Quarter-comma meantone

Construction

In a meantone tuning, we have different chromatic and diatonic semitones; the chromatic semitone is the difference between C and C^♯, and the diatonic semitone the difference between C and D♭. In Pythagorean tuning, the diatonic semitone is often called the Pythagorean limma and the chromatic semitone Pythagorean apotome, but in Pythagorean tuning the apotome is larger, whereas in ⁠ 1 / 4 ⁠ comma meantone the limma is larger. Put another way, in Pythagorean tuning we have that C^♯ is higher than D^♭, whereas in ⁠ 1 / 4 ⁠ comma meantone we have C^♯ lower than D^♭.

In any meantone or Pythagorean tuning, where a whole tone is composed of one semitone of each kind, a major third is two whole tones and therefore consists of two semitones of each kind, a perfect fifth of meantone contains four diatonic and three chromatic semitones, and an octave seven diatonic and five chromatic semitones, it follows that:

• Five fifths down and three octaves up make up a diatonic semitone, so that the Pythagorean limma is tempered to a diatonic semitone.
• Two fifths up and an octave down make up a whole tone consisting of one diatonic and one chromatic semitone.
• Four fifths up and two octaves down make up a major third, consisting of two diatonic and two chromatic semitones, or in other words two whole tones.

Thus, in Pythagorean tuning, where sequences of just fifths (frequency ratio 3 : 2 ) and octaves are used to produce the other intervals, a whole tone is

${\displaystyle {\frac {~\left({\frac {3}{\ 2\ }}\right)^{2}}{2}}={\frac {\ \left({\frac {9}{\ 4\ }}\right)\ }{2}}={\frac {9}{\ 8\ }}\ ,}$

and a major third is

${\displaystyle {\frac {~\left({\frac {3}{\ 2\ }}\right)^{4}}{2^{2}}}={\frac {\ \left({\frac {81}{\ 16\ }}\right)\ }{4}}={\frac {81}{\ 64\ }}\approx {\frac {80}{\ 64\ }}={\frac {5}{\ 4\ }}\$ ;}

the ratio of the different values is the syntonic comma, ⁠ 81 / 80 ⁠ .

An interval of a seventeenth, consisting of sixteen diatonic and twelve chromatic semitones, such as the interval from D₄ to F^♯
₆, can be equivalently obtained using either

• a stack of four fifths (e.g. D₄ A₄ E₅ B₅ F^♯
  ₆), or
• a stack of two octaves and one major third (e.g. D₄ D₅ D₆ F^♯
  ₆).

This large interval of a seventeenth contains 5 + (5 − 1) + (5 − 1) + (5 − 1) = 20 − 3 = 17 staff positions. In Pythagorean tuning, the size of a seventeenth is defined using a stack of four justly tuned fifths (frequency ratio 3 : 2 ):

${\displaystyle \left({\frac {3}{\ 2\ }}\right)^{4}={\frac {81}{\ 16\ }}={\frac {80}{\ 16\ }}\cdot {\frac {81}{\ 80\ }}=5\cdot {\frac {81}{\ 80\ }}~.}$

In quarter-comma meantone temperament, where a just major third ( 5 : 4 ) is required, a slightly narrower seventeenth is obtained by stacking two octaves and a major third:

${\displaystyle 2^{2}\cdot {\frac {5}{\ 4\ }}=5~.}$

By definition, however, a seventeenth of the same size ( 5 : 1 ) must be obtained, even in quarter-comma meantone, by stacking four fifths. Since justly tuned fifths, such as those used in Pythagorean tuning, produce a slightly wider seventeenth, in quarter-comma meantone the fifths must be slightly flattened to meet this requirement. Letting x be the frequency ratio of the flattened fifth, it is desired that four fifths have a ratio of 5 : 1 ,

${\displaystyle x^{4}=5\ ,}$

which implies that a fifth is

${\displaystyle x={\sqrt[{4}]{5\ }}=5^{1/4}\ ,}$

a whole tone, built by moving two fifths up and one octave down, is

${\displaystyle {\frac {~x^{2}}{2}}={\frac {\ {\sqrt {5\ }}\ }{2}}\ ,}$

and a diatonic semitone, built by moving three octaves up and five fifths down, is

${\displaystyle {\frac {\;2^{3}\ }{\ x^{5}}}={\frac {8}{~5^{5/4}}}~.}$

Notice that, in quarter-comma meantone, the seventeenth is ⁠ 81 / 80 ⁠ times narrower than in Pythagorean tuning. This difference in size, equal to about 21.506 cents, is called the syntonic comma. This implies that the fifth is a quarter of a syntonic comma narrower than the justly tuned Pythagorean fifth. Namely, this system tunes the fifths in the ratio of

${\displaystyle 5^{1/4}\approx 1.495349\approx {\frac {\ 643\ }{430}}\ }$

which is expressed in the logarithmic cents scale as

${\displaystyle 1200\ \log _{2}{5^{1/4}}\ {\mathsf {cents}}\approx 696.578\ {\mathsf {cents}}\ ,}$

which is slightly smaller (or flatter) than the ratio of a justly tuned fifth:

${\displaystyle {\frac {3}{2}}=1.5}$

which is expressed in the logarithmic cents scale as

${\displaystyle 1200\log _{2}{\left({\frac {3}{2}}\right)}{\text{ cents}}\approx 701.955{\text{ cents}}~.}$

The difference between these two sizes is a quarter of a syntonic comma:

${\displaystyle \approx 701.955-696.578\ {\mathsf {cents}}\approx 5.377\ {\mathsf {cents}}\approx {\frac {\ 21.506\ {\mathsf {cents}}\ }{4}}~.}$

In sum, this system tunes the major thirds to the just ratio of 5 : 4 (so, for instance, if A₄ is tuned to 440  Hz, C^♯
₅ is tuned to 550 Hz), most of the whole tones (namely the major seconds) in the ratio √5:2, and most of the semitones (namely the diatonic semitones or minor seconds) in the ratio ( 8 : 5 )^5⁄4 . This is achieved by tuning the seventeenth a syntonic comma flatter than the Pythagorean seventeenth, which implies tuning the fifth a quarter of a syntonic comma flatter than the just ratio of 3 : 2. It is this that gives the system its name of quarter-comma meantone.

12-tone scale

The whole chromatic scale (a subset of which is the diatonic scale), can be constructed by starting from a given base note, and increasing or decreasing its frequency by one or more fifths. This method is identical to Pythagorean tuning, except for the size of the fifth, which is tempered as explained above. However, meantone temperaments (except for 12 TET) cannot fit into a 12 note keyboard; and like quarter-comma meantone, most require an infinite number of notes (although there is a very close approximation to quarter-comma that can fit into a keyboard with 31 keys per octave). When tuned to a 12 note keyboard many notes must be left out, and unless the tuning is "tempered" to gloss over the missing notes, keyboard players who substitute the available nearest-pitch note (which is always the wrong pitch) for the actual appropriate quarter comma note (which would sound consonant, if it were available) create dissonant notes in place of the consonant quarter-comma note.

The construction table below illustrates how the pitches of the notes are obtained with respect to D (the base note), in a D-based scale (see Pythagorean tuning for a more detailed explanation). For each note in the basic octave, the table provides the conventional name of the interval from D (the base note), the formula to compute its frequency ratio, and the approximate values for its frequency ratio and size in cents.

Note	Interval from D	Formula	Freq. ratio	Size (cents)	Size (31-ET)
A♭	diminished fifth	${\displaystyle x^{-6}\cdot 2^{4}={\frac {\ 16{\sqrt {5\ }}\ }{25}}}$	1.4311	620.5	16.03
E♭	minor second	${\displaystyle x^{-5}\cdot 2^{3}={\frac {\ 8{\sqrt {5\ }}\ x\ }{25}}}$	1.0700	117.1	3.03
B♭	minor sixth	${\displaystyle x^{-4}\cdot 2^{3}={\frac {\ 8\ }{5}}}$	1.6000	813.7	21.02
F	minor third	${\displaystyle x^{-3}\cdot 2^{2}={\frac {\ 4x\ }{5}}}$	1.1963	310.3	8.02
C	minor seventh	${\displaystyle x^{-2}\cdot 2^{2}={\frac {\ 4{\sqrt {5\ }}\ }{5}}}$	1.7889	1006.8	26.01
G	perfect fourth	${\displaystyle x^{-1}\cdot 2^{1}={\frac {\ 2{\sqrt {5\ }}\ x\ }{5}}}$	1.3375	503.4	13.00
D	unison	${\displaystyle x^{0}\cdot 2^{0}=1}$	1.0000	0.0	0.00
A	perfect fifth	${\displaystyle x^{1}\cdot 2^{0}=x}$	1.4953	696.6	18.00
E	major second	${\displaystyle x^{2}\cdot 2^{-1}={\frac {\ {\sqrt {5\ }}\ }{2}}}$	1.1180	193.2	4.99
B	major sixth	${\displaystyle x^{3}\cdot 2^{-1}={\frac {\ {\sqrt {5\ }}\ x\ }{2}}}$	1.6719	889.7	22.98
F♯	major third	${\displaystyle x^{4}\cdot 2^{-2}={\frac {\ 5\ }{4}}}$	1.2500	386.3	9.98
C♯	major seventh	${\displaystyle x^{5}\cdot 2^{-2}={\frac {\ 5\ x\ }{4}}}$	1.8692	1082.9	27.97
G♯	augmented fourth	${\displaystyle x^{6}\cdot 2^{-3}={\frac {\ 5{\sqrt {5\ }}\ }{8}}}$	1.3975	579.5	14.97

In the formulas,   x  =  ⁴√5  =   5^1⁄4  is the size of the tempered perfect fifth, and the ratios x : 1 or 1 : x represent an ascending or descending tempered perfect fifth (i.e. an increase or decrease in frequency by  x ), while 2 : 1 or 1 : 2 represent an ascending or descending octave.

As in Pythagorean tuning, this method generates 13 pitches, with A^♭ and G^♯ nearly a quarter-tone apart. To build a 12 tone scale, typically A^♭ is arbitrarilly discarded.

C-based construction tables

The table above shows a D-based stack of fifths (i.e. a stack in which all ratios are expressed relative to D, and D has a ratio of ⁠1/ 1 ⁠). Since it is centered at D, the base note, this stack can be called D-based symmetric:

A^♭–E^♭–B^♭–F–C–G–D–A–E–B–F^♯–C^♯–G^♯

With the perfect fifth taken as ⁴√5, the ends of this scale are 125 in frequency ratio apart, causing a gap of ⁠125/128⁠ (about two-fifths of a semitone) between its ends if they are normalized to the same octave. If the last step (here, G^♯) is replaced by a copy of A^♭ but in the same octave as G^♯, that will increase the interval C^♯–G^♯ to a discord called a wolf fifth. (Note that in meantone systems there are no wolf intervals when the actual, correct note is played: The wolf discord always is the result of naïvely trying to substitute the flat above for the required sharp below it, or vice-versa.)

Except for the size of the fifth, this is identical to the stack traditionally used in Pythagorean tuning. Some authors prefer showing a C-based stack of fifths, ranging from A^♭ to G^♯. Since C is not at its center, this stack is called C-based asymmetric:

A^♭–E^♭–B^♭–F–C–G–D–A–E–B–F^♯–C^♯–G^♯

Since the boundaries of this stack (A^♭ and G^♯) are identical to those of the D-based symmetric stack, the note names of the 12 tone scale produced by this stack are also identical. The only difference is that the construction table shows intervals from C, rather than from D. Notice that 144 intervals can be formed from a 12 tone scale (see table below), which include intervals from C, D, and any other note. However, the construction table shows only 12 of them, in this case those starting from C. This is at the same time the main advantage and main disadvantage of the C-based asymmetric stack, as the intervals from C are commonly used, but since C is not at the center of this stack, they unfortunately include an augmented fifth (i.e. the interval from C to G^♯), instead of a minor sixth (from C to A^♭). This augmented fifth is an extremely dissonant wolf interval, as it deviates by 41.1 cents (a diesis of ratio 128 : 125 , almost twice a syntonic comma) from the corresponding pure interval of 8 : 5 or 813.7 cents.

On the contrary, the intervals from D shown in the table above, since D is at the center of the stack, do not include wolf intervals and include a pure minor sixth (from D to B^♭), instead of an impure augmented fifth. Notice that in the above-mentioned set of 144 intervals pure minor sixths are more frequently observed than impure augmented fifths (see table below), and this is one of the reasons why it is not desirable to show an impure augmented fifth in the construction table. A C-based symmetric stack might be also used, to avoid the above-mentioned drawback:

G^♭–D^♭–A^♭–E^♭–B^♭–F–C–G–D–A–E–B–F^♯

In this stack, G^♭ and F^♯ have a similar frequency, and G♭ is typically discarded. Also, the note between C and D is called D^♭ rather than C^♯, and the note between G and A is called A^♭ rather than G^♯. The C-based symmetric stack is rarely used, possibly because it produces the wolf fifth in the unusual position of F^♯–D^♭ instead of G^♯–E♭, where musicians accustomed to the previously used Pythagorean tuning might expect it).

Justly intonated quarter-comma meantone

A just intonation version of the quarter-comma meantone temperament may be constructed in the same way as Johann Kirnberger's rational version of 12-TET. The value of 5^1⁄8 · 35^1⁄3 is very close to 4, which is why a 7-limit interval 6144 : 6125 (which is the difference between the 5-limit diesis 128 : 125 and the septimal diesis 49 : 48), equal to 5.362 cents, appears very close to the quarter-comma (⁠81/80⁠)^1⁄4 of 5.377 cents. So the perfect fifth has the ratio of 6125 : 4096, which is the difference between three just major thirds and two septimal major seconds; four such fifths exceed the ratio of 5 : 1 by the tiny interval of 0.058 cents. The wolf fifth there appears to be 49 : 32, the difference between the septimal minor seventh and the septimal major second.

Greater and lesser semitones

As discussed above, in the quarter-comma meantone temperament,

• the ratio of a semitone is S = 8 : 5^5⁄4,
• the ratio of a tone is T = √5 : 2.

The tones in the diatonic scale can be divided into pairs of semitones. However, since S² is not equal to T, each tone must be composed of a pair of unequal semitones, S, and X:

${\displaystyle S\cdot X=T.}$

Hence,

${\displaystyle X={\frac {T}{S}}={\frac {\sqrt {5}}{2}}{\Bigg /}{\frac {8}{5^{5/4}}}={\frac {5^{1/2}\cdot 5^{5/4}}{8\cdot 2}}={\frac {5^{7/4}}{16}}.}$

Notice that S is 117.1 cents, and X is 76.0 cents. Thus, S is the greater semitone, and X is the lesser one. S is commonly called the diatonic semitone (or minor second), while X is called the chromatic semitone (or augmented unison).

The sizes of S and X can be compared to the just intonated ratio 18 : 17 which is 99.0 cents. S deviates from it by +18.2 cents, and X by −22.9 cents. These two deviations are comparable to the syntonic comma (21.5 cents), which this system is designed to tune out from the Pythagorean major third. However, since even the just intonated ratio 18 : 17 sounds markedly dissonant, these deviations are considered acceptable in a semitone.

In quarter-comma meantone, the minor second is considered acceptable while the augmented unison sounds dissonant and should be avoided.

Size of intervals

The table above shows only intervals from D. However, intervals can be formed by starting from each of the above listed 12 notes. Thus, twelve intervals can be defined for each interval type (twelve unisons, twelve semitones, twelve intervals composed of 2 semitones, twelve intervals composed of 3 semitones, etc.).

As explained above, one of the twelve nominal "fifths" (the wolf fifth) has a different size with respect to the other eleven. For a similar reason, each of the other interval types (except for unisons and octaves) has two different sizes in quarter-comma meantone when truncated to fit into an octave that only permits 12 notes (whereas actual quarter-comma meantone requires approximately 31 notes per octave). This is the price paid for attempting to fit a many-note temperament onto a keyboard without enough distinct pitches per octave: The consequence is "fake" notes, for example, one of the so-called "fifths" is not a fifth, but really a quarter-comma diminished sixth, whose pitch is a bad substitute for the needed fifth.

The table shows the approximate size of the notes in cents: The genuine notes are on a light grey background, the out-of-tune substitutes are on a red or orange background; the name for the genuine intervals are at the top or bottom of a column with plain grey background; the interval names of the bad substitutions are at opposite end, printed on a colored background. Interval names are given in their standard shortened form. ^[a] For instance, the size of the interval from D to A, which is a perfect fifth (P5), can be found in the seventh column of the row labeled D. strictly just (or pure) intervals are shown in bold font. Wolf intervals are highlighted in red. ^[b]

Approximate size in cents of the 144 intervals in D-based quarter-comma meantone tuning. Interval names are given in their standard short-form. Purely just intervals (which are only unisons, octaves, and some major thirds and minor sixths) are shown in bold font. Wolf intervals are highlighted in red. The red and gold colored intervals are out-of-tune substitutes for the missing correct meantone pitches, omitted because of the keyboard limit of 12 notes per octave.

Surprisingly, although this tuning system was designed to produce purely consonant major thirds, only eight of the intervals that are thirds in 12 TET are purely just ( 5 : 4 or about 386.3 cents) in the truncated quarter comma shown on the table: The actual quarter-comma notes needed to start or end the interval of a third are missing from among the 12 available pitches, and substitution of nearby available-but-wrong notes leads to dissonant thirds.

The reason why the interval sizes vary throughout the scale is from using substitute notes, whose pitches are correctly tuned for a different use in the scale, instead of the genuine quarter comma notes for the in desired interval, creates out-of-tune intervals. The actual notes in a fully implemented quarter-comma scale (requiring about 31 keys per octave instead of only 12) would be consonant, like all of the uncolored intervals: The dissonance is the consequence of replacing the correct quarter-comma notes with wrong notes that happen to be assigned to the same key on the 12 tone keyboard. As mentioned above, the frequencies defined by construction for the twelve notes determine two different kinds of semitones (i.e. intervals between adjacent notes):

• The minor second (m2), also called the diatonic semitone, with size

${\displaystyle S\ ({\mathsf {\ or\ }}S_{1})={\frac {\ 8\ }{~5^{5/4}}}\approx \ }$ 117.1 cents.

(for instance, between D and E♭)

• The augmented unison (A1), also called the chromatic semitone, with size

${\displaystyle X\ ({\mathsf {\ or\ }}S_{2})={\frac {~5^{7/4}}{\ 16\ }}\approx \ }$ 76.0 cents.

(for instance, between C and C♯)

Conversely, in an equally tempered chromatic scale, by definition the twelve pitches are equally spaced, all semitones having a size of exactly

${\displaystyle S_{\mathsf {ET}}={\sqrt[{12}]{2\ }}=100{\mathsf {\ cents.}}}$

As a consequence all intervals of any given type have the same size (e.g., all major thirds have the same size, all fifths have the same size, etc.). The price paid, in this case, is that none of them is justly tuned and perfectly consonant, except, of course, for the unison and the octave.

For a comparison with other tuning systems, see also this table.

By definition, in quarter-comma meantone 1 so-called "perfect" fifth (P5 in the table) has a size of approximately 696.6 cents ( 700 − ε cents, where ε ≈ 3.422 cents); since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of 700 + 11 ε cents, which is about 737.6 cents (one of the wolf fifths). Notice that, as shown in the table, the latter interval, although used as a substitute for a fifth, the actual interval is really a diminished sixth (d6), which is of course out of tune with the nearby but different fifth it replaces. Similarly,

• 10  major seconds (M2) are ≈ 193.2 cents ( 200 − 2 ε ), 2  diminished thirds (d3) are ≈ 234.2 cents ( 200 + 10 ε ), and their average is 200 cents;
• 9  minor thirds (m3) are ≈ 310.3 cents ( 300 + 3 ε ), 3  augmented seconds (A2) are ≈ 269.2 cents ( 300 − 9 ε ), and their average is 300 cents;
• 8  major thirds (M3) are ≈ 386.3 cents ( 400 − 4 ε ), 4  diminished fourths (d4) are ≈ 427.4 cents ( 400 + 8 ε ), and their average is 400 cents;
• 7 diatonic semitones (m2) are ≈ 117.1 cents( 100 + 5 ε ), 5 chromatic semitones (A1) are ≈ 76.0 cents ( 100 − 7 ε ), and their average is 100 cents.

In short, similar differences in width are observed for all interval types, except for unisons and octaves, and the excesses and deficits in width are all multiples of ε, the difference between the quarter-comma meantone fifth and the average fifth required if one is to close the spiral of fifths into a circle.

Notice that, as an obvious consequence, each augmented or diminished interval is exactly 12 ε cents (≈ 41.1 cents) wider or narrower than its enharmonic equivalent. For instance, the diminished sixth (or wolf fifth) is 12 ε cents wider than each perfect fifth, and each augmented second is 12 ε cents narrower than each minor third. This interval of size 12 ε cents is known as a diesis, or diminished second. This implies that ε can be also defined as one twelfth of a diesis.

Triads in the chromatic scale

The major triad can be defined by a pair of intervals from the root note: a major third (interval spanning 4 semitones) and a perfect fifth (7 semitones). The minor triad can likewise be defined by a minor third (3 semitones) and a perfect fifth (7 semitones).

As shown above, a chromatic scale has twelve intervals spanning seven semitones. Eleven of these are perfect fifths, while the twelfth is a diminished sixth. Since they span the same number of semitones, perfect fifths and diminished sixths are considered to be enharmonically equivalent. In an equally-tuned chromatic scale, perfect fifths and diminished sixths have exactly the same size. The same is true for all the enharmonically equivalent intervals spanning 4 semitones (major thirds and diminished fourths), or 3 semitones (minor thirds and augmented seconds). However, in the meantone temperament this is not true. In this tuning system, enharmonically equivalent intervals may have different sizes, and some intervals may markedly deviate from their justly tuned ideal ratios. As explained in the previous section, if the deviation is too large, then the given interval is not usable, either by itself or in a chord.

The following table focuses only on the above-mentioned three interval types, used to form major and minor triads. Each row shows three intervals of different types but which have the same root note. Each interval is specified by a pair of notes. To the right of each interval is listed the formula for the interval ratio. The intervals diminished fourth, diminished sixth and augmented second may be regarded as wolf intervals, and have their backgrounds set to pale red. S and X denote the ratio of the two abovementioned kinds of semitones (minor second and augmented unison).

3 semitones (m3 or A2)		4 semitones (M3 or d4)		7 semitones (P5 or d6)
Interval	Ratio	Interval	Ratio	Interval	Ratio
C–E♭	S 2 · X	C–E	S 2 · X 2	C–G	S 4 · X 3
C♯–E	S 2 · X	C♯–F	S 3 · X	C♯–G♯	S 4 · X 3
D–F	S 2 · X	D–F♯	S 2 · X 2	D–A	S4 · X 3
E♭–F♯	S · X 2	E♭–G	S 2 · X 2	E♭–B♭	S 4 · X 3
E–G	S 2 · X	E–G♯	S 2 · X 2	E–B	S 4 · X 3
F–G♯	S · X 2	F–A	S 2 · X 2	F–C	S 4 · X 3
F♯–A	S 2 · X	F♯–B♭	S 3 · X	F♯–C♯	S 4 · X 3
G–B♭	S 2 · X	G–B	S 2 · X 2	G–D	S 4 · X 3
G♯–B	S 2 · X	G♯–C	S 3 · X	G♯–E♭	S 5 · X 2
A–C	S 2 · X	A–C♯	S 2 · X 2	A–E	S 4 · X 3
B♭–C♯	S · X 2	B♭–D	S 2 · X 2	B♭–F	S 4 · X 3
B–D	S 2 · X	B–E♭	S 3 · X	B–F♯	S 4 · X 3

First, look at the last two columns on the right. All the 7 semitone intervals except one have a ratio of

${\displaystyle S^{4}\cdot X^{3}\approx 1.4953\approx 696.6{\text{ cents}}}$

which deviates by −5.4 cents from the just 3:2 of 702.0 cents. Five cents is small and acceptable. On the other hand, the diminished sixth from G^♯ to E^♭ has a ratio of

${\displaystyle S^{5}\cdot X^{2}\approx 1.5312\approx 737.6{\text{ cents}}}$

which deviates by +35.7 cents from the just perfect fifth. Thirty-five cents is beyond the acceptable range.

Now look at the two columns in the middle. Eight of the twelve 4-semitone intervals have a ratio of

${\displaystyle S^{2}\cdot X^{2}=1.25\approx 386.3{\text{ cents}}}$

which is exactly a just 5:4. On the other hand, the four diminished fourths with roots at C♯, F♯, G♯ and B have a ratio of

${\displaystyle S^{3}\cdot X=1.28\approx 427.4{\text{ cents}}}$

which deviates by +41.1 cents from the just major third. Again, this sounds badly out of tune.

Major triads are formed out of both major thirds and perfect fifths. If either of the two intervals is substituted by a wolf interval (d6 instead of P5, or d4 instead of M3), then the triad is not acceptable. Therefore, major triads with root notes of C♯, F♯, G♯ and B are not used in meantone scales whose fundamental note is C.

Now look at the first two columns on the left. Nine of the twelve 3-semitone intervals have a ratio of

${\displaystyle S^{2}\cdot X\approx 1.1963\approx 310.3{\text{ cents}}}$

which deviates by −5.4 cents from the just 6 : 5 of 315.6 cents. Five cents is acceptable. On the other hand, the three augmented seconds whose roots are E^♭, F and B^♭ have a ratio of

${\displaystyle S\cdot X^{2}\approx 1.1682\approx 269.2{\text{ cents}}}$

which deviates by −46.4 cents from the just minor third. It is a close match, however, for the 7 : 6 septimal minor third of 266.9 cents, deviating by +2.3 cents. These augmented seconds, though sufficiently consonant by themselves, will sound "exotic" or atypical when played together with a perfect fifth.

Minor triads are formed out of both minor thirds and fifths. If either of the two intervals are substituted by an enharmonically equivalent interval (d6 instead of P5, or A2 instead of m3), then the triad will not sound good. Therefore, minor triads with root notes of E^♭, F, G^♯ and B^♭ are not used in the meantone scale defined above.

Summary of utility of notes for triads

Usable major triad tonics		C, D, E♭, E, F, G, A, B♭
Usable minor triad tonics		C, C♯, D, E, F♯, G, A, B
[c] Usable for tonics of both major and minor triads		C, D, E, G, A
Only usable for major triad tonics		E♭, F, B♭
Only usable for minor triad tonics		C♯, F♯, B
Not usable as a major nor minor triad tonic		G♯

Note carefully that the limitations of what triads are feasible is determined by the choice to only allow 12 notes per octave, to conform with a standard piano keyboard. It is not a limitation of meantone tuning, per se, but rather the fact that sharps are different from the flats of the notes above them, and standard 12 note keyboards are built on the false assumption that they should be the same. As discussed above, G^♯ is a different pitch that A^♭, as are all other "enharmonic" pairs of sharps and flats in quarter comma meantone: Each requires a separate key on the keyboard and neither can substitute for the other. This is, in fact a property of all other tuning systems, with the exception of 12 tone equal temperament (alone among all equal temperaments) and well temperaments of all types. The limited chordal options is not a fault in meantone tunings; it is the consequence of needing more notes in the octave than is available on some modern equal tempered instruments. ^[d]

Alternative construction

As discussed above, in the quarter-comma meantone temperament truncated to only 12 notes,

• the ratio of a greater (diatonic) semitone is S = 8 : 5^5⁄4,
• the ratio of a lesser (chromatic) semitone is X = 5^7⁄4 : 16,
• the ratio of most whole tones is T = √5 : 2,
• the ratio of most fifths is P = ⁴√5.

It can be verified through calculation that most whole tones (namely, the major seconds) are composed of one greater and one lesser semitone:

${\displaystyle T=S\cdot X={\frac {8}{5^{5/4}}}\cdot {\frac {5^{7/4}}{16}}={\frac {\sqrt {5}}{2}}.}$

Similarly, a fifth is typically composed of three tones and one greater semitone:

${\displaystyle P=T^{3}\cdot S={\frac {5^{3/2}}{2^{3}}}\cdot {\frac {8}{5^{5/4}}}={\sqrt[{4}]{5}},}$

which is equivalent to four greater and three lesser semitones:

${\displaystyle P=T^{3}\cdot S=S^{4}\cdot X^{3}.}$

Diatonic scale

A diatonic scale can be constructed by starting from the fundamental note and multiplying it either by a meantone T to move up by one large step or by a semitone S to move up by a small step.

C    D    E    F    G    A    B    C′   D′ ‖----|----|----|----‖----|----|----‖----|   TTSTTTST

The resulting interval sizes with respect to the base note C are shown in the following table. To emphasize the repeating pattern, the formulas use the symbol P ≡ T³S to represent a perfect fifth (penta):

Note name	Formula	Ratio	Quarter comma (cents)	Pythagorean (cents)	12 TET (cents)
C	1	1.0000	0.0	0.0	0
D	T	1.1180	193.2	203.9	200
E	T 2	1.2500	386.3	407.8	400
F	T 2S	1.3375	503.4	498.0	500
G	P1	1.4953	696.6	702.0	700
A	PT	1.6719	889.7	905.9	900
B	PT 2	1.8692	1082.9	1109.8	1100
C′	PT 2S	2.0000	1200.0	1200.0	1200

Play tonic major chord ^ⓘ Play major third ^ⓘ Play perfect fifth ^ⓘ

Chromatic scale

Construction of a quarter-comma meantone chromatic scale can proceed by stacking a sequence of 12 semitones, each of which may be either the longer diatonic ( S ) or the shorter chromatic ( Χ ).

C    C♯   D    E♭   E    F    F♯  G    G♯   A    B♭   B    C′   C′♯ ‖----|----|----|----|----|----|----‖----|----|----|----|----‖----|   ΧSSΧSΧSΧSSΧ SΧ

Notice that this scale is an extension of the diatonic scale shown in the previous table. Only five notes have been added: C♯, E♭, F♯, G♯ and B♭ (a pentatonic scale).

As explained above, an identical scale was originally defined and produced by using a sequence of tempered fifths, ranging from E♭ (five fifths below D) to G♯ (six fifths above D), rather than a sequence of semitones. This more conventional approach, similar to the D-based Pythagorean tuning system, explains the reason why the Χ and S semitones are arranged in the particular and apparently arbitrary sequence shown above.

The interval sizes with respect to the base note C are presented in the following table. The frequency ratios are computed as shown by the formulas. Delta is the difference in cents between meantone and 12 TET; the column titled "⁠1/ 4 ⁠-c" is the difference in quarter-commas between meantone and Pythagorean tuning. Note that Χ ≡ ⁠T/ S ⁠ , so that Χ S = T ; most of the Χ steps appearing in the chart above disappear in the table below, because they combine with a preceding S and become a T.

Note name	Formula	Frequency ratio	Quarter comma (cents)	12 TET (cents)	Delta (cents)	⁠1/ 4 ⁠-c
C	1	1.0000	0.0	0	0.0	0
C♯	Χ	1.0449	76.0	100	−24.0	−7
D	T	1.1180	193.2	200	−6.8	−2
E♭	TS	1.1963	310.3	300	+10.3	+3
E	T 2	1.2500	386.3	400	−13.7	−4
F	T 2S	1.3375	503.4	500	+3.4	+1
F♯	T 3	1.3975	579.5	600	−20.5	−6
G	P1	1.4953	696.6	700	−3.4	−1
G♯	PΧ	1.5625	772.6	800	−27.4	−8
A	PT	1.6719	889.7	900	−10.3	−3
B♭	PTS	1.7889	1006.8	1000	+6.8	2
B	PT 2	1.8692	1082.9	1100	−17.1	−5
C′	PT 2S	2.0000	1200.0	1200	0.0	0
C′♯	PT 3	2.0898	1276.0	1300	−24.0	−7

Comparison with 31-tone equal temperament

The perfect fifth of quarter-comma meantone, expressed as a fraction of an octave, is ⁠1/4⁠ log₂ 5. Since log₂ 5 is an irrational number, a chain of meantone fifths never closes (i.e. never equals a chain of octaves). However, the continued fraction approximations to this irrational fraction number allow us to find equal divisions of the octave which do close; the denominators of these are 1, 2, 5, 7, 12, 19, 31, 174, 205, 789, ... From this we find that 31 quarter-comma meantone fifths come close to closing, and conversely 31 equal temperament represents a good approximation to quarter-comma meantone.

Footnotes

1. A for augmented (sharpened); d for diminished (flattened perfect, or flattened minor); M for major; m for minor ; P for perfect (even though P4 and P5 are not actually "perfect" fourths or fifths in meantone).
2. "Wolf" intervals are defined here in practice as any presumably consonant interval (which are all composed of 3, 4, 5, 7, 8, or 9 diatonic or chromatic semitones) whose size differs from just intonation by a syntonic comma, or more. Those are: Major and minor thirds or sixths, perfect fourths or fifths, and all their enharmonic equivalents. On the other hand, presumed dissonant intervals (made up of 1, 2, 6, 10, or 11 semitones) are major and minor seconds or sevenths, tritones, and their enharmonic equivalents. Since they are already expected to be dissonant, whether they are justly tuned or not, they are not marked as "wolf" intervals even when they do deviate from just intonation by more than one syntonic comma.
3. It is interesting that the five notes (C, D, E, G, A) that are usable as tonics for both a major and minor triads, themselves constitute a major pentatonic scale.
4. Many instruments are able to accommodate more distinct notes in an octave than pianos. For example, an orchestral harp has distinct pitches for every diatonic note, and all of its sharps, and all of its flats, giving 21 notes for each octave, and each octave can be tuned independently from every other. Fretless stringed instruments like the violin family, fretless guitars, and many lute-family are continuously tunable, as is the human voice.
   There are some rare piano keyboards made with split black keys that have either 17 or 19 notes per octave. Less rare are double-manual harpsichords, multi-manual acoustic organs, and other instruments incorporating multiple-keyboards (such as stacked MIDI keyboards) which conveniently provide 24 notes per octave, or 36, or 48, and so on.
   Having instruments with a larger number of possible notes allows using more of the individual pitches on the meantone scale, extending into all sharps and flats and even some double sharps and double flats, at a minimum (i.e., a 17 note per octave keyboard). The feasible triads for instruments with more notes per octaves is wider, and for quarter comma meantone and tunings close to it, can even add harmonic sevenths (a "lost chord" that comes gratis in quarter-comma as an augmented sixth, e.g. C-E-G-A^♯) to a few triads – and of course, with more notes per octave come more extended harmonies (see 31 tone equal temperament for a circulating tuning system close to quarter-comma that has equally well-tuned extended harmonic notes addable to all triads, e.g. all 7th harmonics and limited 11th and 13th harmonics).

References

1. Lindley, Mark (1974). "Early 16th century keyboard temperaments". Musica Disciplina . 28: 129–151. JSTOR   20532169.

External links

• " Quarter-comma meantone ", on Xenharmonic Wiki.