Editing Tritone (section)

== Size in different tuning systems ==
{{multiple image
 | align = right
 | direction = vertical
 | width = 220
 | header = Tritones
 | image1 = Just augmented fourth on C.png
 | caption1 = [[major fourth|just augmented fourth]] between C and F{{music|sharp}}+ – 45:32 (590.22 cents)[[File:Just augmented fourth on C.mid]]
 | image2 = Pythagorean augmented fourth on C.png
 | caption2 = Pythagorean augmented fourth between C and F{{music|sharp}}++ – 729:512 (611.73 cents)[[File:Pythagorean augmented fourth on C.mid]]
 | image3 = Augmented fourth on C.png
 | caption3 = Classic augmented fourth between C and F{{music|sharp}} – 25:18 (568.72 cents)[[File:Classic augmented fourth on C.mid]]
 | image4 = Diminished fifth on C.png
 | caption4 = Classic diminished fifth between C and G{{music|flat}} – 36:25 (631.28 cents) [[File:Classic diminished fifth on C.mid]]
 | image5 = Lesser septimal tritone on C.png
 | caption5 = Lesser septimal tritone between C and G{{music|7}}{{music|b}}<ref name="Fonville">{{cite journal|last1=Fonville|first1=John|author-link=John Fonville|title=Ben Johnston's Extended Just Intonation: A Guide for Interpreters|journal=[[Perspectives of New Music]]|date=1991|volume=29|issue=2|pages=106–137|doi=10.2307/833435|jstor=833435}}</ref> – 7:5 (582.51 cents)[[File:Lesser septimal tritone on C.mid]]
 | image6 = Comparison of tritones.png
 | caption6 = Comparison of intervals near or enharmonic with the tritone
}}

{{stack|[[File:Tritone1.mid|thumb|Tritone]]}}
In twelve-tone equal temperament, the Aug&nbsp;4 is exactly half an [[octave]] (i.e., a ratio of [[square root of 2|{{sqrt|2}}]]:1 or 600&nbsp;[[cent (music)|cents]]. The inverse of 600&nbsp;cents is 600&nbsp;cents. Thus, in this tuning system, the Aug&nbsp;4 and its inverse (dim&nbsp;5) are [[Enharmonic|equivalent]].

The half-octave or equal tempered Aug&nbsp;4 and dim&nbsp;5 are unique in being equal to their own inverse (each to the other). In other [[meantone temperament|meantone]] tuning systems, besides 12&nbsp;tone equal temperament, Aug&nbsp;4 and dim&nbsp;5 are distinct intervals because neither is exactly half an octave. In any meantone tuning near to {{sfrac|2|9}}-comma meantone the Aug&nbsp;4 is near to the ratio&nbsp;7:5 (582.51) and the dim&nbsp;5 to 10:7 (617.49), which is what these intervals are in [[septimal meantone temperament]]. In [[31 equal temperament]], for example, the Aug&nbsp;4 is 580.65&nbsp;cents, whereas the dim&nbsp;5 is 619.35 cents. This is perceptually indistinguishable from septimal meantone temperament.

Since they are the inverse of each other, by definition Aug&nbsp;4 and dim&nbsp;5 always add up (in cents) to exactly one [[octave|perfect octave]]:
: ''Aug&nbsp;4 + dim&nbsp;5 = Perf&nbsp;8''.
On the other hand, two Aug&nbsp;4 add up to six whole tones. In equal temperament, this is equal to exactly one perfect octave:
: ''Aug&nbsp;4 + Aug&nbsp;4 = Perf&nbsp;8''.
In [[quarter-comma meantone]] temperament, this is a [[diesis]] (128:125) less than a perfect octave:
: ''Aug&nbsp;4 + Aug&nbsp;4 = Perf&nbsp;8 − [[diesis]]''.

{{stack|[[File:Just diminished fifth on C.mid|thumb|Just diminished fifth on C]]}}
In [[just intonation]] several different sizes can be chosen both for the Aug&nbsp;4 and the dim&nbsp;5. For instance, in [[5-limit tuning]], the Aug&nbsp;4 is either 45:32<ref name=Partch>{{cite book |author-link=Harry Partch |last=Partch |first=H. |year=1979 |orig-year=1974 |title=Genesis of a Music: An account of a creative work, its Roots and its fulfillments |title-link=Genesis of a Music |edition=2nd |place=New York, NY |publisher=Da Capo Press |page=69 |ISBN=0-306-80106-X }}
:{{cite book |section=''Genesis of a Music'' |year=1979 |title=scanned copy |edition=2nd |type=online |url=https://pearl-hifi.com/06_Lit_Archive/02_PEARL_Arch/Vol_16/Sec_51/4555_Genesis_of_a_Music_2nd_Edn.pdf |via=pearl-hifi.com |access-date=22 July 2021 }}</ref><ref name=Renold>{{cite book |last=Renold |first=Maria |year=2004 |title=Intervals, Scales, Tones, and the Concert Pitch {{nobr|&hairsp; C {{=}} 128 Hz }} |quote=translated from German |lang=en |others=Stevens, Bevis (translator) ; Meuss, Anna R. (additional editing) |place=Forest Row |publisher=Temple Lodge |ISBN=1-902636-46-5 |pages=15–16 }}</ref><ref name=Helmholtz>{{cite book |last=Helmholtz |first=H. |author-link=Hermann von Helmholtz |editor-first=A.J. |editor-last=Ellis |editor-link=Alexander J. Ellis |year=2005 |orig-year=1875, 1st&nbsp;Engl. |title=On the Sensations of Tone as a Physiological Basis for the Theory of Music |title-link=On Sensations of Tone |edition=reprint |page=457 |ISBN=1-4191-7893-8 |quote=Name of interval: ''Just Tritone'', cents in interval: 590, number to an octave: 2&nbsp;; Name of interval: ''Pyth. Tritone'',  cents in interval: 612, number to an octave: 2&nbsp; }}</ref> or 25:18,<ref name="Haluska 24">{{cite book |last=Haluska |first=Ján |year=2003 |title=The Mathematical Theory of Tone Systems |series=Pure and Applied Mathematics Series |volume=262 |place=New York / London |publisher=Marcel Dekker / Momenta |ISBN=0-8247-4714-3 |page={{mvar|xxiv}}  |quote=25:18 classic augmented fourth }}</ref> and the dim&nbsp;5 is either 64:45 or 36:25.<ref name="Haluska 25">{{harvp|Haluska|2003|p=&nbsp;{{mvar|xxv}} }} "36:25 classic diminished fifth".</ref> The 64:45 just diminished fifth arises in the C [[major scale]] between B and F, consequently the 45:32 augmented fourth arises between F and B.<ref>{{cite book |last=Paul |first=Oscar |year=1885  |quote=musical interval 'pythagorean major third' |title=A Manual of Harmony for use in Music-Schools and Seminaries, and for Self-Instruction |page=165 |publisher=Theodore Baker |translator-first=((Gustav, Sr.)) |translator-last=Schirmer |translator-link=G. Schirmer, Inc. |url=https://archive.org/details/bub_gb_4WEJAQAAMAAJ |via=archive.org }}</ref>

These ratios are not in all contexts regarded as [[Five-limit tuning#The just ratios|strictly just]] but they are the justest possible in 5-limit tuning. [[7-limit]] tuning allows for the justest possible ratios (ratios with the smallest numerator and denominator), namely 7:5 for the Aug&nbsp;4 (about 582.5 cents, also known as [[septimal tritone]]) and 10:7 for the dim&nbsp;5 (about 617.5 cents, also known as Euler's tritone).<ref name=Partch/><ref name="Haluska 23">{{harvp|Haluska|2003|p={{mvar|xxiii}} }} "7:5 septimal or Huygens' tritone, Bohlen-Pierce fourth", "10:7 Euler's tritone".</ref><ref name=Strange>{{cite book |last1=Strange |first1=Patricia |last2=Patricia |first2=Allen |year=2001 |title=The Contemporary Violin: Extended performance techniques |page=147 |ISBN=0-520-22409-4 |quote= ...&nbsp;septimal tritone, 10:7; smaller septimal tritone, 7:5; ... This list is not exhaustive, even when limited to the first sixteen partials. Consider the very narrow augmented fourth, 13:9. ... just intonation is not an attempt to generate necessarily consonant intervals. }}</ref>
These ratios are more consonant than 17:12 (about 603.0&nbsp;[[musical cents|cents]]) and 24:17 (about 597.0&nbsp;cents), which can be obtained in 17&nbsp;limit tuning, yet the latter are also fairly common, as they are closer to the equal-tempered value of 600&nbsp;cents.