Xenharmonic Wiki

Harmonotonic tuning: Difference between revisions

Newer edit →
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,160 edits
just getting something down on the page
 
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,160 edits
→Gallery of monotonic tunings: 2nd wave of first major putting it out there
Line 31: Line 31:
# the subharmonic series has equal steps of length (to play the first four steps of the subharmonic 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)
# the subharmonic series has equal steps of length (to play the first four steps of the subharmonic 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 ===
Other arithmetic tunings can be found by changing the step size. For example, if you vary the harmonic 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 tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.  
Other arithmetic tunings can be found by changing the step size. For example, if you vary the harmonic 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 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: EFS, for equal 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 EFSφ.  
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 EFS are equivalent to taking a harmonic 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.
OS and AFS are equivalent to taking a harmonic 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 subharmonic 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 equal length sequence (ELS). Analogously, by shifting the subharmonic series by a constant amount of string length, the step sizes remain equal in terms of length, but alter their relationship in pitch.
The same principles that were just described for frequency are also possible for length. By varying the subharmonic 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 subharmonic series by a constant amount of string length, the step sizes remain equal in terms of length, but alter their relationship in pitch.
 
=== Divisions ===
If an arithmetic tuning has equal step sizes of some kind of quantity, then an arithmetic tuning can also be produced by taking an interval and equally dividing it according to that kind of quantity.
 
The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EDPO, for 12 equal divisions of the pitch of the octave. Whenever pitch is the pertinent kind of quality, we can assume it, and skip pointing it out. That's why 12-EDO is the better name.
 
But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by frequency, or length. In the former case, you will have 12-EFDO, and in the latter case, you will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12-EFDφ.  


== Non-arithmetic tunings ==
== Non-arithmetic tunings ==
Line 49: Line 57:
For example, we could start with the harmonic series, then take the square root of all the frequencies. This results in something like the harmonic 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.
For example, we could start with the harmonic series, then take the square root of all the frequencies. This results in something like the harmonic 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.
Adding frequency is called shifting a tuning. Exponentiating frequency (or multiplying pitch) is called stretching (or compressing) a tuning.
Here is a table to illustrate:
{| class="wikitable"
|+
!
!frequency
!pitch
|-
|shifting
|addition
|[[wikipedia:Gaussian_logarithm|Gaussian logarithm]]
|-
|transposition
|multiplication
|addition
|-
|stretching
|exponentiation
|multiplication
|-
|...
|[[wikipedia:Tetration|tetration]]
|exponentiation
|}
All powharmonic tunings are monotonic, but non-arithmetic and ir-rational.   
All powharmonic tunings are monotonic, but non-arithmetic and ir-rational.   


== Gallery of monotonic tunings ==
== Table of monotonic tunings ==


{| class="wikitable"
{| class="wikitable"
|+ Caption text
|+ Caption text
!
!
! colspan="3" |tuning type
|-
!
!
! arithmetic
rational
! arithmetic
irrational
! non-arithmetic
irrational
|-
| rowspan="17" |'''tuning'''
'''shape'''
| rowspan="7" |'''decreasing'''
'''step size'''
| '''harmonic series''' || '''shifted harmonic series'''
(± frequency)
''(equivalent to AFS)''
| '''stretched/compressed harmonic series''' (exponentiated frequency, multiplied pitch) ''(equivalent to powharmonic series)''
|-
|'''harmonic mode''' ''(equivalent to n-ODO)''
|
|
|-
|'''n-ODp:''' <u>n</u> <u>o</u>tonal <u>d</u>ivisions of interval <u>p</u>
|'''n-EFDp:''' <u>n</u> <u>e</u>qual <u>f</u>requency <u>d</u>ivisions of interval <u>p</u>
|
|-
|'''(n-)OSp:''' (<u>n</u> pitches of an) <u>o</u>tonal <u>s</u>equence adding by <u>p</u>
|'''(n-)AFSp:''' (n pitches of an) arithmetic frequency sequence
adding by p
|
|-
|
|
|c-powharmonic series exponent c
|-
|
|
|b-logharmonic series base b
|-
|n-ADO: arithmetic division of octave (equivalent to n-ODO)
|
|
|-
| rowspan="3" |'''equal'''
'''step size'''
| '''1D JI lattice'''
| rank-1 temperament || rowspan="3" |
|-
|
|n-EDp: n equal (pitch) divisions of interval p (e.g. 12-EDO) (equivalent to rank-1 temperament of p/n)
|-
|'''(n-)ASp:''' (n pitches of an) ambitonal sequence adding by p ''(equivalent to 1D JI lattice of p)''
|(n-)APSp: (n pitches of an) arithmetic pitch sequence adding by p (equivalent to rank-1 temperament with generator p)
|-
| rowspan="7" |'''increasing'''
'''step size'''
| subharmonic series || shifted subharmonic series (±Hz) (equivalent to ALS) || stretched/compressed subharmonic series (multiplied by cents) (equivalent to subpowharmonic series)
|-
|subharmonic mode (equivalent to n-UDO)
|
|
|-
|n-UDp: n utonal divisions of interval p
|n-ELDp: n equal length divisions of interval p
|
|-
|-
! Header text !! Header text !! Header text
|(n-)USp: (n pitches of a) utonal sequence adding by p
|(n-)ALSp: (n pitches of an) arithmetic length sequence adding by p
|
|-
|-
| Example || Example || Example
|
|
|c-subpowharmonic series exponent c
|-
|-
| Example || Example || Example
|
|
|b-sublogharmonic series base b
|-
|-
| Example || Example || Example
|EDL: equal division of length (equivalent to n-UDn)
|
|
|}
|}


Retrieved from "https://en.xen.wiki/w/Harmonotonic_tuning"