). We could call this the 1/2-powharmonic series. | ||
The next operation above exponentiation is tetration. Tetrating frequency is equivalent to exponentiating pitch. This operation and beyond will not be explored here. | The next operation above exponentiation is tetration. Tetrating frequency is equivalent to exponentiating pitch. This operation and beyond will not be explored here. | ||
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! rowspan="7" |'''decreasing''' | ! rowspan="7" |'''decreasing''' | ||
'''step size''' | '''step size''' | ||
|'''[[Overtone series|overtone series, or harmonic series]]'''||'''shifted | |'''[[Overtone series|overtone series, or harmonic series]]'''||'''shifted overtone series''' (± frequency) | ||
''(equivalent to AFS)'' | ''(equivalent to AFS)'' | ||
| '''stretched/compressed | | '''stretched/compressed overtone series''' (exponentiated frequency, multiplied pitch) ''(equivalent to powharmonic series)'' | ||
|- | |- | ||
|'''[[Overtone scale#Over-n Scales| | |'''[[Overtone scale#Over-n Scales|overtone mode, or over-n scale]]''' ''(equivalent to n-ODO)'' | ||
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! rowspan="7" |'''increasing''' | ! rowspan="7" |'''increasing''' | ||
'''step size''' | '''step size''' | ||
| '''[[wikipedia:Undertone_series|undertone series, or subharmonic series]]''' || '''shifted | | '''[[wikipedia:Undertone_series|undertone series, or subharmonic series]]''' || '''shifted undertone series''' (± frequency) ''(equivalent to ALS)'' || '''stretched/compressed undertone series''' (exponentiated frequency, multiplied pitch) ''(equivalent to subpowharmonic series)'' | ||
|- | |- | ||
|'''[[Overtone scale#Next Steps| | |'''[[Overtone scale#Next Steps|undertone mode, or under-n scale]]''' ''(equivalent to n-UDO)'' | ||
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== Derivation of OS == | == Derivation of OS == | ||
The tuning OS3/4 is the sequence <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...</math></span> and so on. Any OS is equivalent to shifting the | The tuning OS3/4 is the sequence <span><math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...</math></span> and so on. Any OS is equivalent to shifting the overtone series by a constant amount of frequency. In the case of OS3/4, it is a shift by <span><math>\frac 13</math></span>. Let's show how. | ||
Begin with the | Begin with the overtone series: | ||
<math>1, 2, 3, 4...</math> | <math>1, 2, 3, 4...</math> | ||