# Why are diminished fifths called tritones?

The word "tritone" basically means three whole tones. In the C major scale, we find this between F and B as F-G-A-B. This interval is also called an augmented fourth. However, a diminished fifth is not three whole tones. For example, if we try notating B-F as three whole tones, we get B-C♯-D♯-E♯, and E♯ is not F (but it is enharmonic to F). Why are B-F and similar intervals also called tritones?

• Actually, E♯ is the same sound as F. (At least in well temperament.) E and F are only one semitone from each other, so E♯ is a different name for F and F♭ is a different name for E. Commented Jan 27, 2023 at 11:53
• @VitHenych E♯ is the same sound as F in every 12-tone temperament; it's just that in some unequal temperaments that sound may not be particularly well tuned for use as an E♯. Commented Jan 27, 2023 at 12:04
• Commented Jan 27, 2023 at 17:03
• @Dom Why did you get rid of tritone? Commented Feb 20, 2023 at 0:29
• @Dom If you feel the edit didn't make sense, then the problem of robo-reviewers is one you definitely need to deal with. Commented Feb 20, 2023 at 3:36

The counting is different for interval naming versus interval measuring.

The intervals like unison, second, third, etc. can be thought of as 1 indexed or just ordinal numbers. If you were starting on `F` and moved up a third, you would index the tones starting with `F` as 1, go up 2, 3, (`G`,`A`) to get the third at `A`. Or, ordinally, `F` is first (unison in music terms), `G` is second, and `A` is third.

Measuring the size of intervals is zero indexed. The standard units for interval measure is half steps. An augmented fourth/diminished fifth is 6 half steps, but if you measure in whole tones it is 3 whole tones, a tri-tone. Or, you could think of it as zero indexed counting.

For example, if we try notating B-F as three whole tones, we get B-C♯-D♯-E♯, and E♯ is not F. Why are B-F and similar intervals also called tritones?

Counting up three whole tones in distance will give us the measure of the interval size. `B` to `E♯` and `B` to `F` are both in distance 6 half steps or 3 whole steps.

The important factor is that `E♯` and `F` are enharmonically the same so the distance of the intervals is the same. But the choice of `E♯` or `F` is a matter of naming the interval.

Naming the interval is complicated in your example, because you are inadvertently mixing two series of tones `F-G-A-B` and `B-C♯-D♯-E♯`, and also giving `B-C♯-D♯-E♯-F`, which are not a diatonic group. We need a diatonic group of pitches to get the interval names.

Let's modify `B-C♯-D♯-E♯` to `B-C♯-D♯-E-F♯` and then name the intervals `B E♯` and `B F`.

• `B` diatonically up ordinally to the fourth tone reaches `E` which is a perfect fourth, to take it up one more half step to `E♯` we augment the fourth, so `B E♯` is an augmented fourth, an `A4` has a distance of six half steps or three whole steps.
• `B` diatonically up ordinally to the fifth tone reaches `F♯` which is a perfect fifth, to take it down one half step to `F♮` we diminish the fifth, so `B F` is a diminished fifth, a `d5` has a distance of six half steps or 3 whole steps.

I've given this in `B` major, because your ending example used `C♯-D♯`, but that might obscure the reason for why interval naming is based on a diatonic scale. The reason is because the first part of interval naming is to get the ordinal number from the sequence of letters `ABCDEFGA` which is called the gamut, and those letters spaced in whole steps except for half steps between `E F` and `B C`. The second part of interval naming determines exact qualities like major/minor or augmented/diminished.

• Name intervals diatonically.
• Measure intervals enharmonically.
• Strange that this answer was accepted, since it doesn't seem to address the crux of the OP's question, which was based on the difference of names between (for example) E# and F, not a question of numbering (indexing). At least that's my understanding of the question. Commented Jan 27, 2023 at 16:50
• @LarsH If you think an answer is wrong, write an answer. Commenting on an existing answer doesn't really address the issue. Commented Jan 27, 2023 at 17:14
• @LarsH, I think it's covered in the question and answer, but maybe not explicitly written. When the OP wrote "...if we try notating B-F as three whole tones..." counter 3 whole tones and ended on `E♯ `, they inadvertently switched from talking about a `d5` back to an `A4`. If they had counted up ordinally to the diatonic fifth above to get the interval name, they would have reached the `F` and a `d5`. But, I see your point this may be lost between the lines. I try to add a summary. Commented Jan 27, 2023 at 17:40
• @MichaelCurtis: Thank you. I feel bad that I seem to have put you to a lot of extra work just for my sake, which the OP didn't need. Anyway, your update addresses the issue very clearly for me, especially the last 2 bullet points. Commented Jan 27, 2023 at 19:42
• @LarsH, no problem. If hope it might be helpful for anyone who reads it in the future. Commented Jan 29, 2023 at 22:32

If you go back far enough, the distinction you seek was maintained. For example, Johann Joseph Fux (Gradus ad Parnassum, 1725) describes the fourth thus:

Quarta triplex est : Quarta vera, minuta, & major, sive tritonus. Quarta vera constat duobus tonis, & semitonio. ... Minuta constat duobus semitoniis, & tono, ... Tritonus constat tribus tonis, ... .

Translated:

The fourth is threefold: the true fourth, the diminished fourth, and the bigger fourth or tritone. The true fourth consists of two tones and a semitone. The diminished fourth consists of two semitones and a tone. The tritone consists of three tones.

Then

Quinta duplex est, vera, & falsa. Vera constat tribus tonis, & semitonio. ... Quinta falsa constat duobus semitoniis, & tono.

Translated

The fifth is twofold, true and false. The true consists of three tones and a semitone. The false fifth consists of two semitones and a tone.

This is of course an error. In both the French and German translations available on IMSLP, it has been corrected to say "two semitones and two tones."

This degree of precision has obviously fallen into disuse, as Michael Curtis explains. This may be because of the move away from just intonation with its multiple sizes of semitones and tones. (Fux spends a good deal of space at the beginning of the book covering the mathematics of rational numbers.) Another possible reason is the standardization and simplification of the note-name-and-accidental system. I omitted the examples in the source because they confusingly use mi to refer to both B and C♯. As this system became more precise, the need for precision in the other system waned.

• It could be argued that a "degree of precision" which distinguishes between the greater tone and lesser tone "has ... fallen into disuse". However, we should still use enough precision not to muddle major seconds and minor seconds. This entails quantifying an interval in terms of two intervals, such as the major second and minor second. A tritone is three major seconds; a diminished fifth is two major seconds and two minor seconds, just as Fux said (apart from misprints). Commented Jan 28, 2023 at 11:26
• @RosieF my point is that we have started to use semitones and tones as an absolute measure of interval distance, using only ordinal-and-quality names to convey precise information about enharmonic spelling. How many people do you think would react to the statement "a major third is four half-steps" by saying "no, it's two whole steps"? Or "a diminished third is a whole step" by saying "no, it's two half steps"? Not many. This distinction is simply not in use in modern music theory. Commented Jan 28, 2023 at 21:13
• @phoog Who's "we" and "How many people"? People discussing music theory? Or people teaching beginner instrumentalists who still have to look for pitches by counting keys or frets? Reducing every interval to a number of semitones might help such beginners. And at least it serves for Schoenbergian dodecaphony. However, people discussing music theory need a richer model which explains scales. Commented Jan 29, 2023 at 9:08

This a thing of enharmonicity. A tritone is in fact an interval of three whole tones, so an augmented 4th. A diminished fifth is not technically a true tritone (although it is the complementing interval of a true tritone), but enharmonically it is the same interval. And this is sufficient for modern practise to consider a diminished fifth to be tritone.

• How do you come to the conclusion that D5 isn't really a tritone?
– Tim
Commented Jan 27, 2023 at 11:37
• @Tim see my answer. It comes from the notion that "tone" means "major second." A diminished third in this framework is not a whole tone but rather two (diatonic) semitones whereas a whole tone comprises a chromatic semitone and a diatonic semitone. (A chromatic semitone is an augmented unison such as C to C sharp while a diatonic semitone is a minor second such as C sharp to D.) Three tones of an augmented fourth might be F to G to A to B while the two tones and two semitones of a diminished fifth are B to C to D to E to F. Commented Jan 27, 2023 at 11:53
• @phoog - o.k. Theory turns round and bites us on the bum - again!
– Tim
Commented Jan 27, 2023 at 11:55
• @Tim well it's bound to happen when one talks about over 1000 years of European music theory across different places and with different bodies of theory developing for different types of music, especially in the last century or two. The theoretical framework evolves, and statements that are true in one version of the framework won't necessarily be true in another. Commented Jan 27, 2023 at 11:59
• @Tim Did you bother to read the answer? It clearly states that a diminished fifth is not a true tritone (as a fifth is made up of 4 steps), but due to enharmonicity it is still considered a tritone.
– Lazy
Commented Jan 27, 2023 at 15:46

A slightly different approach: In equal temperament, it is the distance of 600 cents. It depends on the harmonic context whether that is (to be written as) a diminished 5th or an augmented 4th.