1/6-comma meantone: Difference between revisions

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'''1/6 comma meantone''' is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the syntonic comma ([[81/80]] ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to [[55edo]] in terms of its tuning.
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| de = Sechstelkomma-mitteltönig
| en = 1/6-comma meantone
| es =
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}}
'''1/6-comma meantone''' is the tuning of [[meantone]] temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the [[81/80|syntonic comma (81/80)]] of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning [[tempering out]] 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly. Leopold Mozart and Wolfgang Amadeus Mozart recommended this tuning (implemented as 55edo or something close to it), with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.<ref>Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]</ref>


== Fractional projection matrix ==
== Tuning profile ==
The [[Fractional monzos|fractional projection matrix]] defining [[7-limit]] 1/6 comma meantone is
[[Projection map]]:


{| class="wikitable"
{| class="right-all"
|-
|-
| [1
| [⟨ || 1 || 2/3 || -4/3 || -19/3 || ]
| 0
| 0
| 0&gt;
|-
|-
| [2/3
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
| 1/3
| 1/6
| 0&gt;
|-
|-
| [-4/3
| ⟨ || 0 || 1/6 || 2/3 || 5/3 || ]
| 4/3
| 2/3
| 0&gt;
|-
|-
| [-19/3
| ⟨ || 0 || 0 || 0 || 0 || ]]
| 10/3
| 5/3
| 0&gt;
|}
|}
[[Tuning map]]: {{val| 1200 1898.3706 2793.4825 3383.7062 }}
[[Error map]]: {{val| 0 -3.5844 +7.1688 +14.8803 }}


== Links ==
== Links ==
* [http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]
* ''Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone'' by [[Ross W. Duffin]] ([https://www.webcitation.org/5zW8FuybZ WebCite])
* [http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joe Monzo]] [http://www.webcitation.org/5zW910Jax permalink]
* ''Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone'' by [[Joseph Monzo]] ([https://web.archive.org/web/20120214163510/sonic-arts.org/monzo/55edo/55edo.htm Internet Archive] [https://www.webcitation.org/5zW910Jax WebCite])
 
== References ==
<references />


[[Category:1/6-comma]]
[[Category:Meantone]]
[[Category:Meantone]]
[[Category:Historical]]

Latest revision as of 21:24, 15 February 2025

1/6-comma meantone is the tuning of meantone temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. 55edo and 67edo approximate it flatly and sharply, respectively, while 122edo using the c val does so near perfectly. Leopold Mozart and Wolfgang Amadeus Mozart recommended this tuning (implemented as 55edo or something close to it), with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.[1]

Tuning profile

Projection map:

[⟨ 1 2/3 -4/3 -19/3 ]
⟨ 0 1/3 4/3 10/3 ]
⟨ 0 1/6 2/3 5/3 ]
⟨ 0 0 0 0 ]]

Tuning map: ⟨1200 1898.3706 2793.4825 3383.7062]

Error map: ⟨0 -3.5844 +7.1688 +14.8803]

Links

References

  1. ↑ Chesnut, John (1977) Mozart's Teaching of Intonation, Journal of the American Musicological Society Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) doi.org/10.2307/831219, https://www.jstor.org/stable/831219
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