In classical music from Western culture, the tritone ( Play (help·info), tri- "three" and tone) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones. In this context, a tritone may also be defined as any interval spanning six semitones.
Since a chromatic scale is formed by 12 pitches, it contains 12 distinct tritones, each of which starting from a different pitch. According to a widely used naming convention, six of them are classified as augmented fourths, and the other six as diminished fifths. In a diatonic scale there is only one tritone, classified as an augmented fourth. For instance, in the C major diatonic scale the only interval formed by three adjacent tones (F-G, G-A, and A-B) is that from F to B.
In the above-mentioned naming convention, a fourth is an interval encompassing four staff positions, while a fifth encompasses five staff positions (see interval number for more details). The augmented fourth (A4) and diminished fifth (d5) are defined as the intervals produced by widening and narrowing by one chromatic semitone the perfect fourth and fifth, respectively.[2] They both span six semitones, and they are the inverse of each other, meaning that their sum is exactly equal to one perfect octave (A4 + d5 = P8). In 12-tone equal temperament, the most commonly used tuning system, the A4 is equivalent to a d5, as both have the size of exactly half an octave. In most other tuning systems, they are not equivalent, and neither is equal to half an octave. The d5 is also called semidiapente.
The tritone is often used as the main interval of dissonance in Western harmony, and is important in the study of musical harmony. "Any tendency for a tonality to emerge may be avoided by introducing a note three whole tones distant from the key note of that tonality.
Definitions
A tritone (abbreviation: TT) is traditionally defined as a musical interval composed of three whole tones. As the symbol for whole tone is T, this definition may be also written as follows:
TT = T+T+T
Only if the three tones are of the same size (which is not the case for many tuning systems) can this formula be simplified to:
TT = 3T
This definition, however, has two different interpretations (broad and strict).
Broad interpretation (chromatic scale)
If a chromatic scale is used, with its 12 notes it is possible to define 12 different tritones, each of which starting from a different note. Six of them are A4, and the other six are d5. Therefore, in this case both A4 and d5 are considered to be tritones. Since each whole tone, in a chromatic scale, can be divided into two semitones:
T = S+S
then three tones are equal to six semitones. In this case, we can generalize the definition of tritone as follows:
TT = T+T+T = S+S+S+S+S+S.
This means that a tritone can be also defined as any musical interval spanning six semitones (indeed, both A4 and d5 are intervals spanning 6 semitones).
Only when the semitones (and the tones formed by pairs of semitones) are equal in size can this formula be simplified to:
TT = 3T = 6S.
Strict interpretation (diatonic scale)
In a diatonic scale, whole tones are regarded as incomposite intervals (that is, they do not divide into smaller intervals). Therefore, in this context the above mentioned "decomposition" of the tritone into six semitones is typically not allowed.
If a diatonic scale is used, with its 7 notes it is possible to form only one sequence of three adjacent whole tones (T+T+T). This interval is an A4, and it is sometimes called the proper tritone. For instance, in the C major diatonic scale (C-D-E-F-G-A-B-...), the only tritone is from F to B. It is a tritone because F-G, G-A, and A-B are three adjacent whole tones. It is a fourth because the notes from F to B are four (F, G, A, B). It is augmented (i.e., widened) because most of the fourths found in the scale have smaller size (they are perfect fourths).
According to this interpretation, the d5 is not a tritone. Indeed, in a diatonic scale, there's only one d5, and this interval does not meet the strict definition of tritone, as it is formed by one semitone, two whole tones, and another semitone:
d5 = S+T+T+S.
For instance, in the C major diatonic scale, the only d5 is from B to F. It is a fifth because the notes from B to F are five (B, C, D, E, F). It is diminished (i.e. narrowed) because most of the fifths found in the scale have larger size (they are perfect fifths).
Alternative definition
Some contemporary authors define a tritone as any interval spanning exactly half an octave, including both the A4 and d5 as tuned in 12-tone equal temperament. This is not consistent with the above mentioned traditional definition (TT = T+T+T).
In this case, context may resolve the tritone to more an A4, a d5, or a "neutral" interval with no clear conventional classification.
Size in different tuning systems
In 12-tone equal temperament, the A4 is exactly half an octave (i.e., a ratio of √2:1 or 600 cents; play (help·info)). The inverse of 600 cents is 600 cents. Thus, in this tuning system, the A4 and its inverse (d5) are equivalent.
The half-octave A4 is unique in being equal to its own inverse. In other meantone tuning systems, besides 12-tone equal temperament, A4 and d5 are distinct intervals because neither is exactly half an octave. In any meantone tuning near to 2⁄9-comma meantone the A4 will be near to the ratio 7⁄5 and the d5 to 10⁄7, which is what these intervals are taken to be in septimal meantone temperament. In 31 equal temperament, for example, the A4, a 10⁄7, or tritone proper, is 617.49 cents, whereas a 7⁄5 is 582.51 cents. This is perceptually indistinguishable from septimal meantone temperament.
Since they are the inverse of each other, by definition A4 and d5 always add up to exactly one perfect octave:
A4 + d5 = P8.
On the other hand, two A4 (proper tritones) add up to six whole tones. In equal temperament, this is equal to exactly one perfect octave:
A4 + A4 = P8.
In quarter-comma meantone temperament, this is a diesis (128/125) less than a perfect octave:
A4 + A4 = P8 - diesis.
In just intonation several different sizes can be chosen both for the A4 and the d5. For instance, in 5-limit tuning, the A4 is either 45/32[4][5][6] or 25/18,[7] and the d5 is either 64/45 Play (help·info) or 36/25,[8] or 1024:729 Play (help·info). 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.[9]
These ratios are not in all contexts regarded as strictly just, but they are the justest possible in 5-limit tuning. 7-limit tuning allows for the justest possible ratios, namely 7/5 for the A4 (about 582.5 cents, also known as septimal tritone) and 10/7 for the d5 (about 617.5 cents, also known as Euler's tritone).[4][10][11] These ratios are more consonant than 17/12 (about 603.0 cents) and 24/17 (about 597.0 cents), which can be obtained in 17-limit tuning, yet the latter are also fairly common, as they are closer to the equal-tempered value of 600.0 cents.
Dissonance and expressiveness
Compared to other commonly occurring intervals like the major second or the minor third, the augmented fourth and the diminished fifth (both two valid enharmonic interpretations of the tritone) are considered awkward intervals to sing. Western composers have traditionally avoided using it explicitly in their melody lines, often preferring to use passing tones or extra note skipping instead of using a direct leap of an augmented fourth or diminished fifth in their melodies. However, as time went by, composers have gradually used the tritone more and more in their music, disregarding its awkwardness and exploiting its expressiveness.[citation needed]
The unstable character of the tritone sets it apart, as discussed in [28] [Paul Hindemith. The Crafts of Musical Composition, Book I. Associated Music Publishers, New York, 1945]. It can be expressed as a ratio by compounding suitable superparticular ratios. Whether it is assigned the ratio 64/45 or 45/32, depending on the musical context, or indeed some other ratio, it is not superparticular, which is in keeping with its unique role in music.[12]
Although this ratio [45/32] is composed of numbers which are multiples of 5 or under, they are excessively large for a 5-limit scale, and are sufficient justification, either in this form or as the tempered "tritone," for the epithet "diabolic," which has been used to characterize the interval. This is a case where, because of the largeness of the numbers, none but a temperament-perverted ear could possibly prefer 45/32 to a small-number interval of about the same width.[13]
In the Pythagorean ratio 81/64 both numbers are multiples of 3 or under, yet because of their excessive largeness the ear certainly prefers 5/4 for this approximate degree, even though it involves a prime number higher than 3. In the case of the 45/32, 'tritone' our theorists have gone around their elbows to reach their thumbs, which could have been reached simply and directly and non-'diabolically' via number 7.[13]
THE TRIAD AND THE TRITONE
The Triad is a chord constructed of three notes, spelled out in thirds, 1-3-5. There are four triads in a major key. They are major (M=1-3-5), minor (m=1-3b-5), diminished (d=1-3b-5b) and augmented (A=1-3-5#). The quality of the chords are the same in every key, and are shown in the table below.
I mention the Triad because it makes it easier to understand the Tritone which is another interval used widely in music. The Tritone (also called the Devil's interval) means 3 tones (3 whole steps apart) according to Sheet Music Magazine, Winter 2005.vol 29, No 1, New York, Noreen Lienhard, pg. 43. It is dissonant and divides the octave perfectly in half, 3 tones up or 3 tones down, it is the same. The diminished chord uses Tritones. The Tritone also makes chord substitutions possible. E and Bb are the 3rd and 7th of the C7 chord while they are also 7th and 3rd of the Gb7 chord. I won't go in to these details here but one thing I was puzzled about was the fact that a 1-3-5 of a C Chord ascending is the same as a 1-3-5 of a G chord descending. I have included the tritone in the Chart below in the last column. Note that different chord roots can have the same tritone. The keys of F# and Gb both have C as the Tritone. The Key of C has F# ascending and Gb descending, which is the same tone.
KEY (M)1-3-5 (m)1-3b-5 (d)1-3b-5b (A)1-3-5# Tritone
C C-E-G C-Eb-G C-Eb-Gb C-E-G# F#-Gb
G G-B-D G-Bb-D G-Bb-Db G-B-D# C#-Db
D D-F#-A D-F-A D-F-Ab D-F#-A# G#-Ab
A A-C#-E A-C-E A-C-Eb A-C#-E# D#-Eb
E E-G#-B E-G-B E-G-Bb E-G#-B# A#-Bb
B B-D#-F# B-D-F# B-D-F B-D-F## F
F# F#-A#-C# F#-A-C# F#-A-C F#-A#-C## C
C# C#-E#-G# C#-E-G# C#-E-G C#-E#-G## G
F F-A-C F-Ab-C F-Ab-Cb F-A-C# B
Bb Bb-D-F Bb-Db-F Bb-Db-Fb Bb-D-F# E
Eb Eb-G-Bb Eb-Gb-Bb Eb-Gb-Bbb Eb-G-B A
Ab Ab-C-Eb Ab-Cb-Eb Ab-Cb-Ebb Ab-C-E D
Db Db-F-Ab Db-Fb-Ab Db-Fb-Abb Db-F-A G
Gb Gb-Bb-Db Gb-Bbb-Db Gb-Bbb-Dbb Gb-Bb-D C
Cb Cb-Eb-Gb Cb-Ebb-Gb Cb-Ebb-Gbb Cb-Eb-G F
Here is basically my notes of the Youtube video.
Video - http://www.youtube.com/watch?v=KYcCq8YadLM
Key of C
Chord Progression
I-IV Groove in C
CMaj9-FMaj9
CMaj9 LH(C,G) RH(B,D,E,G)
FMaj9 LH(F,G,A) RH(C,E,G)
Substitute CMaj9 with Key of C Tritone (E-Bb)
Substitute FMaj9 with Key of F Tritone (F-Eb)
Triads in the right hand over tritones in the left hand.
Amin (E,A,C) triad over C Tritone
Gmin (D,G,Bb) triad over C Tritone
D7b5 (D,Ab,C) triad over C Tritone
Abmaj (Eb,Ab,C) triad over C Tritone
Amaj (E,A,Db) triad over C Tritone
F#maj (C#,F#,A#) triad over C Tritone
Ebmaj (Eb,G,Bb) play together - triad over C Tritone
Dbmaj (Db,F,Ab) play together - triad over C Tritone
Ebmin (Eb,Gb,Bb) triad over C Tritone
Dmin (D,F,A) triad over C Tritone
Gdim (G,Bb,Db) triad over C Tritone
F#min (C#,F#,A) triad over C Tritone
Dmaj (D,Gb,A) triad over C Tritone
Fmin (C,F,Ab) triad over C Tritone
DbAug (Db,F,A) triad over C Tritone
Dbmin (Db,E,Ab) triad over C Tritone
E7sus4 (E,A,D) triad over C Tritone
Eb7sus4 (Eb,Ab,Db) triad over C Tritone
A7sus4 (A,D,G) triad over C Tritone
Now the triads are played over the C Tritone, but it was mentioned that you can basically mix and match triads interchangeably over the F Tritone as well. I find this works particularly well with the Sus4 chords. If correct, you can play for example:
A7sus4 over the C tritone and then back to E7sus4 over the F tritone to substitute the entire progression. It sounds good to my ear and kind of funky as well.
Thanks for any input. Much appreciated.
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