Tone structure (mathematical description)
A tone structure describes a tone system with the help of tones and intervals . Since ancient times, the sound supply of a musical culture has been reproduced on the one hand by specifying pitches and on the other hand by the term interval.
Nowadays, heights and intervals are described using frequencies and frequency relationships. The music theory of Pythagoras is known with the help of proportions (= string ratios on the monochord = reciprocal of the frequency ratios).
The mathematical teaching of tones and intervals is, however, also possible without these physical terms (see description of hearing psychology ). The first known hearing psychological descriptions of a sound system come from Aristoxenos .
The ordered pitch space
A frequency can be assigned to each tone .
- Example: c ' (the dashed c ) has the frequency 264 Hz , e' the frequency 330 Hz, g ' the frequency 396 Hz and c' ' the frequency 528 Hz.
Tones can be distinguished in height. The following applies: The higher a tone sounds, the greater its frequency. From a mathematical point of view, it is a (transitive and trichotomic) strict total order .
- Transitive means: From a higher than b and b higher than c follows a higher than c .
- Trichotomic means: For tones a and b the following applies: Either a = b or a higher than b or b higher than a .
The ordered additive interval space
Every two tones and (with the frequencies and ) is clearly assigned an interval (with the frequency ratio ).
- For example, the octave c'c '' has the frequency ratio 528: 264 = 2, the pure fifth C'G 'the frequency ratio 396: 264 = 3: 2, the major third c'e' the frequency ratio 330: 264 = 5: 4 and the minor third e'g 'the frequency ratio 396: 330 = 6: 5.
An end tone (with the frequency ) of the interval is clearly assigned to each initial tone (with the frequency ) and to each interval (with the frequency ratio ) .
- Example: If a 'has the frequency , then the tone c' ', which sounds a minor third with the frequency ratio higher, has the frequency .
In the language of musicians, intervals are added when performing one after the other. In this sense, the interval space has an additive structure.
- Example: major third + minor third = fifth .
- 12 fifths are roughly equal to 7 octaves . The difference is called the Pythagorean comma . One writes: Pythagorean comma = 12 fifths - 7 octaves . If you perform three pure major thirds one after the other (for example ce-gis-his ), you get an interval (from c to his ) that is slightly smaller than the octave . The difference is called little Diësis . Thus: minor Diësis = octave - 3 major thirds .
The addition of intervals corresponds to the multiplication of the frequency ratios and the subtraction of intervals corresponds to the division of the frequency ratios.
- Example: The addition of the minor third + major third = fifth corresponds to multiplication .
- The frequency ratio of the Pythagorean comma is calculated to and that of the small Diësis to .
Intervals can be compared in terms of size. The following applies: the larger the interval, the greater its frequency ratio .
The frequency ratio increases exponentially.
interval | Frequency ratio |
---|---|
1 octave | 2 |
2 octaves | 4th |
3 octaves | 8th |
••• | |
1 fifth | 3 / 2 |
2 fifths | 9 / 4 |
3 fifths | 27 / 8 |
••• |
From a mathematical point of view, an interval space is an Archimedean ordered commutative group .
Measure the size of intervals
Intervals can be given as multiples of an octave. Mostly, however, the subunit cents is used.
It is a logarithmic measure of the frequency relationships. The sub-unit cents with the definition 1200 cents = 1 octave (or 1 semitone with equal steps = 100 cents ) enables a very precise comparison of the interval size in whole numbers. This octave or cent measure is then proportional to the size of the interval.
interval | Frequency ratio | size |
---|---|---|
1 octave | 2 | 1200 cents |
2 octaves | 4th | 2400 cents |
3 octaves | 8th | 3600 cents |
... | ||
Octaves | ||
Octaves | ||
equal semitone = 1 ⁄ 12 octave | 100 cents | |
minor third | 6: 5 | |
major third | 5: 4 | |
Fifth | 3: 2 | |
Pythagorean comma | 531441: 524288 | |
little Diësis | 128: 125 |
- ( = Logarithm to base 10, = logarithm to base 2).
By using the logarithm when calculating cents, the multiplicative structure of the frequency ratios becomes the additive structure of the intervals again.
- Example:
- Quinte = minor third + major third = 315.641 + 386.314 cents cents = 701.955 cents.
- Pythagorean comma = 12 fifths - 7 octaves = 12 701.955 cents - 7 1200 cents = 23.460 cents.
- Small Diësis = octave - 3 major thirds = 1200 cents - 3 386.3137 cents = 41.059 cents (rounded).
Calculation of the interval size and the frequency ratio
If the frequency ratio of the interval, the size of the interval is calculated as:
Example: The fifth has the frequency ratio of . Then their size is calculated
On the other hand, if the interval is, the frequency ratio is calculated as:
Example 1: The interval of the size has the frequency ratio of:
Example 2: The fifth is 702 cents, exactly . The frequency ratio is then calculated as:
Examples of interval spaces
An interval space consists of the set of all intervals of the tone structure to be considered combined with the combination of the addition of the associated intervals.
In the following tables:
- Ok = octave (frequency ratio ),
- H = semitone (frequency ratio ),
- Q = fifth (frequency ratio ),
- Q m = ¼ decimal point mean fifth (frequency ratio ),
- T = third octave (frequency ratio ).
Name of the interval space | Interval space |
---|---|
The system of fifths interval space in Pythagorean tuning |
|
The ¼ decimal point mean-tone system of fifths Interval space of the mean-tone tuning |
|
The fifth-third system interval space of pure tuning |
|
The twelve-level interval space = interval space of equal tuning (error on pure tuning up to 23 cents) | |
The 53-step interval space (error on pure tuning up to 2 cents) | |
The all-encompassing interval space (all intervals can be divided as required.) |
Divisibility of intervals
In general, you cannot “divide” intervals by hearing. The "half fifth" 1 / 2 Q would be somewhere between small and large third and is in the tuning system of the Pythagorean neither nor on the mean, pure or equal to stage a mood occurring interval. Also, the half octave 1 / 2 Ok (600 cents) does not exist in the tuning system of the Pythagorean, on the mean or just intonation.
Pythagorean mood
The basis of the Pythagorean tuning is the fifth system with the following intervals:
interval | presentation | Frequency ratio | Size in cents |
---|---|---|---|
octave | Ok (basic interval) | 2: 1 | 1200 |
Fifth | Q (basic interval) | 3: 2 | 702 |
Whole tone | 2Q - Ok | 9: 8 | 204 |
Pythagorean major third ( Ditonos ) | 2 whole tones = 4Q - 2Ok | 81:64 | 408 |
Fourth | Ok - Q | 4: 3 | 498 |
Pythagorean semitone ( Limma ) | Quart-Ditonos = 3Ok - 5Q | 256: 243 | 90 |
pythagorean chromatic semitone ( apotome ) | Whole tone Limma = 7Q - 4Ok | 2187: 2048 | 114 |
Pythagorean comma | 12Q - 7Ok | 531441: 524288 | 23 |
detailed table |
Mid-tone mood
The basis of the ¼-point mean-tone tuning is the ¼-point-mean-tone fifth system with the following intervals:
interval | presentation | Frequency ratio | Indicated in cents |
---|---|---|---|
Octave ok | Ok (basic interval) | 2: 1 | 1200 |
Fifth Q m | Q m (basic interval) | 697 | |
Major third | 4Q m - 2OK = T | 5: 4 | 386 |
Fourth | Ok - Q m | 503 | |
Little sext | 3Ok - 4Q m = Ok - T | 8: 5 | 814 |
Minor third | 2ok - 3Q m | 310 | |
Great sext | 3Q m - Ok | 890 | |
Whole tone | 2Q m - Ok | 193 | |
Minor seventh | 2ok - 2Q m | 1007 | |
halftone | 3OK - 5Q m | 117 | |
Major seventh | 5Q m - 2Ok | 1083 | |
detailed table |
Names of the Euler's Tonnetz
In the pure tuning, it is not enough just to specify the tone name after the note image. A designation has to be added that shows whether the occurring fifths and thirds are pure. The names of Euler's clay net are helpful for this:
Pure fifths in the circle of fifths: ... es bfcgdae ...
A syntonic comma lower ..., es, b, c, g, d, a, e ... (deep comma before the tone name)
One syntonic comma higher ... 'es' b' c 'g' d 'a' e ... (apostrophe in front of the tone name)
Example: pure major third: c, e and perfect fifth c g.
Example: pure C major scale: cd, efg, a, h c.
Example: pure, A minor scale:, a, hc, d, efg, a.
Each major key is of the form: 1 2, 3 4 5, 6, 7 8 or '1' 2 3 '4' 5 6 7 '8 etc.
Each minor key is of the form: 1 2 '3 4 5' 6 '7 8 or, 1, 2 3, 4, 5 6 7, 8 etc., with "1" for the first tone and "2" for the second tone etc. the scale is up.
Pure mood
The basis of the pure tuning is the fifth-third system , which consists of the intervals of the form
- with the frequency relationships.
The main intervals are:
Interval (example) | presentation | Frequency ratio | Indicated in cents |
---|---|---|---|
Octave c c ' | Ok (basic interval) | 2: 1 | 1200 |
Fifth cg | Q (basic interval) | 3: 2 | 702 |
Major third c, e | T (basic interval) | 5: 4 | 386 |
Fourth cf. | Ok - Q | 4: 3 | 498 |
Small sext c'as | Ok - T | 8: 5 | 814 |
Minor third c 'es | Q - T | 6: 5 | 316 |
Major sixth c, a | Ok + T - Q | 5: 3 | 884 |
Big whole tone cd | 2Q - Ok | 9: 8 | 204 |
Small whole tone d, e | T - (large whole tone) = Ok + T - 2Q | 10: 9 | 182 |
Minor seventh gf (1st possibility) | Ok - (large whole tone) = 2Ok - 2Q | 16: 9 | 996 |
Minor seventh, ag (2nd possibility) | Ok - (small whole tone) = 2Q - T | 9: 5 | 1018 |
diatonic semitone, ef | Fourth - T = Ok - Q - T | 16:15 | 112 |
chromatic semitone c, cis or d ,, dis |
large whole tone - diatonic semitone = T + 3Q - 2Ok small whole tone - diatonic semitone = 2T - Q |
135: 128 25:24 |
92 71 |
Major seventh ch | Ok - semitone = Q + T | 15: 8 | 1088 |
Syntonic comma , ee | 2 (large whole tones) - T = 4Q - 2Ok - T | 81:80 | 22nd |
detailed table |
Super particulate ratios (engl. Super Particular ratios ) are of the form (n = 1,2,3, ...). The single intervals with such frequency receives nits are in the fifth-third system: octave ( 2 / 1 ), fifth ( 3 / 2 ), fourth ( 4 / 3 ), major third ( 5 / 4 ), minor third ( 6 / 5 ) , large whole tone ( 9 / 8 ), small whole tone ( 10 / 9 ), diatonic halftone ( 16 / 15 ), chromatic halftone ( 25 / 24 ) and syntonic comma ( 81 / 80 ). In the fifth-third system, the numerator and denominator of these fractions are only products of 2, 3 and 5.
It is important in this context: Intervals whose frequency ratio is super-particulate cannot be divided (in particular not halved).
In order to find out from a frequency ratio of the fifth-third system of which basic intervals the interval is composed, one must calculate the triple logarithm.
Example:
the equation
has the unique solution, called "triple logarithm": and .
The relationship (see syntonic comma ) therefore applies to the interval with the frequency ratio 81:80 .
The scales of pure tuning in the circle of fifths
When modulating into a neighboring key, two tones change, one of which is recognizable with a change in sign, the other slightly by a syntonic comma . This can best be represented with the names of Euler's Tone Network : For the tone that sounds a syntonic comma lower than x, the term, x (deep point x) is used. Correspondingly, 'x (apostrophe x) denotes the tone that is one syntonic comma higher than x. The fifths in the circle of fifths… as es bfcgda… are all pure (frequency ratio 3: 2).
The pure scales in the circle of fifths always have the same appearance:
scale | Scale tones listed in a table | ||||||||||||||||
C flat major | ces | of | ,it | fes | total | , as | , b | ces | , A flat minor | , as | , b | ces | ,of | ,it | fes | total | , as |
G flat major | total | as | , b | ces | of | ,it | , f | total | , E flat minor | ,it | , f | total | , as | , b | ces | of | ,it |
D flat major | of | it | , f | total | as | , b | , c | of | , B flat minor | , b | , c | of | ,it | , f | total | as | , b |
A flat major | as | b | , c | of | it | , f | ,G | as | , F minor | , f | ,G | as | , b | , c | of | it | , f |
E flat major | it | f | ,G | as | b | , c | , d | it | , C minor | , c | , d | it | , f | ,G | as | b | , c |
B flat major | b | c | , d | it | f | ,G | , a | b | ,G minor | ,G | , a | b | , c | , d | it | f | ,G |
F major | f | G | , a | b | c | , d | , e | f | , D minor | , d | , e | f | ,G | , a | b | c | , d |
C major | c | d | , e | f | G | , a | ,H | c | ,A minor | , a | ,H | c | , d | , e | f | G | , a |
G major | G | a | ,H | c | d | , e | , fis | G | , E minor | , e | , fis | G | , a | ,H | c | d | , e |
D major | d | e | , fis | G | a | ,H | , cis | d | , B minor | ,H | , cis | d | , e | , fis | G | a | ,H |
A major | a | H | , cis | d | e | , fis | , g sharp | a | , F sharp minor | , fis | , g sharp | a | ,H | , cis | d | e | , fis |
E major | e | f sharp | , g sharp | a | H | , cis | , dis | e | , c sharp minor | , cis | , dis | e | , fis | , g sharp | a | H | , cis |
B major | H | cis | , dis | e | f sharp | , g sharp | , ais | H | , G sharp minor | , g sharp | , ais | H | , cis | , dis | e | f sharp | , g sharp |
F sharp major | f sharp | g sharp | , ais | H | cis | , dis | ,ice | f sharp | , D flat minor | , dis | ,ice | f sharp | , g sharp | , ais | H | cis | , dis |
C sharp major | cis | dis | ,ice | f sharp | g sharp | , ais | , his | cis | , A sharp minor | , ais | , his | cis | , dis | ,ice | f sharp | g sharp | , ais |
The specified minor keys are natural minor. In the harmonic minor, the 6th and 7th degrees are still to be considered:
- For, E-flat minor ,, c and ,, d
- at, b minor ,, g and ,, a
- at, F minor ,, d and ,, e
- in, C minor ,, a and ,, h
- at, G minor ,, e and ,, f sharp
- with, D minor ,, B and ,, C sharp
- in, A minor ,, f sharp and ,, g sharp
- at, E minor ,, c sharp and ,, d flat
- with, B minor ,, G sharp and ,, a sharp
- with, F sharp minor ,, d flat and ,, ice
- in, c sharp minor ,, a sharp and ,, his
- in, G sharp minor ,, eis and ,, fisis
- at, D-flat minor ,, his and ,, cisis
- at, a sharp minor,, f sharp and, g sharp
When transitioning to the minor key, the following tones should be added
- in ces minor the notes' eses / 'asas /' heses
- in G minor the notes' heses / 'eses /' fes
- in D flat minor the notes' fes / 'heses /' ces
- in A flat minor the notes' ces / 'fes /' ges
- in E flat minor the tones' gt / 'ces /' des
- in B flat minor the notes' des / 'gt /' a flat
- in F minor the notes' a flat / 'des /' es
- in c minor the tones' es / 'a flat /' b
- in G minor the notes' b / 'es /' f
- in D minor the notes' f / 'b /' c
- in A minor the notes' c / 'f /' g
- in E minor the notes' g / 'c /' d
- in B minor the notes' d / 'g /' a
- in F sharp minor the notes' a / 'd /' e
- in c sharp minor the notes' e / 'a /' h
The cent values of the tones are calculated as follows:
Adjacent tones that differ only by the schism (2 cents) are marked with *. | |||
volume | Cent value | Occurrence | |
---|---|---|---|
c | 0 * | in C major | |
, his | 2 * | from C sharp major | |
'c | 22nd | from D minor | |
,, cis | 71 | ab, E minor | |
of | 90 * | from A flat major | |
, cis | 92 * | from D major | |
'of | 112 * | from F minor | |
cis | 114 * | from B major | |
,, d | 161 | from, F minor | |
, d | 182 | from F major | |
'eses | 202 * | in G minor | |
d | 204 * | in C major | |
'd | 225 | from E minor | |
,it | 273 * | from G flat major | |
,, dis | 275 * | from, F sharp minor | |
it | 294 * | from B flat major | |
, dis | 296 * | from E major | |
'it | 316 * | in c minor | |
dis | 318 * | from C sharp major | |
,, e | 365 | from, G minor | |
fes | 384 * | from C flat major | |
, e | 386 * | in C major | |
'fes | 406 * | from a flat minor | |
e | 408 * | from D major | |
'e | 429 | from F sharp minor | |
, f | 477 * | from A flat major | |
,,ice | 478 * | off, G sharp minor | |
f | 498 * | in C major | |
,ice | 500 * | from F sharp major | |
'f | 520 | from g minor | |
,, f sharp | 569 | from, A minor | |
total | 588 * | from D flat major | |
, fis | 590 * | from G major | |
'total | 610 * | from B flat minor | |
f sharp | 612 * | from E major | |
,,G | 659 | ab, B flat minor | |
,G | 680 * | from B flat major | |
,, fisis | 682 * | ab, a sharp minor | |
G | 702 | in C major | |
'G | 723 | from A minor | |
, as | 771 * | from C flat major | |
,, g sharp | 773 * | from, B minor | |
as | 792 * | from E flat major | |
, g sharp | 794 * | from A major | |
'as | 814 * | in c minor | |
g sharp | 816 * | from F sharp major | |
,, a | 863 | ab, C minor | |
, a | 884 * | in C major | |
,, gisis | 886 * | ab, a sharp minor | |
'heses | 904 * | from d flat minor | |
a | 906 * | from G major | |
'a | 927 | from B minor | |
, b | 975 * | from D flat major | |
,, ais | 977 * | ab, C sharp minor | |
b | 996 * | from F major | |
, ais | 998 * | from B major | |
'b | 1018 | in c minor | |
,,H | 1067 | ab, D minor | |
ces | 1086 * | from G flat major | |
,H | 1088 * | in C major | |
'ces | 1108 * | from G flat major | |
H | 1110 * | from A major | |
'H | 1131 | from c sharp minor | |
,, c | 1157 | from, E flat minor | |
, c | 1178 * | from E flat major | |
,, his | 1180 * | ab, D flat minor | |
c ' | 1200 | in C major |
The calculation of the cent values here can be carried out according to the following scheme. With p = 1/12 Pythagorean comma ≈ 2.0 cents, the Pythagorean circle of fifths results in ... es = 300-3p b = 1000-2p f = 500-pc = 0 g = 700 + pd = 200 + 2p a = 900 + 3p ... arranged according to semitones:
Equal | Pythagorean | enharmonic |
---|---|---|
0 | c = 0 | his = 12p |
100 | cis = 100 + 7p | des = 100-5p |
200 | d = 200 + 2p | eses = 200-10p |
300 | dis = 300 + 9p | es = 300-3p |
400 | e = 400 + 4p | fes = 400-8p |
500 | f = 500-p | ice = 500 + 11p |
600 | f sharp = 600 + 6p | tot = 600-6p |
700 | g = 700 + p | asas = 700-11p |
800 | g sharp = 800 + 8p | as = 800-4p |
900 | a = 900 + 3p | heses = 900-9p |
1000 | ais = 1000 + 10p | b = 1000-2p |
1100 | h = 1100 + 5p | ces = 1100-7p |
1200 | c = 1200 | deses = 1200-12p |
With p = 1 / 12 Pythagorean comma ≈ 2.0 cents and K = syntonic point ≈ 21.5 cents is calculated as:
- ,, cis = (100 + 7p-2K) Cent = 71 cents (= interval c ,, cis = interval from c to ,, cis)
- 'as = 800-4p + K = 814 cents (= interval from c' as )
- Interval ,, cis' as = (700-11p + 3K) cents = 743 cents.
Frequency ratio 2 (700-11p + 3K) / 1200 = 192 / 125
Equal mood
The basis of equal tuning is the 12-step interval space with the following intervals:
interval | presentation | Size in cents |
---|---|---|
halftone | H | 100 cents |
Whole tone | 2H | 200 cents |
minor third | 3H | 300 cents |
major third | 4H | 400 cents |
... | ||
detailed table |
The division of the octave into 53 pitches
The basis of this mood is the 53-step interval space . The octave is divided into 53 equal parts.
At the time of Zarlinus (1517–1590) it was taught in music schools that the major third can be intone only and that there are deviations from the Pythagorean tuning. It has been taught that the scale should be intoned in such a way that parts can be assigned to the following intervals.
- cd = fg = 9 parts (large whole tone)
- de = ga = 8 parts (small whole tone)
- ef = hc = 5 parts (diatonic semitone)
If you note the distance of the scale from C in brackets and the distance between the notes written lower down, the C major scale reads:
c(0) 9 d(9) 8 ,e(17) 5 f(22) 9 g(31) 8 ,a(39) 9 ,h(48) 5 c(53)
, e ("low point e") means here in a modification of Euler's notation : ", e sounds 1/53 octave lower than e" etc.
The scale is divided into 53 parts here, with
große Terz c,e = 17 Teile Quinte = cg = 31 Teile
The scales in the circle of fifths are then written as follows (for the sake of clarity, all notated from c):
C-Dur: c(0) d(9) ,e(17) f(22) g(31) ,a(39) ,h(48) c(53) G-Dur: c(0) d(9) ,e(17) ,fis(26) g(31) a(40) ,h(48) c(53) D-Dur: ,cis(4) d(9) e(18) ,fis(26) g(31) a(40) ,h(48) ,cis(57) A-Dur: ,cis(4) d(9) e(18) ,fis(26) ,gis(35) a(40) h(49) ,cis(57) E-Dur: ,cis(4) ,dis(13) e(18) fis(27) ,gis(35) a(40) h(49) ,cis(57) H-Dur: cis(5) ,dis(13) e(18) fis(27) ,gis(35) ,ais(44) h(49) cis(58) Fis-Dur: cis(5) ,dis(13) ,eis(22) fis(27) gis(36) ,ais(44) h(49) cisis(61) Cis-Dur: cis(5) dis(14) ,eis(22) fis(27) gis(36) ,ais(44) ,his(53) cis(58)
C-Dur: c(0) d(9) ,e(17) f(22) g(31) ,a(39) ,h(48) c(53) F-Dur: c(0) ,d(8) ,e(17) f(22) g(31) ,a(39) b(44) c(53) B-Dur: c(0) ,d(8) es(13) f(22) ,g(30) ,a(39) b(44) c(53) Es-dur: ,c(52) ,d(8) es(13) f(22) ,g(30) as(35) b(44) ,c(52) As-dur: ,c(52) des(4) es(13) ,f(21) ,g(30) as(35) b(44) ,c(52) Des-dur: ,c(52) des(4) es(13) ,f(21) ges(26) as(35) ,b(43) ,c(52) Ges-dur: ces(48) des(4) ,es(12) ,f(21) ges(26) as(35) ,b(43) ces(48) Ces-dur: ces(48) des(4) ,es(12) fes(17) ges(26) ,as(34) ,b(43) ces(48)
Hermann von Helmholtz writes the following in his theory of tone sensations: “If you want to produce a scale in an almost exact natural tuning, which allows unlimited modulation, ... this can be achieved by dividing the octave into 53 equal intervals, as suggested by Mercator . "
The 53-level uniform tuning
step | Distance from c in cents | Tones of the circle of fifths in cents |
---|---|---|
00 | 0 | c = 0, his = 2 |
01 | 23 | 'c = 22 his = 23 |
02 | 45 | |
03 | 68 | ,, cis = 71 |
04 | 91 | des = 90, cis = 92 |
05 | 113 | 'des = 112 cis = 114 |
06 | 136 | '' des = 133 |
07 | 158 | ,, d = 161 |
08 | 181 | , d = 182 ,, cisis = 184 |
09 | 204 | 'eses = 202 d = 204, cisis = 206 |
10 | 226 | 'd = 225 cisis = 227 |
11 | 249 | |
12 | 272 | , es = 273 ,, dis275 |
13 | 294 | es = 294, dis = 296 |
14th | 317 | 'es = 316 dis = 318 |
15th | 340 | '' es = 337 |
16 | 362 | ,, e = 365 |
17th | 385 | fes = 384, e = 386 |
18th | 408 | 'fes = 406 e = 408 |
19th | 430 | 'e = 429 |
20th | 453 | |
21st | 475 | , f = 477 ,, ice = 478 |
22nd | 498 | f = 498, ice = 500 |
23 | 521 | 'f = 520 ice = 522 |
24 | 543 | |
25th | 566 | ,, f sharp = 569 |
26th | 589 | ges = 588, f sharp = 590 |
27 | 611 | 'ges = 610 f sharp = 612 |
28 | 634 | |
29 | 657 | ,, g = 659 |
30th | 679 | , g = 680 ,, fisis = 682 |
31 | 702 | g = 702, fisis = 704 |
32 | 725 | 'g = 723 fisis = 725 |
33 | 747 | |
34 | 770 | , as = 771 ,, g sharp = 772 |
35 | 792 | as = 792, g sharp = 794 |
36 | 815 | 'as = 814 g sharp = 816 |
37 | 838 | |
38 | 860 | ,, a = 863 |
39 | 883 | , a = 884 ,, g sharp = 886 |
40 | 906 | 'heses = 904 a = 906 |
41 | 928 | 'a = 927 g sharp = 929 |
42 | 951 | |
43 | 974 | , b = 975 ,, ais = 977 |
44 | 996 | b = 996, ais = 998 |
45 | 1019 | 'b = 1018 ais = 1020 |
46 | 1042 | |
47 | 1064 | ,, h = 1067 |
48 | 1087 | ces = 1086, h = 1088 |
49 | 1109 | 'ces = 1108 h = 1110 |
50 | 1132 | 'h = 1131 |
51 | 1155 | ,, c = 1157 |
52 | 1177 | , c = 1178 ,, his1180 |
53 | 1200 | c = 1200 |
Interval table with comparison with the pure mood
interval | Size in cents | Level in the 53 system | Size in cents | Difference exactly |
---|---|---|---|---|
diet. halftone | 112 | 05 | 113 | −1.48 |
small whole tone | 182 | 08 | 181 | +1.29 |
large whole tone | 204 | 09 | 204 | +0.13 |
minor third | 316 | 14th | 317 | −1.34 |
major third | 386 | 17th | 385 | +1.40 |
Fourth | 498 | 22nd | 498 | −0.07 |
Tritone | 590 | 26th | 589 | +0.07 |
Fifth | 702 | 31 | 702 | −1.41 |
little sext | 814 | 36 | 815 | −1.01 |
great sext | 884 | 39 | 883 | +1.34 |
Minor Seventh I. | 996 | 44 | 996 | −0.14 |
Minor seventh II | 1018 | 45 | 1019 | −1.27 |
major seventh | 1088 | 48 | 1087 | +1.47 |
octave | 1200 | 53 | 1200 | 0.00 |
Schismatic mix-up
You can see here: All notes of the circle of fifths are reached with a tolerance of a schism . The tones c and, his, des and, cis etc. differ by the schism of 1.95 cents (see third column in the first table with two tones each).
If you look at all the tones that you get when you look at the circle of fifths to infinity, then these tones are recorded on the 53-point scale with a maximum deviation from a Kleisma. According to Tanaka , a Kleisma (Greek: κλεῖσμα "closure") is the distance from '' 'fes to ,,, eis or' '' ges to ,,, fisis or '' 'ces to ,,, his etc. With' ' 'fes = 448.89 cents and ,,, eis = 456.99 cents results in: Kleisma = 8.11 cents .
Examples in detail
Intervals of equal tuning
Frequency ratio | Interval size in cents | Interval designation |
---|---|---|
1 | 0 | Prim |
100 | equal semitone | |
200 | equal whole tone | |
300 | equal minor third | |
400 | equal major third | |
500 | equal fourths | |
600 | equal tritone | |
700 | equal fifth | |
800 | equal minor sixth | |
900 | equal major sixth | |
1000 | equal minor seventh | |
1100 | equal major seventh | |
2 | 1200 | octave |
Pythagorean tuning intervals
The following table gives an overview of the intervals that can occur in Pythagorean tuning. Each of the intervals was calculated: C -Cis, C-Des *, CD, C-Dis *, C-Es, CE, ..., Cis -Dis *, Cis-Es, Cis-E, Cis-F, Cis-Fis ,…, Des * -Es, Des * -E,…, D -Dis *, D-Es, DE,… The intervals were then sorted according to size in cents. At the same intervals, only one representative was selected.
In the Pythagorean tuning, the fifths of the sequence Gb * -Des * -As * -Es-BFCGDAEH-F sharp-C sharp-G sharp-D sharp * -A sharp * are pure (frequency ratio 3: 2).
Note: The tones Gb *, Db *, A-flat *, D-sharp * and A sharp * are not available on a 12-point scale. They differ from their enharmonically confused ones by the Pythagorean comma.
Each interval can be clearly represented as the sum of the two basic intervals octave and fifth.
- Ok = octave (frequency ratio 2: 1)
- Q = fifth (frequency ratio 3: 2).
interval | from C to | Frequency ratio | in cents | calculation | Interval designation |
---|---|---|---|---|---|
C sharp * | Deses | 524288/531441 | −23.460 | −12Q + 7Ok | Pythagorean diminished second = - Pythagorean comma |
EF | Of | 256/243 | 90.225 | −5Q + 3Ok | Pythagorean Limma = Pythagorean small second |
C-Cis | Cis | 2187/2048 | 113.685 | 7Q - 4Ok | Pythagorean apotome = Pythagorean excessive prim |
C # Eb | Eses | 65536/59049 | 180.450 | −10Q + 6Ok | Pythagorean minor third |
CD | D. | 9/8 | 203.910 | 2Q - Ok | large whole tone = Pythagorean second |
Des * -Dis * | Cisis | 4782969/4194304 | 227,370 | 14Q - 8Ok | Pythagorean double excessive prim |
Dis * -Ges * | Feses | 16777216/14348907 | 270.675 | −15Q + 9Ok | Pythagorean double diminished fourth |
DF | It | 32/27 | 294.135 | −3Q + 2Ok | Pythagorean minor third |
Eb-F sharp | Dis | 19683/16384 | 317,595 | 9Q - 5Ok | Pythagorean excessive second |
C sharp F | Fes | 8192/6561 | 384,360 | −8Q + 5Ok | Pythagorean diminished fourth |
CE | E. | 81/64 | 407.820 | 4Q - 2Ok | Pythagorean major third = Ditonos |
Ges * -Ais * | Disis | 43046721/33554432 | 431.280 | 16Q - 9Ok | Pythagorean double excessive second |
C sharp Ges * | Geses | 2097152/1594323 | 474,585 | −13Q + 8Ok | Pythagorean double diminished fifth |
CF | F. | 4/3 | 498.045 | −Q + Ok | Fourth |
Eb-G sharp | ice | 177147/131072 | 521.505 | 11Q - 6Ok | Pythagorean excessive third |
EB | Ges | 1024/729 | 588.270 | −6Q + 4Ok | Pythagorean diminished fifth |
C-F sharp | F sharp | 729/512 | 611.730 | 6Q - 3Ok | Pythagorean excessive fourth = Pythagorean tritone |
G sharp es | Asas | 262144/177147 | 678.495 | −11Q + 7Ok | Pythagorean diminished sixth |
CG | G | 3/2 | 701.955 | Q | Fifth |
Eb-Ais * | Fisis | 1594323/1048576 | 725,415 | 13Q - 7Ok | Pythagorean double augmented fourth |
Ais * -ges * | Heseses | 67108864/43046721 | 768.720 | −16Q + 10Ok | Pythagorean double diminished seventh |
Ec | As | 128/81 | 792.180 | −4Q + 3Ok | Pythagorean minor sext |
C G sharp | G sharp | 6561/4096 | 815.640 | 8Q - 4Ok | Pythagorean excessive fifth |
C sharp B | Heses | 32768/19683 | 882,405 | −9Q + 6Ok | Pythagorean diminished seventh |
CA | A. | 27/16 | 905.865 | 3Q - Ok | Pythagorean major sixth |
Des * -Ais * | Gisis | 14348907/8388608 | 929,325 | 15Q - 8Ok | Pythagorean double superficial fifth |
Dis * -des * | ceses | 8388608/4782969 | 972.630 | −14Q + 9Ok | Pythagorean double diminished octave |
CB | B. | 16/9 | 996.090 | −2Q + 2Ok | Pythagorean minor seventh |
Es-cis | Ais | 59049/32768 | 1019,550 | 10Q - 5Ok | Pythagorean excessive sixth |
C sharp c | ces | 4096/2187 | 1086.315 | −7Q + 5Ok | Pythagorean diminished octave |
CH | H | 243/128 | 1109.775 | 5Q - 2Ok | Pythagorean major seventh |
C sharp * | deses | 1048576/531441 | 1176.540 | −12Q + 8Ok | Pythagorean diminished ninth (= Ok - Pythagorean diminished second) |
Cc | c | 2/1 | 1200 | OK | octave |
Intervals of the ¼ decimal point mean pitch
The following table gives an overview of the intervals that can occur in the mid-tone tuning. Each of the intervals was calculated: (C) - (Cis), (C) - (Des *), (C) - (D), (C) - (Dis *), (C) - (Es), (C ) - (E), ..., (Cis) - (Dis *), (Cis) - (Eb), (Cis) - (E), (Cis) - (F), (Cis) - (Fis), ... , (Des *) - (Es), (Des *) - (E),…, (D) - (Dis *), (D) - (Es), (D) - (E), ... The intervals became then sorted by size (in cents). At the same intervals, only one representative was selected.
The fifths of the sequence (Gb *) - (Db *) - (As *) - (Es) - (B) - (F) - (C) - (G) - (D ) - (A) - (E) - (H) - (F sharp) - (C sharp) - (G sharp) - (D sharp *) - (A sharp *) by a quarter of the syntonic comma (frequency ratio 81:80) smaller (or closer) than the perfect fifth. So these fifths have the frequency ratio
Note: The tones (Ges *), (Des *), (As *), (Dis *) and (Ais *) are not available on a 12-point scale. They differ from their enharmonically confused ones by the small Diësis (41 cents). Intervals of the form, for example (Cis) - (Des *), however, give an impression of the impurities that occur in enharmonic mix-ups.
The frequency ratio in the third column is often algebraic-irrational. Here means
Each interval can be clearly represented as the sum of the two basic intervals of the mean-tone fifth system.
- Ok = octave
- Q m = mid-tone fifth.
The major third T = (C) - (E) can be represented here as T = 4Q m - 2Ok. The respective calculation appears in the 4th column.
interval | from C to | Frequency ratio | in cents | calculation | Interval designation |
---|---|---|---|---|---|
(Cis) - (Des *) | (Deses) | 128: 125 | 41.059 | −12Q m + 7Ok = −3T + Ok | (larger) diminished second = small Diësis |
(C) - (Cis) | (Cis) | (5:16) w 3 | 76.049 | 7Q m - 4Ok = 2T - Q m | chromatic mid-tone semitone |
(E) - (F) | (Of) | (8:25) w 3 | 117.108 | −5Q m + 3Ok = −T - Q m + Ok | diatonic mid-tone semitone |
(Des *) - (Dis *) | (Cisis) | (125: 256) w 2 | 152.098 | 14Q m - 8Ok = 4T - 2Q m | medium-tone double excessive prim |
(CD) | (D) | (1: 2) w 2 | 193.157 | 2Q m - Ok | mid-tone whole tone |
(Cis) - (Es) | (Eses) | (64: 125) w 2 | 234,216 | −10Q m + 6Ok = −3T + 2Q m | mid-tone diminished third |
(Es) - (F sharp) | (Dis) | (25:32) w | 269.206 | 9Q m - 5Ok = 2T + Q m - Ok | mean-tone excessive second |
(D) - (F) | (It) | (4: 5) w | 310.265 | −3Q m + 2Ok = −T + Q m | medium-tone minor third |
(Ges *) - (Ais *) | (Disis) | 625: 512 | 345.255 | 16Q m - 9Ok = 4T - Ok | mid-tone double excessive second |
(Dis *) - (Ges *) | (Feses) | (512: 625) w | 351,324 | −15Q m + 9Ok = −4T + Q m + Ok | mid-tone double diminished fourth |
(C) - (E) | (E) | 5: 4 | 386,314 | 4Q m - 2Ok = T | major third |
(Cis) - (F) | (Fes) | 32:25 | 427,373 | −8Q m + 5Ok = −2T + Ok | diminished fourth |
(Es) - (G sharp) | (Ice) | (25:64) w 3 | 462,363 | 11Q m - 6Ok = 3T - Q m | medium-tone excessive third |
(C) - (F) | (F) | (2: 5) w 3 | 503,422 | −Q m + Ok | medium-tone fourth |
(Cis) - (Gb *) | (Geses) | (256: 625) w 3 | 544,480 | −13Q m + 8Ok = −3T - Q m + 2Ok | mid-tone doubly diminished fifth |
(F) - (H) | (F sharp) | (5: 8) w 2 | 579,471 | 6Q m - 3Ok = T + 2Q m - Ok | medium-tone excessive fourth, medium-tone tritone |
(Cis) - (G) | (Ges) | (16:25) w 2 | 620.529 | −6Q m + 4Ok = −2T + 2Q m | medium-tone diminished fifth |
(Des *) - (G sharp) | (Fisis) | (125: 128) w | 655.520 | 13Q m - 7Ok = 3T + Q m - Ok | mid-tone double excessive fourth |
(C) - (G) | (G) | w | 696,578 | Q m | mean fifth |
(G sharp) - (es) | (Asas) | (128: 125) w | 737.637 | −11Q m + 7Ok = −3T + Q m + Ok | mid-tone diminished sixth |
(C) - (G sharp) | (G sharp) | 25:16 | 772.627 | 8Q m - 4Ok = 2T | small excessive fifth, double third |
(E) - (c) | (As) | 8: 5 | 813.686 | −4Q m + 3Ok = −T + Ok | small sixth |
(Des *) - (Ais *) | (Gisis) | (125: 256) w 3 | 848.676 | 15Q m - 8Ok = 4T - Q m | mid-tone doubly excessive fifth |
(Ais *) - (ges *) | (Broom) | 1024: 625 | 854.745 | −16Q m + 10Ok = 4T + 2Ok | mid-tone double diminished seventh |
(C) - (A) | (A) | (1: 2) w 3 | 889.735 | 3Q m - Ok = T - Q m + Ok | medium-tone major sixth |
(Cis) - (B) | (Bes) | (64: 125) w 3 | 930.794 | −9Q m + 6Ok = −2T - Q m + 2Ok | mid-tone diminished seventh |
(Es) - (cis) | (Ais) | (25:32) w 2 | 965.784 | 10Q m - 5Ok = 2T + 2Q m - Ok | medium-tone excessive sixth |
(D) - (c) | (B) | (4: 5) w 2 | 1006,843 | −2Q m + 2Ok | medium-tone minor seventh |
(G sharp) - (total *) | (ceses) | (512: 625) w 2 | 1047.902 | −14Q m + 9Ok = −4T + 2Q m + Ok | mid-tone double diminished octave |
(C) - (H) | (H) | (5: 4) w | 1082.892 | 5Q m - 2Ok = T + Q m | mean major seventh |
(Cis) - (c) | (ces) | (32:25) w | 1123,951 | −7Q m + 5Ok = −2T + Q m + Ok | mid-tone diminished octave |
(Es) - (dis *) | (his) | 125: 64 | 1158.941 | 12Q m - 6Ok = 3T | excessive seventh |
(C) - (c) | (c) | 2: 1 | 1200 | OK | octave |
Intervals of pure mood
The following table gives an overview of the intervals that can occur with pure tuning. Starting from the chromatic scale C 'D' D 'E -flat , EF, F sharp G' A -flat , A 'B, HC, each of the intervals is calculated: C -, C sharp / C-' Des / CD / C - ,, Dis / C- 'Es / C-, E /… / , Cis - ,, Dis /, Cis-'Es /, Cis-, E /, Cis-F /, Cis-, Fis /… / D - ,, Dis / D- 'Es / D-, E / ... (for the designations see Euler's Tonnetz : "Low point x" with the designation ", x" means ", x" is a syntonic comma lower than "x". "Quotation x" with the designation "' x "is a syntonic comma higher than" x ". The pure C major scale is written as" CD, EFG, A, B c ". The pure C minor scale is written as" CD 'Es FG' A 'B c'). The intervals were then sorted by size (in cents). At the same intervals, only one representative was selected.
The interval reference is C major and C minor with the pure chords C-, EG / C-'Es-G / F-, Ac / F-'As-c / G-, HD and G-'Bd / supplemented by more Intermediate tones with the diatonic semitone steps (frequency ratio 16/15) C-'Des /, C sharp-D / ,, D-sharp, E / F-'Ges /, F sharp-G / ,, G sharp, A and ,, A sharp, H.
Each interval can be clearly represented as the sum of the three basic intervals of the fifth-third system.
- Ok = octave
- Q = fifth and
- T = major third.
The respective calculation appears in the 5th column.
interval | from C to | Frequency ratio | in cents | calculation | Interval designation |
---|---|---|---|---|---|
Des-, Cis | , His | 32805: 32768 | 1.954 | T + 8Q - 5Ok | minor augmented seventh - octave, schism |
, Cis-'Des | 'Deses | 2048: 2025 | 19,553 | −2T - 4Q + 3Ok | (smaller) diminished second, diaschism |
,, Dis-'Es | '' 'Deses | 128: 125 | 41.059 | −3T + Ok | (larger) diminished second, little Diësis |
D - ,, Dis | ,, Cis | 25:24 | 70.672 | 2T - Q | (minor) excessive prim, minor chromatic halftone , minor chroma |
C-, Cis | , Cis | 135: 128 | 92.179 | T + 3Q - 2Ok | (larger) excessive prim, large chromatic halftone , large chroma |
, EF | 'Of | 16:15 | 111.731 | −T - Q + Ok | small seconds , diatonic semitone |
,FROM | ''Of | 27:25 | 133.238 | −2T + 3Q - Ok | (bigger) small seconds, big Limma, |
'Des - ,, Dis | ,,, Cisis | 1125: 1024 | 162.851 | 3T + 2Q - 2Ok | double excessive prim |
D-, E | , D | 10: 9 | 182,404 | T - 2Q + Ok | small whole tone (smaller major second) |
CD | D. | 9: 8 | 203.910 | 2Q - Ok | large whole tone = Pythagorean whole tone (larger major second) |
, E-total | 'Eses | 256: 225 | 223,463 | −2T - 2Q + 2Ok | (minor) diminished third |
,, G sharp-'B | '' 'Eses | 144: 125 | 244,969 | −3T + 2Q | (major) diminished third |
C - ,, Dis | ,, Dis | 75:64 | 274,582 | 2T + Q - Ok | excessive second |
DF | It | 32:27 | 294.135 | −3Q + 2Ok | Pythagorean minor third (impure minor third of the 2nd degree) |
C-'Es | 'It | 6: 5 | 315.641 | −T + Q | minor third |
,, Dis-'Ges | '' 'Feses | 4096: 3375 | 335.194 | −3T - 3Q + 3Ok | double diminished fourth |
'Ges - ,, Ais | ,,, Disis | 10125: 8192 | 366.761 | 3T + 4Q - 3Ok | double excessive second |
C-, E | , E | 5: 4 | 386,314 | T | major third |
D-Total | 'Fes | 512: 405 | 405.866 | −T - 4Q + 3Ok | (smaller) diminished fourth |
, A-, cis | E. | 81:64 | 407.820 | 4Q - 2Ok | Pythagorean major third = Ditonos |
, E-'As | '' Fes | 32:25 | 427,373 | −2T + Ok | diminished fourth |
'Es - ,, G sharp | ,,,Ice | 125: 96 | 456.986 | 3T - Q | (minor) excessive third |
F - ,, Ais | ,,Ice | 675: 512 | 478,492 | 2T + 3Q - 2Ok | (major) excessive third |
CF | F. | 4: 3 | 498.045 | −Q + Ok | Fourth |
, C sharp 'total | 'Geses | 8192: 6075 | 517,598 | −2T - 5Q + 4Ok | doubly diminished fifth |
, Ad | 'F | 27:20 | 519,551 | −T + 3Q - Ok | impure fourths (in C major, 2nd degree ad) |
,, Dis-'As | '' 'Geses | 512: 375 | 539.104 | −3T - Q + 2Ok | doubly diminished fifth |
D - ,, G sharp | ,, Fis | 25:18 | 568.717 | 2T - 2Q + Ok | (minor) excessive fourths |
'Ges-, cis | ,, Fisis | 6075: 4096 | 682,402 | 2T + 5Q - 3Ok | double diminished fourth |
C-, Fis | , Fis | 45:32 | 590.224 | T + 2Q - Ok | Tritone , excessive fourth |
, F sharp c | 'Ges | 64:45 | 609.776 | −T - 2Q + 2Ok | (minor) diminished fifth |
, A-'es | '' Ges | 36:25 | 631.283 | −2T + 2Q | (major) diminished fifth |
'It - ,, Ais | ,,, Fisis | 375: 256 | 660.896 | 3T + Q - Ok | double excessive fourth |
THERE | ,G | 40:27 | 680,449 | T - 3Q + 2Ok | impure fifth (in C major because of the second degree chord) |
CG | G | 3: 2 | 701.955 | Q | Fifth |
, H-'ges | 'Asas | 1024: 675 | 721.508 | −2T - 3Q + 3Ok | (smaller) diminished sixth |
,, Dis-'B | '' 'Asas | 192: 125 | 743.014 | −3T + Q + Ok | (larger) diminished sixth |
C - ,, G sharp | ,, G sharp | 25:16 | 772.627 | 2T | small excessive fifth, double third |
, C sharp, A | As | 128: 81 | 792.180 | −4Q + 3Ok | Pythagorean minor sixth |
F-, cis | , G sharp | 405: 256 | 794.134 | T + 4Q - 2Ok | (larger) excessive fifth |
, Ec | 'As | 8: 5 | 813.686 | −T + Ok | small sixth |
,, Ais-'ges | '' 'Broom | 16384: 10125 | 833.239 | −3T - 4Q + 4Ok | double diminished seventh |
'Des - ,, Ais | ,,, Gisis | 3375: 2048 | 864,806 | 3T + 3Q - 2Ok | double excessive fifth |
C-, A | , A | 5: 3 | 884.359 | T - Q + Ok | major sixth |
Fd | A. | 27:16 | 905.865 | 3Q - Ok | pyth. major sixth (in 2nd chord) |
, E-'des | '' Bes | 128: 75 | 925,418 | −2T - Q + 2Ok | (major) diminished seventh |
'B - ,, g sharp | ,,, Ais | 125: 72 | 955.031 | 3T - 2Q + Ok | (minor) excessive sixth |
C - ,, Ais | ,, Ais | 225: 128 | 976,537 | 2T + 2Q - Ok | (larger) excessive sixth |
Dc | B. | 16: 9 | 996.090 | −2Q + 2Ok | minor minor seventh (= octave - major whole tone) |
C-'B | 'B | 9: 5 | 1017,596 | −T + 2Q | major minor seventh (= octave - small whole tone) |
,, Dis-'des | '' 'ceses | 2048: 1125 | 1037.149 | −3T - 2Q + 3Ok | double diminished octave |
'B - ,, ais | ,,, his | 125: 64 | 1158.941 | 3T | excessive seventh |
'B-, a | ,,H | 50:27 | 1066,762 | 2T - 3Q + 2Ok | (minor) major seventh |
C-, H | ,H | 15: 8 | 1088.269 | T + Q | major seventh |
, C sharp-c | 'ces | 256: 135 | 1107.821 | −T - 3Q + 3Ok | (smaller) probably octave |
,, Dis-d | '' ces | 48:25 | 1129,328 | −2T + Q + Ok | (larger) diminished octave |
'Des-, cis | ,, his | 2025: 1024 | 1180,447 | 2T + 4Q - 2Ok | (larger) overm. Seventh |
Cc | c | 2: 1 | 1200 | OK | octave |
Intervals sorted by size
Designations:
C-Cis-Des * -D-D-flat * -Es-E ... Pythagorean scale supplemented by semitones, based on perfect fifths.
(C) - (Cis) - (Des *) - (D) - (Dis *) - (Es) - (E) - (F) -… ¼-point mean-tone scale supplemented by semitones, based on mean-tone fifths ( 696.6 cents).
C-, Cis-'Des-D - ,, D-'Es-, E ... Pure scale supplemented by semitones (for names see Euler's Tonnetz : "Low point x" with the name ", x" means ", x" is a syntonic one Comma lower than "x". "Quotation x" with the designation "'x" is a syntonic comma higher than "x").
- Ok = octave (frequency ratio 2)
- Q = fifth (frequency ratio 3: 2)
- Q m = mid-tone fifth (frequency ratio )
- T = major third (frequency ratio 5: 4).
Intervals | from C to |
Frequency ratio | in cents | calculation | Interval designation |
---|---|---|---|---|---|
CC | C. | 1: 1 | 0 | Prim | |
, His | 32805: 32768 | 1.954 | 8Q + T - 5Ok | Schism = difference between Pythagorean and syntonic commas | |
, Cis-'Des | 'Deses | 2048: 2025 | 19,553 | −2T - 4Q + 3Ok | (smaller) diminished second, diaschism |
'C | 81:80 | 21.506 | 4Q - T - 2Ok | syntonic comma : difference d (C major) and, d (F major) | |
Des * -Cis | His | 531441: 524288 | 23,460 | 12Q - 7Ok | Pythagorean comma |
(Dis) - (Es) = ,, Dis-'Es |
(Deses) = '' 'Deses |
128: 125 | 41.059 | −12Q m + 7Ok = −3T + Ok | (in the pure tuning: larger) diminished second = minor diësis (difference from octave to 3 major thirds). |
'' '' Deses | 648: 625 | 62.565 | 4Q - 4T - Ok | major Diësis = difference of four minor thirds to the octave | |
D - ,, Dis | ,, Cis | 25:24 | 70.672 | 2T - Q | (minor) excessive prim, minor chromatic halftone , minor chroma |
(C) - (Cis) | (Cis) | (5:16) w 3 | 76.049 | 7Q m - 4Ok | chromatic mid-tone semitone |
EF | Of | 256: 243 | 90.225 | −5Q + 3Ok | Pythagorean Limma = Pythagorean small second |
C-, Cis | , Cis | 135: 128 | 92.179 | T + 3Q - 2Ok | (larger) excessive prim, large chromatic halftone , large chroma |
100 | (1:12) Ok | small equal second | |||
, EF | 'Of | 16:15 | 111.731 | −T - Q + Ok | small seconds, diatonic semitone |
C-Cis | Cis | 2187: 2048 | 113.685 | 7Q - 4Ok | Pythagorean apotome = Pythagorean excessive prim |
(E) - (F) | (Of) | (8:25) w 3 | 117.108 | −5Q m + 3Ok | diatonic mid-tone semitone |
,FROM | ''Of | 27:25 | 133.238 | −2T + 3Q - Ok | (bigger) small seconds, big Limma, |
(Des *) - (Dis *) | (Cisis) | (125: 256) w 2 | 152.098 | 14Q m - 8Ok | medium-tone double excessive prim |
'Des - ,, Dis | ,,, Cisis | 1125: 1024 | 162.851 | 3T + 2Q - 2Ok | double excessive prim |
C # Eb | Eses | 65536: 59049 | 180.450 | −10Q + 6Ok | Pythagorean minor third |
D-, E | , D | 10: 9 | 182,404 | T - 2Q + Ok | small whole tone |
(CD) | (D) | (1: 2) w 2 | 193.157 | 2Q m - Ok | mid-tone whole tone |
200 | (2:12) Ok | large equal second | |||
CD | D. | 9: 8 | 203.910 | 2Q - Ok | large whole tone = Pythagorean second |
, E-total | 'Eses | 256: 225 | 223,463 | −2T - 2Q + 2Ok | (minor) diminished third |
Des * -Dis * | Cisis | 4782969: 4194304 | 227,370 | 14Q - 8Ok | Pythagorean double excessive prim |
(Cis) - (Es) | (Eses) | (64: 125) w 2 | 234,216 | −10Q m + 6Ok | mid-tone diminished third |
,, G sharp-'B | '' 'Eses | 144: 125 | 244,969 | −3T + 2Q | (major) diminished third |
(Es) - (F sharp) | (Dis) | (25:32) w | 269.206 | 9Q m - 5Ok | mean-tone excessive second |
Dis * -Ges * | Feses | 16777216: 14348907 | 270.675 | −15Q + 9Ok | Pythagorean double diminished fourth |
C - ,, Dis | ,, Dis | 75:64 | 274,582 | 2T + Q - Ok | excessive second |
DF | It | 32:27 | 294.135 | −3Q + 2Ok | Pythagorean minor third (impure minor third of the 2nd degree) |
300 | (3:12) Ok | minor equal third | |||
(D) - (F) | (It) | (4: 5) w | 310.265 | −3Q m + 2Ok | medium-tone minor third |
C-'Es | 'It | 6: 5 | 315.641 | −T + Q | minor third |
Eb-F sharp | Dis | 19683: 16384 | 317,595 | 9Q - 5Ok | Pythagorean excessive second |
,, Dis-'Ges | '' 'Feses | 4096: 3375 | 335.194 | −3T - 3Q + 3Ok | double diminished fourth |
(Ges *) - (Ais *) | (Disis) | 625: 512 | 345.255 | 16Q m - 9Ok = 4T - Ok | mid-tone double excessive second. (Disis) = ,,,, Disis. |
(Dis *) - (Ges *) | (Feses) | (512: 625) w | 351,324 | −15Q m + 9Ok | mid-tone double diminished fourth |
'Ges - ,, Ais | ,,, Disis | 10125: 8192 | 366.761 | 3T + 4Q - 3Ok | double excessive second |
C sharp F | Fes | 8192: 6561 | 384,360 | −8Q + 5Ok | Pythagorean diminished fourth |
(C) - (E) = C-, E |
(E) =, E. |
5: 4 | 386,314 | 4Q m - 2Ok = T | major third |
400 | (4:12) Ok | major equal third | |||
D-Total | 'Fes | 512: 405 | 405.866 | −T - 4Q + 3Ok | (smaller) diminished fourth |
, A-, cis | E. | 81:64 | 407.820 | 4Q - 2Ok | Pythagorean major third = Ditonos |
(Cis) - (F) =, E-'As |
(Fes) = '' Fes |
32:25 | 427,373 | −8Q m + 5Ok = Ok - 2T | diminished fourth |
Ges * -Ais * | Disis | 602409: 469571 | 431.280 | 16Q - 9Ok | Pythagorean double excessive second |
'Es - ,, G sharp | ,,,Ice | 125: 96 | 456.986 | 3T - Q | (minor) excessive third |
(Es) - (G sharp) | (Ice) | (25:64) w 3 | 462,363 | 11Q m - 6Ok | medium-tone excessive third |
C sharp Ges * | Geses | 2097152: 1594323 | 474,585 | −13Q + 8Ok | Pythagorean double diminished fifth |
F - ,, Ais | ,,Ice | 675: 512 | 478,492 | 2T + 3Q - 2Ok | (major) excessive third |
CF | F. | 4: 3 | 498.045 | −Q + Ok | Fourth |
500 | (5:12) Ok | equal fourths | |||
(C) - (F) | (F) | (2: 5) w 3 | 503,422 | −Q m + Ok | medium-tone fourth |
, C sharp 'total | 'Geses | 8192: 6075 | 517,598 | −2T - 5Q + 4Ok | doubly diminished fifth |
, Ad | 'F | 27:20 | 519,551 | −T + 3Q - Ok | impure fourths (in C major, 2nd degree ad) |
Eb-G sharp | ice | 177147: 131072 | 521.505 | 11Q - 6Ok | Pythagorean excessive third |
,, Dis-'As | '' 'Geses | 512: 375 | 539.104 | −3T - Q + 2Ok | doubly diminished fifth |
(Cis) - (Gb *) | (Geses) | (256: 625) w 3 | 544,480 | −13Q m + 8Ok | mid-tone doubly diminished fifth |
11: 8 | 551,318 | Just as a supplement: The Alphorn Fa (the 11th natural tone) | |||
D - ,, G sharp | ,, Fis | 25:18 | 568.717 | 2T - 2Q + Ok | (minor) excessive fourths |
(F) - (H) | (F sharp) | (5: 8) w 2 | 579,471 | 6Q m - 3Ok | medium-tone excessive fourth, medium-tone tritone |
EB | Ges | 1024: 729 | 588.270 | −6Q + 4Ok | Pythagorean diminished fifth |
C-, Fis | , Fis | 45:32 | 590.224 | T + 2Q - Ok | Tritone, excessive fourth |
600 | (6:12) Ok | equal tritone, excessive equal fourth, diminished equal fifth | |||
, F sharp c | 'Ges | 64:45 | 609.776 | −T - 2Q + 2Ok | (minor) diminished fifth |
C-F sharp | F sharp | 729: 512 | 611.730 | 6Q - 3Ok | Pythagorean excessive fourth = Pythagorean tritone |
(Cis) - (G) | (Ges) | (16:25) w 2 | 620.529 | −6Q m + 4Ok | medium-tone diminished fifth |
, A-'es | '' Ges | 36:25 | 631.283 | −2T + 2Q | (major) diminished fifth |
(Des *) - (G sharp) | (Fisis) | (125: 128) w | 655.520 | 13Q m - 7Ok | medium-tone double excessive fourth |
'It - ,, Ais | ,,, Fisis | 375: 256 | 660.896 | 3T + Q - Ok | double excessive fourth |
G sharp es | Asas | 262144: 177147 | 678.495 | −11Q + 7Ok | Pythagorean diminished sixth |
THERE | ,G | 40:27 | 680,449 | T - 3Q + 2Ok | impure fifth (in C major because of the second degree chord) |
'Ges-, cis | ,, Fisis | 6075: 4096 | 682,402 | 2T + 5Q - 3Ok | double diminished fourth |
(C) - (G) | (G) | w | 696,578 | Q m | mean fifth |
700 | (7:12) Ok | equal fifth | |||
CG | G | 3: 2 | 701.955 | Q | Fifth |
, H-'ges | 'Asas | 1024: 675 | 721.508 | −2T - 3Q + 3Ok | (smaller) diminished sixth |
Eb-Ais * | Fisis | 1594323: 1048576 | 725,415 | 13Q - 7Ok | Pythagorean double augmented fourth |
(G sharp) - (es) | (Asas) | (128: 125) w | 737.637 | −11Q m + 7Ok | mid-tone diminished sixth |
,, Dis-'B | '' 'Asas | 192: 125 | 743.014 | −3T + Q + Ok | (larger) diminished sixth |
Ais * -ges * | Broom | 67108864: 43046721 | 768.720 | −16Q + 10Ok | Pythagorean double diminished seventh |
(C) - (G sharp) = C - ,, G sharp |
(G sharp) = ,, G sharp |
25:16 | 772.627 | 8Q m - 4Ok = 2T | (In the pure tuning smaller) excessive fifth, double third |
Ec | As | 128: 81 | 792.180 | −4Q + 3Ok | Pythagorean minor sixth |
F-, cis | , G sharp | 405: 256 | 794.134 | T + 4Q - 2Ok | (larger) excessive fifth |
800 | (8:12) Ok | small equal sixth | |||
, Ec | 'As | 8: 5 | 813.686 | −T + Ok | small sixth |
C G sharp | G sharp | 6561: 4096 | 815.640 | 8Q - 4Ok | Pythagorean excessive fifth |
,, Ais-'ges | '' 'Broom | 16384: 10125 | 833.239 | −3T - 4Q + 4Ok | double diminished seventh |
(Des *) - (Ais *) | (Gisis) | (125: 256) w 3 | 848.676 | 15Q m - 8Ok | medium-tone doubly excessive fifth |
(Ais *) - (ges *) | (Broom) | 1024: 625 | 854.745 | −16Q m + 10Ok = −4T + 2Ok | mid-tone double diminished seventh. (Broom) = '' '' broom. |
'Des - ,, Ais | ,,, Gisis | 3375: 2048 | 864,806 | 3T + 3Q - 2Ok | double excessive fifth |
C sharp B | Bes | 32768: 19683 | 882,405 | −9Q + 6Ok | Pythagorean diminished seventh |
C-, A | , A | 5: 3 | 884.359 | T - Q + Ok | major sixth |
(C) - (A) | (A) | (1: 2) w 3 | 889.735 | 3Q m - Ok | medium-tone major sixth |
900 | (9:12) Ok | large equal sixth | |||
CA | A. | 27:16 | 905.865 | 3Q - Ok | Pythagorean major sixth |
, E-'des | '' Bes | 128: 75 | 925,418 | −2T - Q + 2Ok | (major) diminished seventh |
Des * -Ais * | Gisis | 14348907: 8388608 | 929,325 | 15Q - 8Ok | Pythagorean double superficial fifth |
(Cis) - (B) | (Bes) | (64: 125) w 3 | 930.794 | −9Q m + 6Ok | mid-tone diminished seventh |
'B - ,, g sharp | ,,, Ais | 125: 72 | 955.031 | 3T - 2Q + Ok | (minor) excessive sixth |
(Es) - (cis) | (Ais) | (25:32) w 2 | 965.784 | 10Q m - 5Ok | medium-tone excessive sixth |
7: 4 | 968.826 | i | Just to complement: The natural septime , the 7th natural tone, sometimes referred to as i. | ||
Dis * -des * | Ceses | 8388608: 4782969 | 972.630 | −14Q + 9Ok | Pythagorean double diminished octave |
C - ,, Ais | ,, Ais | 225: 128 | 976,537 | 2T + 2Q - Ok | (larger) excessive sixth |
Dc | B. | 16: 9 | 996.090 | −2Q + 2Ok | Pythagorean minor seventh |
1000 | (10:12) Ok | small equal seventh | |||
(D) - (c) | (B) | (4: 5) w 2 | 1006,843 | −2Q m + 2Ok | medium-tone minor seventh |
C-'B | 'B | 9: 5 | 1017,596 | −T + 2Q | minor seventh |
Es-cis | Ais | 59049: 32768 | 1019,550 | 10Q - 5Ok | Pythagorean excessive sixth |
,, Dis-'des | '' 'ceses | 2048: 1125 | 1037.149 | −3T - 2Q + 3Ok | double diminished octave |
(G sharp) - (total *) | (ceses) | (512: 625) w 2 | 1047.902 | −14Q m + 9Ok | mid-tone double diminished octave |
'B-, a | ,,H | 50:27 | 1066,762 | 2T - 3Q + 2Ok | (minor) major seventh |
(C) - (H) | (H) | (5: 4) w | 1082.892 | 5Q m - 2Ok | mean major seventh |
C sharp c | Ces | 4096: 2187 | 1086.315 | −7Q + 5Ok | Pythagorean diminished octave |
C-, H | ,H | 15: 8 | 1088.269 | T + Q | major seventh |
1100 | (11:12) Ok | major seventh of the same order | |||
, C sharp-c | 'ces | 256: 135 | 1107.821 | −T - 3Q + 3Ok | (smaller) diminished octave |
CH | H | 243: 128 | 1109.775 | 5Q - 2Ok | Pythagorean major seventh |
(Cis) - (c) | (ces) | (32:25) w | 1123,951 | −7Q m + 5Ok | mid-tone diminished octave |
,, Dis-d | '' ces | 48:25 | 1129,328 | −2T + Q + Ok | (larger) diminished octave |
(Es) - (dis *) = 'B - ,, ais |
(his) = ,,, his |
125: 64 | 1158.941 | 12Q m - 6Ok = 3T | excessive seventh |
C sharp * | deses | 1048576: 531441 | 1176.540 | −12Q + 8Ok | Pythagorean diminished ninth (= Ok - Pythagorean diminished sec.) |
'Des-, cis | ,, his | 2025: 1024 | 1180,447 | 2T + 4Q - 2Ok | (major) excessive seventh |
Cc | 2: 1 | 1200 | OK | octave |
Description of the tone structure in terms of hearing psychology without acoustics
The understanding of tones and intervals can be conveyed without physical terms. The first known auditory-psychological mathematical descriptions of a sound system come from Aristoxenus . The pitch of a certain tone can be determined and passed on by a "original" tuning fork without specifying its frequency (similar to how the unit meter can be determined by the original meter ). A teacher can “show” his student what an octave, a fifth, a major third, etc. is without going into the frequency relationship of the vibrations. The underlying theory is explained below.
Description of the tone structure as an algebraic structure
With a tone structure you have a set of tones on the one hand and a set of intervals on the other , for which the following rules apply:
A unique interval of is assigned to each tone pair .
Conversely, if the fundamental tone and the interval are known, the final tone is uniquely determined.
The successive execution of intervals defines an addition: is and , then is .
Intervals can be compared: We write when the final note of is higher than the final note of with the same root note.
Everyday calculating with quantities applies to intervals on the additive musical level . From a mathematical point of view, the interval space is an Archimedean ordered commutative group . From a purely auditory psychological perspective, this results from the experience of musical practice.
To measure the interval size suitable as a unit , the octave with the subunit cents 1200 cents = 1 octave.
Example 1 (octave = 12 semitones)
- If you go up 12 fifths, you get the starting tone again with an octave (approximately): 12 fifths = 7 octaves. The result is a fifth = 7 ⁄ 12 octave = 700 cents . Corresponding:
- If you go up three major thirds, you get (roughly) an octave. So major third = 1 ⁄ 3 octave = 400 cents . You can now continue to calculate :
- Minor third = fifth - major third = 1 ⁄ 4 octave = 300 cents and
- Semitone = major third - minor third = 1 ⁄ 12 octave = 100 cents .
- From a purely psychological point of view, you can divide the octave (approximately) into 12 semitones and represent each interval as a multiple of semitones.
Example 2 (octave = 53 commas)
At the time of Zarlino (16th century) one taught in music schools: The large whole tone has a size of 9 parts , the small whole tone of 8 parts and the diatonic semitone of 5 parts .
It follows from this:
- Octave = 1200 cents = 3 large whole tones + 2 small whole tones + 2 diatonic semitones = 53 parts
- Major third = major whole tone + small whole tone = 17 parts = 385 cents
- minor third = major whole tone + diatonic semitone = 14 parts = 317 cents
- Fifth = major third + minor third = 31 parts = 702 cents
With this classification, the proportions for the pure intonation of tone steps can be easily described.
- diatonic semitone = 5 parts
- small whole tone = 8 parts
- Large whole tone = 9 parts
- diminished third (see example B - G sharp = BA (5 parts) + A G sharp (5 parts) = 10 parts
This division of the octave into 53 parts can be derived purely mathematically from two integer relationships for the three intervals Ok = octave, Q = fifth and gT = major third without reference to the frequency relationships. (Confirmed on the spinet by Neumaier)
- 53 Q = 31 Ok (no difference between initial tone and octaves audible after 53 fifths)
- 12 Q - 7Ok = 4Q - 2Ok -gT (no difference between syntonic comma and Pythagorean comma audible)
Dissolved This system of equations with k = 1 / 53 Ok:
- Ok = 53k
- Q = 31k
- gT = 17k
Now you can define further intervals and represent them as multiples of k: For example:
- Fourth = Ok - Q = 22k
- minor third = Q - gT = 14k
- large whole tone = 2Q - Ok = 9k
- small whole tone = gT - large whole tone = 8k
- diatonic semitone = gT - minor third = 5k
Example 3 (the fifth third system)
Axiom : There is a homomorphism f from the additive group of the interval space with the intervals Ok = octave, Q = fifth and gT = major third into the multiplicative group of real numbers, for which applies:
- f (Ok) = 2
- f (Q) = 3 / 2 and
- f (GT) = 5 / 4
Homomorphism says: f (i 1 + i 2 ) = f (i 1 ) • f (i 2 ) and f (r • i) = f (i) r for intervals i 1 , i 2 and i as well as for a real number r.
For the calculation of r and s for Q = r • Ok and gT = s • Ok it follows with the subunit Ok = 1200 cents:
- f (r • Ok) = 2 r = 3 / 2 ie Q = log 2 ( 3 / 2 ) Ok = 701.955 cents
- f (s • Ok) = 2 s = 5 / 4 so gT = log 2 ( 5 / 4 ) Ok = 386.314 cents.
Web links
Remarks
- ↑ Sources: Rudolf Wille : "Mathematics and Music Theory", in Music and Numbers, Bonn - Bad Godesberg 1976, pp. 233–264 and "Mathematical Language in Music Theory", in the Yearbook Overviews Mathematics 1980, pp. 167–184. Wilfried Neumaier: "What is a tone system? A historical-systematic theory of the western tone systems, based on the ancient theorists Aristoxenus, Eucleides and Ptolemaios, represented by means of modern algebra." Verlag Peter Lang, Frankfurt / Main ISBN 3-8204-9492-8
- ↑ The information relates to the pure tuning , with which intervals can be assigned whole-number relationships.
- ↑ Euclid calculated with proportions, namely with string ratios that correspond to the reciprocal of the frequency ratios.
- ↑ Except for powers of two (integer multiples of the octave) logarithms of two of pure ( rational ) frequency ratios are irrational and even transcendent .
- ↑ The cent unit is so small that the difference from, for example, a schism (2 cents) is to be located at the “limit of perceptible tone differences”.
- ↑ The Pythagorean Archytas of Taranto (approx. 400 BC) proved that the octave, the fifth and fourth, etc., cannot be halved if commensurable quantities are used as a basis.
- ↑ https://eclass.uoa.gr/modules/document/index.php?course=MATH379&download=/5122bb9396op/5122bf834msr.pdf
- ↑ Note: 700-11p has the frequency ratio: (2/3) 11 • 2 7 (11 fifths octaved down, see asas) ⇒ 2 (700-11p + 3K) / 1200 = (2/3) 11 • 2 7 • (81/80) 3 = 192 / 125
- ↑ In Euler's notation - a notation for the pure mood , the deep decimal means a decrease by the syntonic comma = 21.5 cents. Here the low point means a decrease of 1200/53 cents = 22.6 cents. A deviation of 1 cent cannot be distinguished by hearing.
- ↑ The approximation of the octave by fifths (12 fifths corresponds to approximately 7 octaves) led to equal tempering by dividing the octave into 12 equal intervals. It has the disadvantage of very rough major thirds. The closest approximation (41 fifths corresponds to approximately 24 octaves) is better for an equal division of the octave into 41 parts, but not satisfactory with regard to the major third and the displacements around a syntonic comma. The following approximation of the octave (53 fifths corresponds to almost exactly 31 octaves) has a convincing advantage: If you divide the octave into 53 equal intervals, then the 31st degree (701.887 cents) corresponds very precisely to the perfect fifth (701.955 cents) and - that is particularly important and not to be expected - the 17th step (384.906 cents) of the major third (386.314 cents) and the shift by a syntonic comma (21.506 cents) by almost exactly one step (22.642 cents) of this tempering.
- ↑ Hermann von Helmholtz : The theory of tone sensations as a physiological basis for the theory of music . Vieweg, Braunschweig 1863, p. 531 (reprint: Minerva-Verlag, Frankfurt am Main 1981), ISBN 3-8102-0715-2 , ( excerpt ). Helmholtz continues: “Mr. Bosanquet has recently used such a tuning for a harmonium with a symmetrium arranged keyboard. [An elementary Treatease on Musical Intervals and Temperament by. RHM Bosanquet, London. Macmillan 1875] ”.
- ↑ In contrast to the pure or medium-tone tuning, in the Pythagorean tuning the tone C sharp is higher than Des or - better known - His is higher than c. Therefore the note Deses is lower than C and the interval Cis-Deses * or C-Deses is notated negatively. The interval Cis-des * or C-deses, increased by one octave, is notated here as a Pythagorean reduced ninth. To get from C sharp to D flat or from His to C you have to go twelve fifths down and seven octaves up. The Pythagorean comma is known as an interval = twelve fifths up and seven octaves down.
-
↑ a b c Winfried Neumaier p. 64ff shows: Already Aristoxenus calculated in the 3rd century BC as described in this section. He calculated with octave, fifth, fourth = octave - fifth, whole tone = fifth - fourth and with the help of the axiom that the whole tone can still be divided, with semitones and even with quarter tones (but not with pure major thirds). As an empirical value he “heard”: fourth = 2½ whole tones and based on this a coherent theory. (Euclid recognized: 2½ whole tones are slightly smaller than the fourth.)
According to Neumaier, for example, on the spinet you can still verify: 53 fifths = 31 octaves (no more hearing difference) and this then results in: fifth = 31 ⁄ 53 octave = 702 cents . So you can determine very precise values for interval sizes without acoustics. - ↑ In addition to clarity, this is important for the interpretation of historical descriptions of the pitch system. According to Wilfried Neumaier What is a sound system. A historical-systematic theory of the occidental sound systems, based on the ancient theorists Aristoxenus, Eucleides and Ptolemaios, presented with the means of modern algebra (= sources and studies on the history of music from antiquity to the present. Vol. 9). Peter Lang, Frankfurt am Main a. a. 1986, ISBN 3-8204-9492-8
- ↑ The next better approximation would be: 28 major thirds = 9 octaves (hardly comprehensible to the ear), so major third = 9 ⁄ 28 octaves = 386 cents .
-
↑ The exact values of the intervals in the pure tuning , which are calculated with the help of the frequency ratios, differ only slightly from the values determined here:
- major third (neat) = 1200 • log 2 ( 5 / 4 ) = 386 cents
- minor third (neat) = 1200 • log 2 ( 6 / 5 ) = 316 cents
- Fifth (neat) = 1200 • log 2 ( 3 / 2 ) = 702 cents
-
↑ The deviation from the pure mood is smaller than a schism (2 cents ).
- Ok = 1200 cents (So k = 1,200 / 53 cents = 22.642 cents)
- Q = 1200 * log 2 ( 3 / 2 ) = 701.955 cent cents. 31k = 701.887 cents
- gT = 1200 * log 2 ( 5 / 4 )) = 386.3137 cent cents. 17k = 384.906 cents
- ↑ If no scalar multiplication is assumed in the interval space, the definition applies . This smallest upper bound does not always have to exist. For example, the interval space of all multiples of Ok, Q and gT does not contain .