The dissonant
tritone and Just Tuning theory

Q. Tuan Pham, 26
June 2002

Unjust
tuning can have unfortunate consequences! |

The simple frequency ratio theory is the oldest theory for explaining consonance and dissonance in Western music theory. Its origins go back to Greek times. Supporters of this hypothesis have included Boethius, Galileo, Leibniz, Euler, Lipps and Polak and some versions of it are still found in many modern textbooks. For example, Galileo stated:

"Agreeable consonances are pairs of tones which strike the ear with a certain regularity; this regularity consists in the fact that the pulses delivered by the two tones, in the same interval of time, shall be commensurable in number, so as not to keep the ear drum in perpetual torment."

A table of ratio between the frequency of each note in the scale and that of the root is shown below, as frequently listed in texts and articles. This table is based on Nicola Vicentino (1511-c.1576). It is also known as "just tuning", a clearly pre-judgmental word:

note |
frequency ratio |
pitch (cents) |

C |
2/1 |
1200 |

B |
15/8 |
1088 |

Bb |
16/9 |
0996 |

A |
5/3 |
0884 |

Ab |
8/5 |
0814 |

G |
3/2 |
0702 |

F# |
45/32 |
0590 |

F |
4/3 |
0498 |

E |
5/4 |
0386 |

Eb |
6/5 |
0316 |

D |
9/8 |
0204 |

C# |
16/15 |
0112 |

C |
1/1 |
0000 |

On the surface, this seems to confirm the simple ratio theory: intervals that the Western musician considers as consonant have simple ratios (5th = G = 3/2, 4th = F = 4/3, major 3rd = E = 5/4, minor 3rd = Eb = 6/5), those considered dissonant have complex ratio (major 7th = 15/8, minor 2nd = C# = 16/15, and especially the tritone = F# = 45/32). Seems to explain perfectly why the tritone is called the Devil's interval in Western music.

To my knowledge, however,
nobody seems to have realised that this table is rigged! It has, in fact, been
deliberately designed to justify preconceptions of what is consonant and what
is dissonant. While the ratios for 4th, 5th and 3rd have been *deliberately*
chosen to be simple (after all, the ratios can be varied up and down a little
since no *standard tuning* such as equal tempered is assumed, and the only
strict requirement is that each tone is clearly distinguishable from its neighbours),
that of the tritone, 45/52, has been *deliberately* left as a complex ratio
rather than adjusted to the nearest simple ratio. Although the fraction 45/52
could be justified on various grounds (see Appendix
1: How To Generate a Justly Tuned Tritone: The Alternatives), none of them
is entirely convincing.

A much simpler ratio can be found for the tritone: 7/5. This has the value 1.400, as compared with the tabled value of 45/32 = 1.406. The difference corresponds to 1/13 of a semitone (8 cents), or 1/3 of a comma, an interval that is almost undetectable by most human ears. (On average, professional piano tuners fail to tune notes more accurately than about 8 cents.) (see Appendix 2: How to convert frequency ratios to cents.)

If the tritone interval is imperceptibly changed from 45/32 to 7/5 then, miraculously, according to the simple ratio theory, the most dissonant interval in Western music becomes one of the most consonant, about as much so as the minor third (6/5) or minor 6th (8/5)! This is sufficient to dispose of the simple ratio theory.

It must be said that similar
numerical manipulation can be made to change any "consonant" interval into any
"dissonant" interval. Merely using the equal temperament, for example, changes
ALL the frequency ratios into *irrational* numbers, numbers that cannot
be expressed by any ratio at all, or alternatively, that can only be expressed
by ratios involving an infinite number of digits. This is because all equal
tempered frequency ratios involve some root of 2: 2^{(1/12)}, 2^{(2/12)},
2^{(3/12)}, etc. Thus the equal tempered major 3rd, 2^{(4/12)}
= 1.25992..., cannot be expressed exactly by any ratio at all, let alone a simple
ratio. (Notice that the equal tempered major 3rd differs from the "simple ratio"
3rd by 1.25992/1.25000, a difference of 14 cents, almost twice that between
Vicentino's tritone 45/52 and the "simple ratio" tritone 7/5; the equal tempered
tritone differs from Vicentino's tritone by 10 cents). If the simple ratio theory
was correct, then an equal temperament major 3rd (as played on a well tuned
piano), or *any* equal temperament interval apart from the octave, would
be infinitely more dissonant than a "just tuning" tritone or minor second! Yet
Joos Vos (1987) tested 18 western musicians and found that they actually *preferred*
equal temperament tuning to just tuning (see Ohio
State University website). Attributing the perceived consonance of equal
temperament thirds to listening habit begs the question, since listening habit
can then explain the consonance of the "just tuning" third as well, so why have
two explanations when one will do?

This kind off numerological
manipulation has actually been used in the past to justify musicologists' preconception
of consonance and dissonance. For example, in the table above, the major 2nd
(whole tone) has been defined by *two* different ratios, 9/8 (between C
and D) and 10/9 (between D and E) in order to make it looks simpler. The difference
between these ratios, by the way, is 1/5 of a semitone (21 cents), which is
clearly audible. If a single whole tone was defined, say, by dividing the major
3rd into two equal intervals (a far more obvious approach since the major 3rd
is the only multiple of a whole tone in the Western scale), a ratio of about
28/25 would have been found. As another example, 3rds and 6ths were considered
dissonances by Boethius (480-524 AD) because of their complex ratios (major
3rd = 81/64 and minor 3rd = 32/27 or 19/16) until later theorists came and "fixed"
these intervals. No doubt Boethius chose these complex ratios for the 3rd and
6th to justify the musical preconceptions of his own time.

In conclusion, the "simple ratio" theory of consonance is nothing more than musical numerology.

(For a detailed discussion of Just Tuning systems in antiquity, see Bill Alves, The Just Intonation System of Nicola Vicentino. See also Daniel White's The 12 Golden notes is all it takes.

How to Generate a Justly Tuned Tritone: the Alternatives

Originally the Just Tuning system was derived for the diatonic major scale only, based on simple frequency ratios. The simplest ratios in the range 1 (unison) to 2 (octave) is 3/2 which generated the fifth (G in the scale of C). The next simplest is 4/3 which generated the fourth (F). The ratio between F and G is (3/2) / (4/3) = 9/8, which was used to define the major second or whole tone. This immediately gave D, a whole tone above C, as 9/8.

Finding E by adding another
whole tone to D would give E = (9/8) x (9/8) = 81/64, a far from simple ratio
which would therefore not explain the consonance of the major third in the mind
of the Just Tuning adherents. Therefore E was defined by a simple ratio, 5/4.
This means, however, that there are now *two* different whole tones: 9/8
and (5/4)/(9/8) = 10/9. In numerical terms these are 9/8 = 1.125 and 10/9 =
1.111 respectively, a difference of 22 cents (about 1/5 of a semitone).

The semitone is defined as the ratio between F and E, (4/3)/(5/4) = 16/15. This gives the note B since it is just below the octave (2) of the root, B = (2) / (16/15) = 15/8. In other words B is defined in such a way that the interval BC is the same as the interval EF. Note that the semitone (16/15 = 112 cents) is not equal to half of either the 9/8 semitone (204 cents) or the 10/9 semitone (182 cents). This gives an ideas of the irregularities inherent in the Diatonic Just Tuning system.

The above pertains to the
*diatonic* Just Tuning table. To generate the *chromatic* Just Tuning
table some generalization must be made, which involves sometimes arbitrary choices.
Since the tritone is what concerns us, let's see how it (F#) could be generated,
although the same discussion would apply to other chromatics. Keep in mind that
we are putting ourselves in the shoes of the designer of the chromatic Just
Tuning table, who had no precedent to follow and hence had to make up rules
as they went, in a manner as consistent with previous rules as possible. We
will examine two possible approaches:

APPROACH A: The simplest
and most fundamental way to proceed is to continue applying the rule of Simple
Ratios: F# should have the simplest possible ratio, as long as it is sufficiently
close to midway between F and G. 7/5 is the simplest possible ratio, but is
it sufficiently close to midway? What do we mean by "sufficiently close"? Since
*two* different whole tones have been accepted by the Just Tuners, as we
have seen above, the discrepancy between them (22 cents) - however disturbing
- can be taken as an acceptable margin of error. Thus we have to determine whether
the distance between 7/5 = 1.400 and 1.414 (the midway between 4/3 and 3/2)
is less than 22 cents. This distance worked out to 17 cents, which is indeed
"acceptable".

APPROACH B: There are a number of other ways to generate F#, based on relationships with the existing notes on the diatonic scale.

F# = major 3rd from D, giving F# = 1.406

F# = major 2nd from E, giving F# = 1.406

F# = semitone above F, giving F# = 1.422

F# = semitone below G, giving F# = 1.406

F# = major 6th from A, giving F# = 1.389

F# = fifth from B giving, F# = 1.406

This approach gives F# ranging from 1.389 to 1.422, a range which encompasses both 7/5 and the true midpoint (also equal-tempered) value of 1.414. Some of these methods give 1.406 = 45/32, which is the value tabled by Vicentino. Incidentally, the simple ratio tritone at 7/5 is closer to Vicentino's just tuning value than today's equal-tempered tritone (8 cents vs 10 cents). The figure below illustrates the differences between tuning methods, using Vicentino's ratio as basis.

**How
to convert frequency ratios to cents**

By definition, the octave interval, which has a frequency ratio of exactly 2, is equal to 1200 cents on the logarithmic scale, where equal frequency ratios correspond to equal number of cents. Mathematically:

1200 cents = log (2)

Therefore, if two notes which are in the frequency ratio of R, and C is the number of cents between them,

C/1200 = log(R) / log(2)

where the logarithm may be to any base. Thus

C = 1200 log(F) / log(2)

For esxample, the frequency ratio for the justly tuned major third is F = 5/4 = 1.25. Substituting into the above equation gives C = 1200 log(1.25)/log(2) = 386 cents. The frequency ratio between the "just tuning" tritone 45/32 and the "simple ratio" tritone 7/5 is (45/32)/(7/5) = 1.004464. Substituting into the above equation gives C = 1200 log(1.004464)/log(2) = 7.7 cents.

To calculate frequency ratio from cents, the above equation can be written as:

F = 2^{C/1200}

In equal tempered
tuning, successive notes on the chromatic scale differ by exactly 100 cents.
Therefore a semitone is 100 cents from the root, corresponding to a frequency
ratio of F = 2^{100/1200} = 1.05946.