Suppose the series of fifth on which Pythagorean tuning is based is extended to 12 or more members. This can be achieved by successively multiplying by 3 / 2 as illustrated in the second column of the following table. The process amounts to raising the numbers 2 and 3 to successively higher powers and the entire sequence is built upon powers of just these two numbers.

To keep all the pitches in the same octave it is necessary to transpose most of them down by one or more octaves, i.e. multiply by 1 / 2, 1 / 4, 1 / 8, etc. as illustrated in the third and fourth columns.

Index | Original fifth | Octaves to drop | Transposed fifth | Approximate decimal value |

1 | 1 / 1 | 0 | 1 / 1 | 1 |

2 | 3 / 2 | 0 | 3 / 2 | 1.5 |

3 | 9 / 4 | 1 | 9 / 8 | 1.125 |

4 | 27 / 8 | 1 | 27 / 16 | 1.6875 |

5 | 81 / 16 | 2 | 81 / 64 | 1.265625 |

6 | 243 / 32 | 2 | 243 / 128 | 1.898437 |

7 | 729 / 64 | 3 | 729 / 512 | 1.423828 |

8 | 2187 / 128 | 4 | 2187 / 2048 | 1.067871 |

9 | 6561 / 256 | 4 | 6561 / 4096 | 1.601806 |

10 | 19683 / 512 | 5 | 19683 / 16384 | 1.201354 |

11 | 59049 / 1024 | 5 | 59049 / 32768 | 1.802032 |

12 | 177147 / 2048 | 6 | 177147 / 131072 | 1.351524 |

13 | 531441 / 4096 | 7 | 531441 / 524288 | 1.013643 |

The decimal value of the interval produced by transposing the 13th fifth down 7 octaves differs from unity by just over 1%. From this point the sequence produces a series of decimal values which are all very close to the first 12 then another series of 12 values slightly higher again and so on. This sequence is known as the *circle of fifths*. The amount by which the 13th value differs from unity, 531441 / 524288 is known as the *Pythagorean comma*. This is the small amount of overlap at the end of the circle of fifths.

If these fifths were sorted by ascending pitch then the following chromatic tuning would result. The first seven members of the series give rise to the white notes of a keyboard and the remaining members of the series give rise to black notes all tuned as sharps.

Index | Transposed fifth | Approximate decimal value | Interval to next step | Approximate decimal step size | Note name |

1 | 1 / 1 | 1 | 2187 / 2048 | 1.067871 | F |

8 | 2187 / 2048 | 1.067871 | 256 / 243 | 1.053498 | F# |

3 | 9 / 8 | 1.125 | 2187 / 2048 | 1.067871 | G |

10 | 19683 / 16384 | 1.201354 | 256 / 243 | 1.053498 | G# |

5 | 81 / 64 | 1.26562 | 2187 / 2048 | 1.067871 | A |

12 | 177147 / 131072 | 1.351524 | 256 / 243 | 1.053498 | A# |

7 | 729 / 512 | 1.423828 | 256 / 243 | 1.053498 | B |

2 | 3 / 2 | 1.5 | 2187 / 2048 | 1.067871 | C |

9 | 6561 / 4096 | 1.601806 | 256 / 243 | 1.053498 | C# |

4 | 27 / 16 | 1.6875 | 2187 / 2048 | 1.067871 | D |

11 | 59049 / 32768 | 1.802032 | 256 / 243 | 1.053498 | D# |

6 | 243 / 128 | 1.898437 | 256 / 243 | 1.053498 | E |

The 13th member of the series would produce the note E# which would differ from F by the Pythagorean comma. It is this discrepancy that gives rise to the wolf fifths and fourths of Pythagorean tuning and just tuning.

A mean tone temperament is a system of tuning which seeks to close the overlap in the circle of fifths by reducing the size of most of the fifths. [SCHULTER]

The interval of a Pythagorean major third arises between the 1st and 5th members of the series of fifths, 81 / 64 when transposed to lie in the same octave. It differs from the ideal harmonic major third, 5 / 4 by the *syntonic comma,* 81 / 80 (81/64 ÷ 5/4 = 81/64 x 4/5 = 324/320 = 81/80). The sequence of notes from the unsorted series, say C - G - D - A - E, includes 4 intervals of a fifth. If the size of each interval is reduced by a quarter of the syntonic comma then the third, C - E, will be a true harmonic third. The two tones of which it is comprised, C - D - E, are equally divided, *mean tone*, as D is half way up the series. The resultant tuning is called *quarter comma mean tone tuning*. It was first documented by Pierto Aaron (1490 - 1545) some time after 1500. Other forms of mean tone temperament include 1/3rd comma, 2/7ths comma, 1/5th comma and 1/6th comma.

But what exactly do we mean by quarter of the syntonic comma? The scale of pitch is logarithmic so we need a 'quarter' in logarithmic terms, (81/80)^{1/4}. the fourth root of 81 (i.e. 3) divided by the fourth root of 80. Because the size of each fifth needs to be reduced this term must be reciprocated, (80/81)^{1/4}. The tuning ratios from a sequence of fifths can be multiplied by the number of quarter commas required, for example 27/16 x (80/81)^{3/4} for the third member of the series. The mean tone itself is adjusted by the square root of the syntonic comma, (80/81)^{1/2}. The arithmetic and harmonic means of Zarlino's system are now accompanied by a *geometric mean*. It is interesting to note that the decimal value of 80/81 is approximately 0.987654321.

An alternative view of the process is possible. Arithmetically, the four intervals involved amount to (3/2)^{4} or 81/16 which evaluates to 5.0625. What is required is to make this value equal to 5 giving the 5/4 major third desired. To do so each (logarithmic) step needs to be equal to the fourth root of 5 or 5^{1/4}. Each following step is some power of this transposed down an octave if necessary, for example the third step is 5^{3/4 }/ 2; the mean tone is 5^{1/2}/2.

Both methods are shown in the table below. They are arithmetically equivalent yielding the same decimal value. F has been calculated as a fourth down from C by reciprocating the ratios. In musical terms, the effect of this tuning is that the fifths have been very slightly narrowed in order to produce precise harmonic major thirds.

Note name | Place in series | Interval from tonic using quarter comma | Interval from tonic using reduced fifth | Approximate decimal value |

C | 0 | 1 / 1 | 1 / 1 | 1 |

D | 2 | 9/8 x (80/81)^{2/4} |
5^{2/4 }/ 2 |
1.118034 |

E | 4 | 81/64 x (80/81)^{4/4} = 5/4 |
5^{4/4 }/ 4 |
1.25 |

F | -1 | 4/3 x (81/80)^{1/4} |
2 / 5^{1/4} |
1.337481 |

G | 1 | 3/2 x (80/81)^{1/4} |
5^{1/4} |
1.495349 |

A | 3 | 27/16 x (80/81)^{3/4} |
5^{3/4 }/ 2 |
1.671851 |

B | 5 | 243/128 x (80/81)^{5/4} |
5^{5/4 }/ 4 |
1.869186 |

C | - | 2 / 1 | 2 / 1 | 2 |

A problem with this method is that it over corrects the fifths. Every 4 fifths are reduced by the syntonic comma so over a full cycle of 12 fifths the total reduction amounts to 3 syntonic commas or (80/81)^{3} which evaluates to about 0.963418. To truly close the circle of fifths requires a value of 1. There is going to be a very large wolf howling at the end of the series.

Undeterred, we might extend the series in both directions. Continuing the series by upward fifths leads into the series of sharps, F#, C#, G#, D#, A#. Continuing by downward fifths leads into the series of flats Bb, Eb, Ab, Db, Gb. It is very clear that the sharps and flats do not represent the same note at all. They all differ by the same amount, almost quarter of a tone. One approach to mean tone temperament on keyboards with only one set of black note keys is to pick and mix members of both the ascending and descending series so as to obtain the most even semitone steps, Db, Eb, F#, G# A#. This is indicated in bold in the following table.

Note name | Place in series | Interval from tonic using quarter comma | Interval from tonic using reduced fifth | Approximate decimal value |

C |
0 |
1 / 1 |
1 / 1 |
1 |

C# | 7 | 2187/2048 x (80/81)^{7/4} |
5^{7/4 }/ 16 |
1.044907 |

Db |
-5 |
256/243 x (81/80)^{5/4} |
8 / 5^{5/4} |
1.069984 |

D |
2 |
9/8 x (80/81)^{2/4} |
5^{2/4 }/ 2 |
1.118034 |

D# | 9 | 19683/16384 x (80/81)^{9/4} |
5^{9/4 }/ 32 |
1.168241 |

Eb |
-3 |
32/27 x (81/80)^{3/4} |
4 / 5^{3/4} |
1.196279 |

E |
4 |
81/64 x (80/81)^{4/4} = 5/4 |
5^{4/4 }/ 4 |
1.25 |

F |
-1 |
4/3 x (81/80)^{1/4} |
2 / 5^{1/4} |
1.337481 |

F# |
6 |
729/512 x (80/81)^{6/4} |
5^{6/4 }/ 8 |
1.397542 |

Gb | -6 | 1024/729 x (81/80)^{6/4} |
16 / 5^{6/4} |
1.431084 |

G |
1 |
3/2 x (80/81)^{1/4} |
5^{1/4} |
1.495349 |

G# |
8 |
6561/4096 x (80/81)^{8/4} = 25/16 |
5^{8/4 }/ 16 |
1.5625 |

Ab | -4 | 128/81 x (81/80)^{4/4} |
8 / 5^{4/4} |
1.6 |

A |
3 |
27/16 x (80/81)^{3/4} |
5^{3/4 }/ 2 |
1.671851 |

A# |
10 |
59049/32768 x (80/81)^{10/4} |
5^{10/4 }/ 32 |
1.746928 |

Bb | -2 | 16/9 x (81/80)^{2/4} |
4 / 5^{2/4} |
1.788854 |

B |
5 |
243/128 x (80/81)^{5/4} |
5^{5/4 }/ 4 |
1.869186 |

C |
- |
2 / 1 |
2 / 1 |
2 |

A convenient analysis of mean tone temperament is facilitated by the modern logarithmic interval measure of a cent.

Cents are a logarithmic measure of intervals defined such that 1200 cents comprise an octave.

To convert the decimal value of a tuning ratio (interval) into cents take its logarithm to the base 2 and multiply by 1200. For example, the tuning ratio of B in the table of mean tone temperament above is 1.869186; to convert this to cents calculate log_{2}(1.869186) x 1200 = 1082.892164.

(If your calculator does not provide logarithms to the base 2 you can make use of the relation log_{a}(x) = log_{b}(x) / log_{b}(a). That is, take the logarithm in some other base, usually *e* the base of natural logarithms or 10, then divide by the log of 2 in that base. For example using natural logarithms, often indicated *ln*, the previous example becomes ln(1.869186) x 1200 / ln(2).)

Because the measure is logarithmic the difference between two intervals may be found by subtracting their values rather than dividing as is required when the intervals are expressed as ratios. To combine two intervals measured in cents simply add the values.

Interval | Ratio | Approximate decimal value | Measure to nearest cent |

Octave | 2 / 1 | 2 | 1200 |

Harmonic fifth | 3 / 2 | 1.5 | 702 |

Harmonic fourth | 4 / 3 | 1.333333 | 498 |

Pythagorean major third | 81 / 64 | 1.265625 | 408 |

Just major third | 5 / 4 | 1.25 | 386 |

Pythagorean minor third | 32 / 27 | 1.185185 | 294 |

Just minor third | 6 / 5 | 1.2 | 316 |

Tone | 9 / 8 | 1.125 | 204 |

Just minor tone | 10 / 9 | 1.111111 | 182 |

Pythagorean chromatic semitone | 2187 / 2048 | 1.067871 | 113 |

Just diatonic semitone | 16 / 15 | 1.066667 | 112 |

Pythagorean diatonic semitone | 256 / 243 | 1.053498 | 90 |

Pythagorean comma | 531441 / 524288 | 1.013643 | 23 |

Syntonic comma | 81 / 80 | 1.0125 | 22 |

Quarter Pythagorean comma | (531441 / 524288)^{1/4} |
1.003394 | 6 |

Quarter syntonic comma | (81/80)^{1/4} |
1.003110 | 5 |

A full chromatic analysis of mean tone temperament (using the same black notes as those selected in the derivation of mean tone temperament above) is given in the table below. Colour is used to indicate similar intervals.

If you have audio support in your browser you can left click to hear the notes in sequence and right click (hold on mobile, two fingers on a MacBook) to hear the notes as a chord.

The first main column of the table repeats the tunings from the table above. Each main column of the table gives the interval from the note named at the head of the column (using modern note names) to the next higher note named at the left of each row. If you would like to swap the rows and columns click the top left cell.

pivot | C | Db | D | Eb | E | F | F# | G | G# | A | A# | B | C |

C | 0 | 1083 | 1007 | 890 | 814 | 697 | 621 | 503 | 427 | 310 | 234 | 117 | 1200 |

Db | 117 | 0 | 1124 | 1007 | 931 | 814 | 738 | 620 | 544 | 427 | 351 | 234 | 117 |

D | 193 | 76 | 0 | 1083 | 1007 | 890 | 814 | 696 | 620 | 503 | 427 | 310 | 193 |

Eb | 310 | 193 | 117 | 0 | 1124 | 1007 | 931 | 813 | 737 | 620 | 544 | 427 | 310 |

E | 386 | 269 | 193 | 76 | 0 | 1083 | 1007 | 889 | 813 | 696 | 620 | 503 | 386 |

F | 503 | 386 | 310 | 193 | 117 | 0 | 1124 | 1006 | 930 | 813 | 737 | 620 | 503 |

F# | 579 | 462 | 386 | 269 | 193 | 76 | 0 | 1082 | 1006 | 889 | 813 | 696 | 579 |

G | 697 | 580 | 504 | 387 | 311 | 194 | 118 | 0 | 1124 | 1007 | 931 | 814 | 697 |

G# | 773 | 656 | 580 | 463 | 387 | 270 | 194 | 76 | 0 | 1083 | 1007 | 890 | 773 |

A | 890 | 773 | 697 | 580 | 504 | 387 | 311 | 193 | 117 | 0 | 1124 | 1007 | 890 |

A# | 966 | 849 | 773 | 656 | 580 | 463 | 387 | 269 | 193 | 76 | 0 | 1083 | 966 |

B | 1083 | 966 | 890 | 773 | 697 | 580 | 504 | 386 | 310 | 193 | 117 | 0 | 1083 |

C | 1200 | 1083 | 1007 | 890 | 814 | 697 | 621 | 503 | 427 | 310 | 234 | 117 | 0 |

Most of the fifths have the interval 696 cents which sometimes appears as 697 cents due to the process of rounding to the nearest cent. This is just slightly less than the size of a 3 / 2 fifth which is about 702 cents. But there are two fifths of 656 cents, nearly a quarter tone too narrow and three of 737 cents, nearly quarter of a tone too wide. A similar problem exists with the fourths. For fifths and fourths this is a serious discrepancy as perception of the tuning of these intervals is quite sensitive.

There are also some problematic thirds. Four of the major thirds are too wide. Four of the minor thirds are too narrow and one is too wide. Similar problems exist with the sixths.

In practical terms the tuning is fine so long as one stays in or close to the key of C. Modulation to distant keys is unwise.

Jean Philippe Rameau (1683 - 1764) established a theory of tonal harmony based on the crucial fourth and fifth scale degrees. The fifth scale degree is known as the *dominant* and its inverse, the fourth, five degrees down from the tonic is called the *subdominant*. The entire diatonic scale can be constructed as common chords on these scale degrees, for example C E G, F A C, G B D.

Suppose one similarly constructs a new scale on the dominant of the existing scale. The dominant of fifth of C is G so the new scale would contain G B D, C E G, D F# A. This requires the introduction of F#. It is the last note in the scale of G and is called the *leading note* because in melody it leads back to the tonic.

The pattern of intervals in the new scale is exactly the same as that in the old (tone, tone, semitone; tone; tone, tone semitone) so this is not a different mode, it is a different *key*, the same mode but from a different starting note. One can *modulate* to a new key via a *pivot chord* which is present in both the old and new keys.

Building a new scale on the dominant of G yields D F# A, G B D, A, C#, E, again introducing a new sharp for the leading note. Continuing this process creates the following sequence of sharps for the leading notes in the same way as they arise from the series of fifths.

Tonic | Dominant | Leading note |

C | G | B |

G | D | F# |

D | A | C# |

A | E | G# |

E | B | D# |

B | F# | A# |

Conversely, one can construct a new scale on the subdominant of C, F. This would contain the triads F A C, Bb D F, C E G. The note Bb must be added as the new subdominant of the scale Again, this is a new key. Construction of a new scale on the subdominant of F, Bb, yields the triads Bb D F, Eb G Bb, F A C. Continuing this process creates the following sequence of flats for the new subdominants.

Tonic | Subdominant |

C | F |

F | Bb |

Bb | Eb |

Eb | Ab |

Ab | Db |

Db | Gb |

Moving from one key to another is *key modulation*. Common forms of key modulation involve moving to the dominant or subdominant. Because the underlying intervals of a fifth and a fourth are the inversion of each other, a new key formed on the dominant will have the key note of the old key as its subdominant. From the perspective of the new key one has come from the subdominant. Moving in this direction is known as the *plagal* orbit analogous to the Medieval plagal modes in which the final is the fourth note of the range. If one moves to the subdominant then this new key will have the key note of the old key as its dominant. Movement in this direction, from the dominant, is known as the *cadential* orbit.

Diatonic intervals are measured as the number of different named notes in the diatonic scale that they include. For example C to G is a fifth, C D E F G. The interval from B to F is also a fifth, B C D E F, even though it is a semitone smaller.

Every fifth in the diatonic major scale involves 7 semitone steps (3 tones and one semitone) except the one from the leading note to the subdominant, B to F in the key of C major. The fifths comprising 3 tones and a semitone are called *perfect fifths*. The one from B to F is a semitone smaller and is called a *diminished fifth* or *tritone *(3 tones*)*.

Similarly, every fourth in the scale involves 5 semitone steps (2 tones and a semitone) except the one from the subdominant to the leading note, F to B in the key of C major. The fourths comprising 2 tones and a semitone are called *perfect fourths*. The one from F to B is a semitone larger and is called an *augmented fourth*. This interval is also the tritone, an interval generally regarded as most inharmonious. Actually, in just tunings, the augmented fourth and diminished fifth differ by the Pythagorean comma, the overlap in the circle of fifths. The tritone thus has two different values which perhaps accounts for its inharmonious and troublesome nature.

Unison, octaves, fifths and fourths are the only perfect intervals. Because the inversion of a fifth is a fourth (subtracting from an octave) the inversion of a perfect interval is also a perfect interval.

The character (size) of all the other intervals depends on their position in the diatonic scale as the size of the interval may vary by one semitone. The smaller of the two possibilities is called a *minor* interval and the larger a *major* interval. The inversion of a minor interval is always a major interval and vice versa.

In general these are the intervals that arise in the major and minor scale modes. All the degrees of a diatonic major scale are related to the tonic by the corresponding major interval. The minor scale always has a major second and sometimes the sixth and seventh degrees are raised to major intervals. In the *melodic minor* scale, suited to melody, the sixth and seventh degrees are raised a semitone to major intervals in an ascending scale but the minor form is used in a descending scale. In the *harmonic minor* scale used to form chords just the seventh degree is raised to a major seventh.

The following table lists all the diatonic intervals with their size in semitone steps and their inversion. (The audio uses Quarter Comma Mean Tone Temperament as above.)

Interval | Semitone steps | Inversion |

Unison | 0 | Octave |

Minor second | 1 | Major seventh |

Major second | 2 | Minor seventh |

Minor third | 3 | Major sixth |

Major third | 4 | Minor sixth |

Perfect fourth | 5 | Perfect fifth |

Tritone | 6 | Tritone |

Perfect fifth | 7 | Perfect fourth |

Minor sixth | 8 | Major third |

Major sixth | 9 | Minor third |

Minor seventh | 10 | Major second |

Major seventh | 11 | Minor second |

Octave | 12 | Unison |

Other augmented and diminished intervals occasionally arise. For example C# to Db is a diminished second; it is a second because there are two note names but it is diminished because it is a semitone smaller than a minor second. C to A# is an augmented sixth; it is a sixth because there are 6 note names, C D E F G A# but it is a semitone larger than a major sixth. Augmented and diminished forms of all intervals may be found.

Tunings which attempt to eradicate the wolf intervals or minimize their impact on a particular style of music are known as *well temperament*. Such temperaments flourished from the late 17th to the late 19th century, a time when the common chord major and minor key system also prevailed. [SCHULTER]

Andreas Werckmeister (1645 - 1706) proposed several such tuning systems. In the one known as Werckmeister Temperament III the black notes, the furthest members of the series of fifths and the most troublesome notes of mean tone temperament, are left in Pythagorean just intonation whilst the white notes are tempered. Werckmeister's temperament uses the Pythagorean comma, 531441 / 524288, which is the usual basis of well temperament. In the following table I have indicated this somewhat unwieldy term as P. The white note tunings are adjusted by P^{3/4} at most.

Note name | Position in series | Interval from tonic | Approximate decimal value | Measure to nearest cent |

C | 0 | 1 / 1 | 1 | 0 |

Db | . | 256 / 243 | 1.053498 | 90 |

D | 2 | 9/8 x 1/P^{2/4} |
1.117403 | 192 |

Eb | . | 32 / 27 | 1.185185 | 294 |

E | 4 | 81/64 x 1/P^{3/4} |
1.252827 | 390 |

F | . | 4 / 3 | 1.333333 | 498 |

Gb | . | 1024 / 729 | 1.404664 | 588 |

G | 1 | 3/2 x 1/P^{1/4} |
1.494927 | 696 |

Ab | . | 128 / 81 | 1.580247 | 792 |

A | 3 | 27/16 x 1/P^{3/4} |
1.670436 | 888 |

Bb | . | 16 / 9 | 1.777778 | 996 |

B | 5 | 243/128 x 1/P^{3/4} |
1.879241 | 1092 |

C | . | 2 / 1 | 2 | 1200 |

A full chromatic analysis of Werckmeister's Well Temperament is given in the table below. Colour is used to indicate similar intervals.

If you have audio support in your browser you can left click to hear the notes in sequence and right click (hold on mobile, two fingers on a MacBook) to hear the notes as a chord.

The first main column of the table repeats the tunings from the table above right. Each main column of the table gives the interval from the note named at the head of the column (using modern note names) to the next higher note named at the left of each row. If you would like to swap the rows and columns click the top left cell.

pivot | C | Db | D | Eb | E | F | Gb | G | Ab | A | Bb | B | C |

C | 0 | 1110 | 1008 | 906 | 810 | 702 | 612 | 504 | 408 | 312 | 204 | 108 | 0 |

Db | 90 | 0 | 1098 | 996 | 900 | 792 | 702 | 594 | 498 | 402 | 294 | 198 | 90 |

D | 192 | 102 | 0 | 1098 | 1002 | 894 | 804 | 696 | 600 | 504 | 396 | 300 | 192 |

Eb | 294 | 204 | 102 | 0 | 1104 | 996 | 906 | 798 | 702 | 606 | 498 | 402 | 294 |

E | 390 | 300 | 198 | 96 | 0 | 1092 | 1002 | 894 | 798 | 702 | 594 | 498 | 390 |

F | 498 | 408 | 306 | 204 | 108 | 0 | 1110 | 1002 | 906 | 810 | 702 | 606 | 498 |

Gb | 588 | 498 | 396 | 294 | 198 | 90 | 0 | 1092 | 996 | 900 | 792 | 696 | 588 |

G | 696 | 606 | 504 | 402 | 306 | 198 | 108 | 0 | 1104 | 1008 | 900 | 804 | 696 |

Ab | 792 | 702 | 600 | 498 | 402 | 294 | 204 | 96 | 0 | 1104 | 996 | 900 | 792 |

A | 888 | 798 | 696 | 594 | 498 | 390 | 300 | 192 | 96 | 0 | 1092 | 996 | 888 |

Bb | 996 | 906 | 804 | 702 | 606 | 498 | 408 | 300 | 204 | 108 | 0 | 1104 | 996 |

B | 1092 | 1002 | 900 | 798 | 702 | 594 | 504 | 396 | 300 | 204 | 96 | 0 | 1092 |

C | 1200 | 1110 | 1008 | 906 | 810 | 702 | 612 | 504 | 408 | 312 | 204 | 108 | 0 |

All the fifths are either true fifths, ~702 cents, or a quarter (Pythagorean) comma less. Similarly the fourths are either true fourths, ~498 cents, or a quarter comma more. Major thirds vary adopting one of the quarter comma positions between 5 / 4 (about 386 cents) and 81 / 64 (about 408 cents). Minor thirds take up positions between 32 / 27 (about 294 cents) and 6 / 5 (about 316 cents). This scheme is tending towards one in which the intervals are equally spaced.

By providing a spectrum of different sized major and minor thirds and sixths Werckmeister's system not only facilitates modulation to more distant keys but imbues each key with its own character, a part of the musical language of the 18th century.

Other forms of temperament include those proposed by Francesco Antonio Vallotti (1697 - 1780), Anton Bemetzrieder and Margo Schulter. [SCHULTER] Typically, these involve leaving the white notes and Bb in Pythagorean just intonation and tempering the remaining black notes by either a 1/4 or a 1/6th of a Pythagorean comma.

One way to close the overlap in the circle of fifths is to divide seven octaves into twelve equal fifths. If we represent such a fifth as T then T^{12} = 2^{7}. Each fifth then becomes 2^{7/12} which evaluates to about 1.498307 or exactly 700 cents. By definition an octave is 1200 cents so a fourth is 500 cents by subtracting a fifth. The difference between a fourth and a fifth, a whole tone, is 200 cents. A semitone is 100 cents. A chromatic scale of 12 equal 100 cent steps can be constructed. Indeed, this is the reason for defining an octave as 1200 cents and the basis of the tuning system called *equal temperament*. Such a tuning, or something close to it, was in use on fretted instruments such as lutes by the middle of the 16th century.

Note name | Interval from tonic | Exact tuning in cents relative to first note |

C | 1 / 1 | 0 |

C# / Db | 1 / 2^{1/12} |
100 |

D | 1 / 2^{2/12} |
200 |

D# / Eb | 1 / 2^{3/12} |
300 |

E | 1 / 2^{4/12} |
400 |

F | 1 / 2^{5/12} |
500 |

F# / Gb | 1 / 2^{6/12} |
600 |

G | 1 / 2^{7/12} |
700 |

G# / Ab | 1 / 2^{8/12} |
800 |

A | 1 / 2^{9/12} |
900 |

A# / Bb | 1 / 2^{10/12} |
1000 |

B | 1 / 2^{11/12} |
1100 |

C | 1 / 2^{12/12} |
1200 |

For completeness a full chromatic analysis of Equal Temperament is given in the table below. Colour is used to indicate similar intervals.

If you have audio support in your browser you can left click to hear the notes in sequence and right click (hold on mobile, two fingers on a MacBook) to hear the notes as a chord.

The first main column of the table repeats the tunings from the table above right. Each main column of the table gives the interval from the note named at the head of the column (using modern note names) to the next higher note named at the left of each row. If you would like to swap the rows and columns click the top left cell.

pivot | C | Db | D | Eb | E | F | Gb | G | Ab | A | Bb | B | C |

C | 0 | 1100 | 1000 | 900 | 800 | 700 | 600 | 500 | 400 | 300 | 200 | 100 | 0 |

Db | 100 | 0 | 1100 | 1000 | 900 | 800 | 700 | 600 | 500 | 400 | 300 | 200 | 100 |

D | 200 | 100 | 0 | 1100 | 1000 | 900 | 800 | 700 | 600 | 500 | 400 | 300 | 200 |

Eb | 300 | 200 | 100 | 0 | 1100 | 1000 | 900 | 800 | 700 | 600 | 500 | 400 | 300 |

E | 400 | 300 | 200 | 100 | 0 | 1100 | 1000 | 900 | 800 | 700 | 600 | 500 | 400 |

F | 500 | 400 | 300 | 200 | 100 | 0 | 1100 | 1000 | 900 | 800 | 700 | 600 | 500 |

Gb | 600 | 500 | 400 | 300 | 200 | 100 | 0 | 1100 | 1000 | 900 | 800 | 700 | 600 |

G | 700 | 600 | 500 | 400 | 300 | 200 | 100 | 0 | 1100 | 1000 | 900 | 800 | 700 |

Ab | 800 | 700 | 600 | 500 | 400 | 300 | 200 | 100 | 0 | 1100 | 1000 | 900 | 800 |

A | 900 | 800 | 700 | 600 | 500 | 400 | 300 | 200 | 100 | 0 | 1100 | 1000 | 900 |

Bb | 1000 | 900 | 800 | 700 | 600 | 500 | 400 | 300 | 200 | 100 | 0 | 1100 | 1000 |

B | 1100 | 1000 | 900 | 800 | 700 | 600 | 500 | 400 | 300 | 200 | 100 | 0 | 1100 |

C | 1200 | 1100 | 1000 | 900 | 800 | 700 | 600 | 500 | 400 | 300 | 200 | 100 | 0 |

In equal temperament all similar intervals remain constant no matter where in the scale they occur. Intervals are equally in tune (and equally out of tune) in all keys and key modulation is unhindered by the tuning system. The consonance of pure intervals is lost as there are no longer any pure intervals except the octave. All basis for different keys having different character is gone. Utility has triumphed over beauty.

It is open to debate whether J. S. Bach (1685 - 1750) in *Well-tempered Clavier* was seeking to demonstrate the freedom of key modulation afforded by equal temperament or the contrasting key characters offered by well temperament. [SCHULTER]

In the following table the interval sizes from tonic to other scale degrees for Pythagorean and just intonation are compared with equal temperament. (Note reference for audio is A.)

Note name | Equal temperament | Pythagorean (F# x B) | Just | |||||

cents | decimal | ratio | cents | decimal | ratio | cents | decimal | |

C | 0 | 1.0 | 1 / 1 | 0 | 1.0 | 1 / 1 | 0 | 1.0 |

C# / Db | 100 | 1.0594 | 256 / 243 | 90.22 | 1.0535 | 16 / 15 | 111.73 | 1.0667 |

D | 200 | 1.1225 | 9 / 8 | 203.91 | 1.125 | 9 / 8 | 203.91 | 1.125 |

D# / Eb | 300 | 1.1892 | 32 / 27 | 294.13 | 1.1852 | 6 / 5 | 315.64 | 1.2 |

E | 400 | 1.2599 | 81 / 64 | 407.82 | 1.2656 | 5 / 4 | 386.31 | 1.25 |

F | 500 | 1.3348 | 4 / 3 | 498.04 | 1.3333 | 4 / 3 | 498.04 | 1.3333 |

F# / Gb | 600 | 1.4142 | 1024 / 729 | 588.27 | 1.4047 | 45 / 32 | 590.22 | 1.4063 |

G | 700 | 1.4983 | 3 / 2 | 701.96 | 1.5 | 3 / 2 | 701.96 | 1.5 |

G# / Ab | 800 | 1.5874 | 128 / 81 | 792.18 | 1.5802 | 8 / 5 | 813.69 | 1.6 |

A | 900 | 1.6818 | 27 / 16 | 905.87 | 1.6875 | 5 / 3 | 884.36 | 1.6667 |

A# / Bb | 1000 | 1.7818 | 16 / 9 | 996.09 | 1.7778 | 9 / 5 | 1017.60 | 1.8 |

B | 1100 | 1.8877 | 243 / 128 | 1109.78 | 1.8984 | 15 / 8 | 1088.27 | 1.875 |

C | 1200 | 2.0 | 2 / 1 | 1200.00 | 2.0 | 2 / 1 | 1200.00 | 2.0 |