Equal-step tuning: Difference between revisions
Cmloegcmluin (talk | contribs) →Equal multiplications: add references to arithmetic tuning terms |
mNo edit summary |
||
| Line 24: | Line 24: | ||
For the mathematically inclined, this kind of diagram is closely related to [[The_Riemann_Zeta_Function_and_Tuning|the Riemann zeta function]]. | For the mathematically inclined, this kind of diagram is closely related to [[The_Riemann_Zeta_Function_and_Tuning|the Riemann zeta function]]. | ||
==Gallery of equal divisions== | == Gallery of equal divisions == | ||
===[[EdVI through edX|…of sixths through tenths]]=== | === [[EdVI through edX|…of sixths through tenths]] === | ||
* [[EdVI|''' | * [[EdVI|'''…of sixths (e. g. 5/3 and 11/7)''']] | ||
* [[EdVII|''' | * [[EdVII|'''…of sevenths (e. g. 9/5)''']] | ||
* [[EDO|''' | * [[EDO|'''...of the Octave/Duple (2/1)''']] | ||
* [[EdIX|'''…of ninths (e. g. 9/4)''']] | |||
* [[EdIX|''' | * [[EdX|'''…of tenths (e. g. 7/3)''']] | ||
* [[EdX|''' | |||
===[[EdXI and beyond|…of elevenths and beyond]]=== | ===[[EdXI and beyond|…of elevenths and beyond]]=== | ||
* [[EdXI|''' | * [[EdXI|'''…of elevenths (e. g. 8/3)''']] | ||
** | ** [[EDN|'''...of the Natave (e/1)''']] | ||
* [[EdXII|''' | * [[EdXII|'''…of twelfths (e. g. 3/1)''']] | ||
** [[ | ** [[EDT|'''...of the Tritave/Triple (3/1)''']] | ||
*** most famously the [[Bohlen-Pierce|BP]] scale, but lots of others, too | *** most famously the [[Bohlen-Pierce|BP]] scale, but lots of others, too | ||
* [[EdXIII|''' | * [[EdXIII|'''…of thirteenths (e. g. 10/3)''']] | ||
* [[EdXIV|''' | * [[EdXIV|'''…of fourteenths (e. g. 7/2)''']] | ||
* [[Ed4|''' | * [[Ed4|'''...of the Double Octave (4/1)''']] | ||
* | * [[EdXVI|'''…of sixteenths (e. g. 9/2)''']] | ||
* | * [[EdXVII|'''…of seventeenths (e. g. 5/1)''']] | ||
** [[Ed5|''' | ** [[Ed5|'''...of the Just Major 17th (5/1)''']] | ||
* | * [[EdXVIII|'''…of eighteenths (e. g. 11/2)''']] | ||
* | * [[EdXIX|'''…of nineteenths (e. g. 6/1)''']] | ||
** [[Ed6|''' | ** [[Ed6|'''...of the hexatave (6/1)''']] | ||
* [[Ed7|''' | * [[Ed7|'''...of the 7th Natural (7/1)''']] | ||
* [[Ed8|''' | * [[Ed8|'''...of the Triple Octave (8/1)''']] | ||
* [[Ed9|''' | * [[Ed9|'''...of the Double Tritave (9/1)''']] | ||
* [[Ed10|''' | * [[Ed10|'''...of the Just Major 24th (10/1)''']] | ||
* [[Ed11|''' | * [[Ed11|'''...of the 11th Natural (11/1)''']] | ||
* [[Ed12|''' | * [[Ed12|'''...of the Just Perfect 26th (12/1)''']] | ||
* [[Ed13|''' | * [[Ed13|'''...of the 13th Natural (13/1)''']] | ||
===[[EdV|...of fifths (e. g. 3/2 and 13/9)]]=== | === [[EdV|...of fifths (e. g. 3/2 and 13/9)]] === | ||
* [[EDF|''' | * [[EDF|'''...of the Perfect Fifth (3/2)''']] | ||
** most famously Carlos Alpha, Beta and Gamma, but lots of others, too | ** most famously Carlos Alpha, Beta and Gamma, but lots of others, too | ||
* ''' | * '''...of the Tridecimal High Tritone (13/9)''' | ||
** 11 - [[11ed13/9|Eleventh root of 13 over 9]] | ** 11 - [[11ed13/9|Eleventh root of 13 over 9]] | ||
* ''' | * '''...of the Septimal High Tritone (10/7)''' | ||
** 8 - [[8ed10/7|Eighth root of 10 over 7]] | ** 8 - [[8ed10/7|Eighth root of 10 over 7]] | ||
===[[EdIV|...of fourths (e. g. 4/3 and 15/11)]]=== | === [[EdIV|...of fourths (e. g. 4/3 and 15/11)]] === | ||
* ''' | * '''...of the Perfect Fourth (4/3)''' | ||
** 3 - [[Cube_Root_of_P4|Cube Root of 4/3]] | ** 3 - [[Cube_Root_of_P4|Cube Root of 4/3]] | ||
** 9 - '[[Noleta]]' Scale | ** 9 - '[[Noleta]]' Scale | ||
* ''' | * '''...of the Septimal Narrow Tritone (7/5)''' | ||
** 4 - [[4ed7/5|Fourth root of 7/5]] | ** 4 - [[4ed7/5|Fourth root of 7/5]] | ||
** 11 - [[11ed7/5|Eleventh root of 7/5]] | ** 11 - [[11ed7/5|Eleventh root of 7/5]] | ||
** 13 - [[13ed7/5|Thirteenth root of 7/5]] | ** 13 - [[13ed7/5|Thirteenth root of 7/5]] | ||
* ''' | * '''...of the Undecimal Semiaugmented Fourth (15/11)''' | ||
** 13 - [[13ed15/11|Thirteenth root of 15/11]] | ** 13 - [[13ed15/11|Thirteenth root of 15/11]] | ||
* ''' | * '''...of the Tridecimal Ultramajor Third (13/10)''' | ||
** 2 - [[Square root of 13 over 10]] | ** 2 - [[Square root of 13 over 10]] | ||
===[[EdIII|...of thirds (e. g. 9/7, 5/4, 11/9, and 6/5)]]=== | ===[[EdIII|...of thirds (e. g. 9/7, 5/4, 11/9, and 6/5)]]=== | ||
* [[Ed5/4|''' | * [[Ed5/4|'''...of the Just Major Third (5/4)''']] | ||
** 2 - [[2ed5/4|Square Root of 5/4]] | ** 2 - [[2ed5/4|Square Root of 5/4]] | ||
** 3 - [[3ed5/4|Cube Root of 5/4]] | ** 3 - [[3ed5/4|Cube Root of 5/4]] | ||
| Line 86: | Line 85: | ||
===[[Equal divisions of the whole tone|...of whole tones (e. g. 8/7, 9/8, and 10/9)]]=== | ===[[Equal divisions of the whole tone|...of whole tones (e. g. 8/7, 9/8, and 10/9)]]=== | ||
* ''' | * '''...of the Septimal Whole Tone (8/7)''' | ||
** 5 - [[5ed8/7|Fifth root of 8/7]] | ** 5 - [[5ed8/7|Fifth root of 8/7]] | ||
* ''' | * '''...of the Pythagorean Whole Tone (9/8)''' | ||
** 9 - [[9ed9/8|Ninth root of 9/8]] | ** 9 - [[9ed9/8|Ninth root of 9/8]] | ||
===[[Equal divisions of the semitone|...of semitones (e. g. 15/14, 16/15, and 25/24)]]=== | ===[[Equal divisions of the semitone|...of semitones (e. g. 15/14, 16/15, and 25/24)]]=== | ||
* ''' | * '''...of the Classic Diatonic Semitone (16/15)''' | ||
** 8 - [[Delta scale]] | ** 8 - [[Delta scale]] | ||
==Equal multiplications== | == Equal multiplications == | ||
An equal multiplication of a rational interval can also be called an [[AS|ambitonal sequence (AS)]]. For example, "25/24s equal temperament" could also be written "AS25/24". | An equal multiplication of a rational interval can also be called an [[AS|ambitonal sequence (AS)]]. For example, "25/24s equal temperament" could also be written "AS25/24". | ||
| Line 102: | Line 100: | ||
=== ...of a given cents value === | === ...of a given cents value === | ||
* [[65cET]] | * [[65cET]] | ||
* [[88cET]] | * [[88cET]] | ||
| Line 109: | Line 106: | ||
=== ...of various whole tones === | === ...of various whole tones === | ||
* ''' | * '''...of true whole tones''' | ||
** [[8/7ths equal temperament]] - multiples of [[8/7]] | ** [[8/7ths equal temperament]] - multiples of [[8/7]] | ||
** [[9/8ths equal temperament]] - multiples of [[9/8]] | ** [[9/8ths equal temperament]] - multiples of [[9/8]] | ||
** [[10/9ths equal temperament]] - multiples of [[10/9]] | ** [[10/9ths equal temperament]] - multiples of [[10/9]] | ||
* ''' | * '''...of wolf whole tones''' | ||
** [[11/10ths equal temperament]] - multiples of [[11/10]] | ** [[11/10ths equal temperament]] - multiples of [[11/10]] | ||
** [[12/11ths equal temperament]] - multiples of [[12/11]] | ** [[12/11ths equal temperament]] - multiples of [[12/11]] | ||
** [[13/12ths equal temperament]] - multiples of [[13/12]] | ** [[13/12ths equal temperament]] - multiples of [[13/12]] | ||
=== ...of various semitones === | === ...of various semitones === | ||
* ''' | * '''...of diatonic semitones''' | ||
** [[14/13ths equal temperament]] - multiples of [[14/13]] | ** [[14/13ths equal temperament]] - multiples of [[14/13]] | ||
** [[15/14ths equal temperament]] - multiples of [[15/14]] | ** [[15/14ths equal temperament]] - multiples of [[15/14]] | ||
** [[16/15ths equal temperament]] - multiples of [[16/15]] | ** [[16/15ths equal temperament]] - multiples of [[16/15]] | ||
** [[17/16ths equal temperament]] - multiples of [[17/16]] | ** [[17/16ths equal temperament]] - multiples of [[17/16]] | ||
* ''' | * '''...of chromatic semitones''' | ||
** [[18/17s equal temperament]] - multiples of [[18/17]] | ** [[18/17s equal temperament]] - multiples of [[18/17]] | ||
** [[19/18s equal temperament]] - multiples of [[19/18]] | ** [[19/18s equal temperament]] - multiples of [[19/18]] | ||
| Line 133: | Line 130: | ||
** [[25/24s equal temperament]] - multiples of [[25/24]] | ** [[25/24s equal temperament]] - multiples of [[25/24]] | ||
=== ...of various microtones === | === ...of various microtones === | ||
* ''' | * '''...of third tones''' | ||
** [[26/25s equal temperament]] - multiples of [[26/25]] | ** [[26/25s equal temperament]] - multiples of [[26/25]] | ||
** [[27/26s equal temperament]] - multiples of [[27/26]] | ** [[27/26s equal temperament]] - multiples of [[27/26]] | ||
** [[28/27s equal temperament]] - multiples of [[28/27]] | ** [[28/27s equal temperament]] - multiples of [[28/27]] | ||
* ''' | * '''...of quarter tones''' | ||
** [[33/32s equal temperament]] - multiples of [[33/32]] | ** [[33/32s equal temperament]] - multiples of [[33/32]] | ||
** [[36/35s equal temperament]] - multiples of [[36/35]] | ** [[36/35s equal temperament]] - multiples of [[36/35]] | ||
==See also== | == See also == | ||
* [[Edo anatomy]] | * [[Edo anatomy]] | ||
* [[Macrotonal edos]] | * [[Macrotonal edos]] | ||