Modulation with pure tuning

Of pure mood is when the triads of the tonic, the subdominant and the dominant are pure. By this is meant that the occurring fifths (frequency ratio ³ / ₂ ), major thirds (frequency ratio ⁵ / ₄ ) and minor thirds (frequency ratio ⁶ / ₅ ) pure.

As a rule of thumb, the following applies: When modulating into a neighboring key (major or minor), two tones change, one of which is recognizable with a change in sign, the other slightly by a syntonic comma .

Each modulation to the next key in the circle of fifths can be explained in two stages, namely first the modulation into a parallel key, in which only one note changes, and then a further modulation into the target key, in which another note changes.

Modulation towards the dominant in major

The 2nd and 7th notes of the target key increase.

When modulating from C major to G major, the A around a syntonic comma and the F in F sharp ,

more accurate:

Modulation from C major via E minor to G major

C major

Name of the tone		C.		D.		E.		F.		G		A.		H		c
frequency		264		297		330		352		396		440		495		528
Frequency ratio			9/8		10/9		16/15		9/8		10/9		9/8		16/15

The chords of the tonic CEG , the subdominant FAc and the dominant GHd are pure.

E minor For better comparison, not in the order E-FIS-GAHcde, but CDE-FIS-GAHc.

Name of the tone		C.		D.		E.		FIS (!)		G		A.		H		c
frequency		264		297		330		371.3 (!)		396		440		495		528
Frequency ratio			9/8		10/9		9/8		16/15		10/9		9/8		16/15

Here the chords of the tonic EGH, the subdominant Ace and the dominant (in minor) Bd-f sharp are pure.

G major

Name of the tone		C.		D.		E.		FIS (!)		G		A (!)		H		c
frequency		264		297		330		371.3 (!)		396		445.5		495		528
Frequency ratio			9/8		10/9		9/8		16/15		9/8		10/9		16/15

Here the chords of the tonic GHd , the subdominant CEG and the dominant D-F sharp-A are pure.

Compared to C major, the F in G major has increased by a semitone to FIS - as with the modulation from C major to E minor - (this is also shown by the note designation), but also the A by a syntonic one comma (frequency ratio ⁸¹ / ₈₀ 21.5 cents) changed. Otherwise the dominant in G major D-FIS-A or in E minor Bd-F sharp would no longer consist of pure thirds and fifths.

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Audio sample


Modulation from C major to G major	Soprano only	The pure third in the soprano of the chord - as here in the second and last chord - appears to the ear, which is used to the equal tuning, to be voiced too low. The clear sound, on the other hand, is convincing: the effect of the modulation as an increase and brightening fades somewhat in the equal-level tuning. For comparison: Are you listening equally ^?^/ⁱ
Listen in ^{? / i}	Listen in ^{? / i}

Choirs that listen to each other particularly well can detonate. With the D major chord, the soprano must hear the a to a around a syntonic comma. In practice, the soprano has to intone the perfect fifth to the bass note D. This sharpens the modulation and prevents detonation. If this is not observed, you fall into the comma trap .

Modulation towards the subdominant in major

The 2nd and 7th tone decrease.

When modulating from C major to F major, the D around a syntonic comma and the B in B ,

more accurate:

Modulation from C major via A minor to F major

C major

Name of the tone		C.		D.		E.		F.		G		A.		H		c
frequency		264		297		330		352		396		440		495		528
Frequency ratio			9/8		10/9		16/15		9/8		10/9		9/8		16/15

The chords of the tonic CEG , the subdominant FAc and the dominant GHd are pure.

A minor (descending) For better comparison, not in the order AHcdefga , but again CDEFGAHc .

Name of the tone		C.		D (!)		E.		F.		G		A.		H		c
frequency		264		293.3 (!)		330		352		396		440		495		528
Frequency ratio			10/9		9/8		16/15		9/8		10/9		9/8		16/15

Here the chords of the tonic Ace, the subdominant DFA ( in C major: the 2nd degree ) and the dominant (in minor) EGH are pure.

As you can see here, in C major the second DFA chord signifies the beginning of a modulation in the direction of the subdominant. (However, this chord can also be interpreted as a double dominant - often clarified as D-F sharp-A.)

F major

Name of the tone		C.		D (!)		E.		F.		G		A.		B (!)		c
frequency		264		293.3 (!)		330		352		396		440		469.3 (!)		528
Frequency ratio			10/9		9/8		16/15		9/8		10/9		16/15		9/8

Here the chords of the tonic FAc , the subdominant Bdf and the dominant CEG are pure.

Compared to C major, the B in F major has decreased by a semitone to B (this is also shown by the note designation), but also - as with the modulation from C major to A minor - the D has decreased by one syntonic point (frequency ratio ⁸¹ / ₈₀ 21.5 cents) changed. Otherwise the subdominant in F major Bdf or in A minor DFA would no longer consist of pure thirds and fifths.

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By the way: In the church key Jonisch (today major) the 2nd and 7th notes have always fluctuated. (Often known was instead the sound B sound H intones, as well as the D was sometimes as a perfect fifth to G , other times as a perfect fourth to A intones (a comma lower).)

Modulation towards the dominant in minor

The same applies here: The 2nd and 7th notes of the target key increase.

When modulating from C minor to G minor, the A -flat in A and F by a syntonic comma,

more accurate:

Example: Modulation from C minor via E flat major to G minor

c minor (descending)

Name of the tone		C.		D.		It		F.		G		As		B.		c
frequency		264		297		316.8		352		396		422.4		475.2		528
Frequency ratio			9/8		16/15		10/9		9/8		16/15		9/8		10/9

The chords of the tonic C-Es-G , the subdominant F-A- flat -c and the dominant GBd are pure.

Eb major (for a better comparison not in the order Eb-FG-As-Bcd-Es, but CD-Eb-FG-As-Bc).

Name of the tone		C.		D.		It		F (!)		G		As		B.		c
frequency		264		297		316.8		356.4		396		422.4		475.2		528
Frequency ratio			9/8		16/15		9/8		10/9		16/15		9/8		10/9

Here the chords of the tonic Eb -GB , the subdominant A-C-Eb and the dominant BDF are pure.

G minor

Name of the tone		C.		D.		It		F (!)		G		A (!)		B.		c
frequency		264		297		316.8		356.4		396		445.5		475.2		528
Frequency ratio			9/8		16/15		9/8		10/9		9/8		16/15		10/9

Here the chords of the tonic GBD , the subdominant C-Eb-G and the dominant DFA are pure.

At the transition from C minor to G minor, the F is increased by a syntonic comma (as with the modulation from C minor to E flat major) and the A flat by a semitone to A (this is also shown by the note designation).

Modulation towards the subdominant in minor

The same applies here: The 2nd and 7th tones decrease.

When modulation of c-minor to minor f the D in In and B is a syntonic comma

more accurate:

Example: Modulation from C minor via A flat major to F minor

c minor (descending)

Name of the tone		C.		D.		It		F.		G		As		B.		c
frequency		264		297		316.8		352		396		422.4		475.2		528
Frequency ratio			9/8		16/15		10/9		9/8		16/15		9/8		10/09

The chords of the tonic C-Es-G , the subdominant F-A- flat -c and the dominant GBd are pure.

A flat major For a better comparison not in the order A-B- C-des-Es- fg- A-flat , but C-Des-Es-FG-As-bc .

Name of the tone		C.		Of(!)		It		F.		G		As		B.		c
frequency		264		281.6		316.8		352		396		422.4		475.2		528
Frequency ratio			16/15		9/8		10/9		9/8		16/15		9/8		10/9

Here the chords of the tonic A-C-Eb, the subdominant Eb-F-A flat and the dominant Eb-GB are pure.

F minor (descending)

Name of the tone		C.		Of(!)		It		F.		G		As		B (!)		c
frequency		264		281.6		316.8		352		396		422.4		469.3		528
Frequency ratio			16/15		9/8		10/9		9/8		16/15		10/9		9/8

Here the chords of the tonic F-A-flat-c , the subdominant B- flat -flat and the dominant C-Eb-G are pure.

Compared to C minor that has been in F Minor D by a semitone to Des reduced - as in the modulation of c-minor to A flat major - (which also shows the note name), but also the B a syntonic comma (frequency ratio ⁸¹ / ₈₀ 21.5 cents) is lowered. Otherwise the subdominant in F minor B flat D flat f or in A flat major D flat F flat no longer consists of pure thirds and fifths.

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Depiction in Euler's clay net

See: Euler's Tonnetz

Modulation from C major to A minor and F major

C major	c	d	, e	f	G	, a	,H	c
A minor	c	, d	, e	f	G	, a	,H	c
F major	c	, d	, e	f	G	, a	b	c

Modulation from C major to E minor and G major

C major	c	d	, e	f	G	, a	,H	c
E minor	c	d	, e	, fis	G	, a	,H	c
G major	c	d	, e	, fis	G	a	,H	c

(For a better comparison always in the order cdefgahc)

Major chords always have the representation x -, y - z:

... f -, a - c c -, e - g g -, a - d ... or a syntonic comma lower

..., f - ,, a - c, c - ,, e - g, g - ,, a - d ...

Fine analysis of intonation problems

A cappella singing

Modulation from A major through G major and E major back to A major

Modulation from A major via G major and E major back to A major according to Heinrich Schütz in pure tuning with a detonation

The different intonation of tones of the same name in modulations in pure tuning harbors some dangers that were recognized from the beginning of polyphony. To avoid these dangers, a detailed analysis is helpful at critical points.

In an unaccompanied section from the five-part sacred choral music So I drive to Jesus Christ ( Schütz-Werke -verzeichnis 379), for example, at the text passage so I fall asleep and rest , the harmony sequence shown in the musical notation appears, somewhat simplified. Max Planck reports that a choir selected with the best singers detonated around a syntonic comma (about a fifth of a semitone), which is clearly perceived by trained ears.

The following table uses the names of Euler's tone network : For example, the (low) comma before 'c sharp' in the chord A -, c sharp '- e' - a 'means that c sharp' sounds a syntonic comma lower than the c sharp in the circle of fifths ... cgdaeh f sharp c sharp ...

Frequencies of the tones from the example after Heinrich Schütz in pure tuning with detonation
chord	Tone 1	Tone 2	Tone 3	Tone 4	Frequency 1 in Hertz	Frequency 2 in Hertz	Frequency 3 in Hertz	Frequency 4 in Hertz	Frequency 4 in equal tuning
A.	A.	'c sharp'	e '	a '	110.0	275.0	330.0	440.0	440.0
D.	d	d '	'fis'	a '	146.7	293.3	366.7	440.0	440.0
G	G	d '	G'	,H'	195.6	293.3	391.1	488.9	493.9
E.	, e	,H	,, g sharp '	,H'	163.0	244.4	407.4	488.9	493.9
d6	, d	,H	,, f '	, a '	144.9	244.4	347.7	434.6	440.0
E.	, e	,H	, e '	,, g sharp '	163.0	244.4	325.9	407.4	415.3
A.	, A	, e	,, c sharp '	, a '	108.6	163.0	271.6	434.6	440.0

Modulation from A major via G major and E major back to A major according to Heinrich Schütz in pure tuning without detonation

The suggestion that an a cappella choir should then better intone in equal numbers must be rejected:

The clear basic tone feeling with pure intonation (because of the harmonic difference tones) and the beat-free sound (because of common overtones), which characterize a top choir, are lost.
An a cappella choir cannot intonate tempered intervals with their ears.

A detailed musical analysis is necessary here. This shows that the 'b' in the 4th chord as a third to the G major chord in the modulation to E major must sound a syntonic comma higher than b ', namely as a perfect fifth to e'.

The soprano must therefore orientate itself on the e 'in the 4th chord and intone the corresponding fifth h' a syntonic comma higher. If this is observed, the modulation sounds much more expressive.

Frequencies of the tones from the example after Heinrich Schütz in pure tuning with detonation
chord	Tone 1	Tone 2	Tone 3	Tone 4	Frequency 1 in Hertz	Frequency 2 in Hertz	Frequency 3 in Hertz	Frequency 4 in Hertz	Frequency 4 in equal tuning
A.	A.	'c sharp'	e '	a '	110.0	275.0	330.0	440.0	440.0
D.	d	d '	'fis'	a '	146.7	293.3	366.7	440.0	440.0
G	G	d '	G'	,H'	195.6	293.3	391.1	488.9	493.9
E , h increases by a comma to h	e	H	'g sharp'	H'	165	247.5	412.5	495	493.9
d6	d	H	, f '	a '	146.7	247.5	352	440	440
E.	e	H	e '	'g sharp'	165	247.5	330	412.5	415.3
A.	A.	e	'c sharp'	a '	110	165	275	440	440

Modulation chain

Modulation chain from C major to C sharp major. The first and last chords have almost the same pitch here in the first version, although they are notated offset by a semitone. In the second version without detonation, the last C sharp major chord is really a half tone higher than the C major chord

Modulation chain from C major to C sharp major in pure tuning with detonation

If several modulations are strung together, in which the same tone has a constant pitch in two consecutive chords and the other two tones are played exactly in pure intervals, the reference pitch can gradually be shifted by a semitone , as shown in the following example :

Frequencies of the tones from the example with the modulation chain in pure tuning
Major chord	Tone 1	Tone 2	Tone 3	Tone 4	Frequency 1 in Hertz	Frequency 2 in Hertz	Frequency 3 in Hertz	Frequency 4 in Hertz	Frequency 4 in equal tuning
C.	c	c '	, e '	G'	132.0	264.0	330.0	396.0	392.0
F.	f	c '	f '	, a '	176.0	264.0	352.0	440.0	440.0
D.	, d	, d '	,, f sharp '	, a '	146.7	293.3	366.7	440.0	440.0
G	,G	, d '	,G'	,,H'	195.6	293.3	391.1	488.9	493.9
G	,G	,G	,,H	, d '	97.8	195.6	244.4	293.3	273.7
C.	, c	,G	, c '	,, e '	130.4	195.6	260.7	325.9	329.6
A.	,, A	,, a	,,, cis'	,, e '	108.6	217.3	271.6	325.9	329.6
D.	,, d	,, a	,, d '	,,, fis'	144.9	217.3	289.7	362.1	370.0
D.	,, d	,, d '	,,, fis'	,, a '	144.9	289.7	362.1	434.6	440.0
G	,,G	,, d '	,,G'	,,,H'	193.1	289.7	386.3	482.9	493.9
E.	,,, e	,,, e '	,,,, g sharp '	,,,H'	161.0	321.9	402.4	482.9	493.9
A.	,,, a	,,, e '	,,, a '	,,,, cis "	214.6	321.9	429.2	536.5	554.4
A.	,,, A	,,, a	,,,, cis'	,,, e '	107.3	214.6	268.3	321.9	329.6
D.	,,, d	,,, a	,,, d '	,,,, fis'	143.1	214.6	286.1	357.7	370.0
H	,,,,H	,,,,H	,,,,, dis'	,,,, fis'	119.2	238.4	298.1	357.7	370.0
E.	,,,, e	,,,,H	,,,, e '	,,,,, g sharp '	159.0	238.4	317.9	397.4	415.3
E.	,,,, e	,,,, e '	,,,,, g sharp '	,,,,H'	159.0	317.9	397.4	476.9	493.9
A.	,,,, a	,,,, e '	,,,, a '	,,,,, cis "	212.0	317.9	423.9	529.9	554.4
F sharp	,,,,, f sharp	,,,,, f sharp '	,,,,,, ais'	,,,,, cis "	176.6	353.3	441.6	529.9	554.4
H	,,,,,H	,,,,, f sharp '	,,,,,H'	,,,,,, dis "	235.5	353.3	471.0	588.8	622.3
H	,,,,,H	,,,,,H	,,,,,, dis'	,,,,, f sharp '	117.8	235.5	294.4	353.3	370.0
E.	,,,,, e	,,,,,H	,,,,, e '	,,,,,, g sharp '	157.0	235.5	314.0	392.5	415.3
Cis	,,,,,, cis	,,,,,, c sharp '	,,,,,,,ice'	,,,,,, g sharp '	130.8	261.7	327.1	392.5	415.3

Modulation chain from C major to C sharp major in pure tuning without detonation (modulation much more expressive)

The musical detailed analysis shows here with the first chords that the chord of the second degree d - f - a, here in major, d - f sharp - a of C major in pure tuning either as, d - ,, f sharp -, a ( Parallel to the subdominant in major) or as d -, f sharp - a (double dominant). Since the modulation in the circle of fifths does not take place in the direction of the subdominant, but rather in the direction of the dominant, the f -, f sharp - a chord is preferable.

At the latest with the following G major chord, a trained ear will hear that 'g -, d' -, g '- ,, h' is too low. The chord g - d '- g' -, b 'is expected. This sounds more expressive. However, the soprano, aware of the modulation in the preceding chord, has to intone the 'a' a syntonic comma expressively higher than a '. This applies accordingly to all further modulations in the modulation chain.

Frequencies of the tones from the example with the modulation chain in pure tuning
Major chord	Tone 1	Tone 2	Tone 3	Tone 4	Frequency 1 in Hertz	Frequency 2 in Hertz	Frequency 3 in Hertz	Frequency 4 in Hertz	Frequency 4 in equal tuning
C.	c	c '	, e '	G'	132.0	264.0	330.0	396.0	392.0
F.	f	c '	f '	, a '	176.0	264.0	352.0	440.0	440.0
D.	d	d '	'fis'	a '	148.5	297.0	371.3	445.5	440.0
G	G	d '	G'	,H'	198.0	297.0	396.0	495.0	493.9
G	G	G	,H	d '	99.0	198.0	247.5	297	293.7
C.	c	G	c '	, e '	132.0	198.0	264.0	330	329.6
A.	A.	a	'c sharp'	e '	111.4	222.8	278.4	334.1	329.6
D.	d	a	d '	'fis'	148.5	222.8	297.0	371.3	370.0
D.	d	d '	'fis'	a '	148.5	297.0	371.3	445.5	440.0
G	G	d '	G'	,H'	198.0	297.0	396.0	495.0	493.9
E.	e	e '	'g sharp'	H'	167.1	334.1	417.7	501.2	493.9
A.	a	e '	a '	, cis "	222.8	334.1	445.5	556.9	554.4
A.	A.	a	'c sharp'	e '	111.4	222.8	278.4	334.1	329.6
D.	d	a	d '	'fis'	148.5	222.8	297.0	371.3	370.0
H	H	H	, dis'	fis'	125.3	250.6	313.2	375.9	370.0
E.	e	H	e '	'g sharp'	167.1	250.6	334.1	417.7	415.3
E.	e	e '	'g sharp'	H'	167.1	334.1	417.7	501.2	493.9
A.	a	e '	a '	, cis "	222.8	334.1	445.5	556.9	554.4
F sharp	f sharp	fis'	'ais'	cis "	187.9	375.9	469.9	563.8	554.4
H	H	fis'	H'	, dis "	250.6	375.9	501.2	626.5	622.3
H	H	H	, dis'	fis'	125.3	250.6	313.2	375.9	370.0
E.	e	H	e '	'g sharp'	167.1	250.6	334.1	417.7	415.3
Cis	cis	cis'	,ice'	g sharp '	141.0	281.9	338.3	422.9	415.3

literature

Hermann von Helmholtz : The theory of tone sensations as a physiological basis for the theory of music . Vieweg, Braunschweig 1863 (reprint: Minerva-Verlag, Frankfurt am Main 1981, ISBN 3-8102-0715-2 , excerpt ).
Bettina Gratzki: The pure intonation in choral singing (= Orpheus series on basic issues in music 70). Publishing house for systematic musicology GmbH, Bonn 1993, ISBN 3-922626-70-X ( excerpt ).
Ross W. Duffin: How Equal Temperament Ruined Harmony (And Why You Should Care). WW Norton & Company , New York NY 2007, ISBN 978-0-393-06227-4 ( excerpt ).

Web links

swell

↑ Bettina Gratzki, p. 90f (see literature) and Max Planck : The natural mood in modern vocal music. In: Quarterly magazine for musicology. Number 9, Berlin (October 1893), pages 418 to 440. At that time there were no precise measuring devices for frequencies. The examinations by Max Planck and Arthur von Oettingen at that time were based solely on hearing.
↑ Max Planck: The natural mood in modern vocal music . In: Quarterly magazine for musicology . tape 9 , no. 4 . Berlin October 1893, p. 418–440 ( DigiZeitschriften - Breitkopf & Härtel).
↑ This designation is sufficient for the detailed analysis. The frequencies are only included here to show how the pitch changes.

[1] Bettina Gratzki, p. 90f (see literature) and Max Planck : The natural mood in modern vocal music. In: Quarterly magazine for musicology. Number 9, Berlin (October 1893), pages 418 to 440. At that time there were no precise measuring devices for frequencies. The examinations by Max Planck and Arthur von Oettingen at that time were based solely on hearing.

[2] Max Planck: The natural mood in modern vocal music . In: Quarterly magazine for musicology . tape 9 , no. 4 . Berlin October 1893, p. 418–440 ( DigiZeitschriften - Breitkopf & Härtel).

[3] This designation is sufficient for the detailed analysis. The frequencies are only included here to show how the pitch changes.