A Handbook on the Middle Tuning

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A Handbook on the Middle Tuning

Second Edition with corrected and improved tuning indications

Bevis Stevens – Maria Renold





The Scale of Twelve Fifths

A Handbook on the Middle Tuning

Second Edition with corrected and improved tuning indications

A temperament founded on the tones c = 128 Hz, gelis1 = 362.41 Hz and a1 = 432 Hz

1 Gelis is Maria’s term for her newly discovered tone corresponding to f sharp. See further theExplanation of Terms



Private Press

© First Edition, Copyright 2006

by Bevis Stevens, Dornach – Switzerland

Second Edition © 2012Havelock North – New Zealand

Email: [email protected]



Cont ents

Forward to the second edition 6

Forward to the first edition 6

........................................................................................Background

8

...............................Why a new concert pitch? Why a new temperament? 10

............................................................................................................Origins 10

.........................................................................................Aural experiments 11

The mathematical-cosmic origin of the frequencies C = 128 Hz and A =.............................................................................................................432 Hz 11

Table 1 ........................................The 10 x 7 factors of the Platonic cosmic year 12Table 2 .......................................................................Intervals of the major scale 13Table 3 ......................................................................Intervals of the minor scale 13

............................................................................Qualitative characteristics 13

...............................................................................New discoveries

16

..............................................................................................‘Open intervals’ 18 Table 4 ....................Measurements of the double octave–Interval sizes in cents 18

..................................................................................The secret of the fi fths 18

Table 5 ................................................Fifths and Fourths–interval sizes in cents 19Table 6 .......................................Oscilloscope pictures of differently sized fifths 20Table 7 ....................................................................(Renold, 2004, p. 136, fig. 4) 20

..........................................................................Tuning instructions

22

..................................................The structure of the scale of twelve fifths 24

...............................................................Tuning instruction s–Maria Renold 2 4

.....................................................................Tuning open fourths and fifths 24 Table 8 ....................................................................Difference tone of the fourth 25

Table 9 .......................................................................Difference tone of the fifth 25Table 10 ...........................................Tuning to the scale of twelve fifths - Renold 28

......................................................The Ideal Mathematical Representation 29 Table 11 ......................................................................Tuning instructions – Davis 29Table 12 .............................................................Offset for Renold 2 for the Tuner 30

...........................................................................................Appendix

31

...........................Experiences in doing Eurythmy with the Middle Tuning 33

....................................................................................Explanation of Terms 35

..................................................................................................Bibliography 36

3



4



5



Forward to the second edition

Since writing the first edition I have learnt to tune the piano myself and Paul Davis’

appendix to Maria Renold’s book (2004) became understandable. This made me aware of the

fact that following her and Thomma’s indications (Renold, 2004) actually results in a third

version of the tuning! Although I like it, as it has even more key colour, it is not what Maria

intended. Now Paul has further refined the mathematically ideal tuning to a point where it

accurately represents Maria’s intentions. This has made a total revision of the whole tuning

section necessary. I have kept Maria’s description of the tuning method, abandoned those

given by Henken and Thomma and laid out the tuning in the style developed by Jorgensen

(1991) which is generally known and accepted by tuners worldwide.

On a historical note: The research of Michael Kurz at the Goetheanum has shown

that Rudolf Steiner could not have given an indication for the pitch of the A = 432 Hz to the

recorder mak er Ziemann-Molitor. However this does not affect the validity of this concert

pitch for Maria’s temperament as it results as a matter of course when beginning from her

original starting point of Steiner’s recommendation for C = 128 Hz. As a result the merits of

A = 432 Hz will be still covered.

– Bevis Stevens March 2012

Forward to the first edition

This booklet is an attempt to satisfy the need for a short introduction into the twelve

fifth–tones scale. It is directed towards all interested people who wish to have their instrument

tuned in this tuning, as well as professional tuners who require a handbook. Although this booklet follows the content of, and in deed adds to Maria Renold’s book (2004), it by no

means replaces it. This booklet deals with the twelve fifth–tones tuning alone, whereas

Renold’s book also goes into the construction and historical development of the Western

scales and the origin of their tones and intervals, researches why the human ear experiences

some intervals to sound false, while others sound genuine, and verifies the existence of an

ethical effect of the pitch of single tones. Renold also looks into various remarks on music

and specifications given by Rudolf Steiner 2.

There are two versions of the tuning. The first was discovered in 1962 (Renold,

2004, p. 57).3 Over the years, the tuning was developed further and new discoveries made a

second more beautiful version of the tuning possible. It is the second version of the tuning,which will be introduced here.

– Bevis Michael Stevens, June 2006

6

2 The Founder of Anthroposophy. See also footnote 4

3 After Renold discovered the 1st tuning method, she found out, more or less by chance, thatHenricus Grammateus had already constructed the scale mathematically in 1518 (Jorgensen, 1991, p.332). Because it has often occurred that Renold has been accused of simply copying this scale, it must

be stressed that it was a new discovery, through hearing and not through mathematical construction.



7



Background

8



9



Why a new concert pitch? Why a new temperament?

One may well ask: why use a different concert pitch, haven’t our ears long become

accustomed to a = 440 Hz? Understandably this question is often posed by musicians who

have the so-called gift of absolute pitch. But the definition absolute pitch is misleading, as the pitch to which this ‘ability’ is set varies greatly. Therefore absolute pitch cannot be used as a

guide to determine the correctness of a concert pitch. The correctness of a pitch is of course a

contentious issue but Maria Renold’s research shows that small variations in the pitch of a

tone give rise to remarkably different qualities. The Concert pitch A = 432 Hz is a result of the

temperament itself which takes its starting point from Rudolf Steiner’s recommendation for

the pitch of C.

But what does one want to achieve with yet another tuning method? Is our equal

tempered tuning not perfect enough already? Yes indeed, but it’s ‘equalness’ ultimately

lessoned and restricted music’s expressiveness and variation in tonal colour. The intention is

therefore not one of wanting to replace something or wishing to find something better, butrather one of expanding the expressive possibilities of music and thereby enriching our

musical experience. Qualitative differences between pitches are often first noticeable when

directly compared. One is then often amazed at the great effect that small differences in the

pitch of a tone can have. Furthermore it is striking how quickly one can get used to a new

temperament and its concert pitch.

It needs to be added at the outset that this new method of tuning arose through

hearing and not through theory. Therefore in order to evaluate the tuning it is necessary to

hear it! Herewith a limitation of this booklet is addressed. However, an attempt to describe

this tuning and talk about its attributes as well as giving tuning instructions will be made

Origins

The experiences that Maria Renold had – on the one hand as a violinist and violist in

the internationally renowned Busch Chamber Orchestra and Busch String Quartet and on the

other hand with eurythmy4 – were important. Her tonal world was one of perfect fifths and so-

called Pythagorean tuning. The discrepancy posed by equal-tempered tuning bothered her.

She wondered if it weren’t possible to tune the piano in a way, which didn’t sound so false to

her string-player ears.

In 1962, beginning with a diatonic scale based on perfect fifths, Maria Renold found

the five chromatic tones between them through inner listening. As these are in the middle of

the other tones, they are mathematically geometric mean tones. The result was a scale in

which all major and minor keys could be played and which sounded aurally genuine, having

more tonal variation and key colour than equal temperament. Renold called the resulting scale

“the scale of twelve fifths”. She continued to experiment and develop the temperament and a

second version was published in 1991. The two versions are often referred to as Renold1 and

Renold2. Renold2 is also known as the “Middle Tuning” by which it is referred to here

because of its grounding, centering quality and the way it speaks directly to the middle of the

human being, to the heart. It has a sun-like, radiant quality.

10

4 The art of movement inaugurated by Rudolf Steiner. For more information on eurythmy goto www.eurythmie.com and for more on Rudolf Steiner and Anthroposophy see www.goetheanum.org or www.anthroposophy.org

http://www.anthroposophy.org/

http://www.anthroposophy.org/

http://www.goetheanum.org/45.html?L=1

http://www.goetheanum.org/45.html?L=1

http://www.eurythmie.com/

http://www.eurythmie.com/



Aural experiments

The first time the piano was tuned in the Middle Tuning, Maria Renold used the

concert pitch a1 = 440 Hz. It sounded nice but there arose an unsocial, hostile mood amongst

the members of the family–a rare occurrence! Some time later, she heard of Rudolf Steiner's

specification c = 128 Hz, which in this method of tuning results in the concert pitch a 1 = 432

Hz. The piano was retuned and harmony reigned amongst those present (Renold, 2004).5

Amazed by this experience of hers, Renold began to make countless aural

experiments in order to test the objectivity of her observation and to verify the importance of

Rudolf Steiner's specification6. She tested musically trained and untrained people in America

and Europe.

She discovered that, although a1 = 440 Hz and c1 = 261.656 Hz were the more

familiar pitches, over 90% of those tested preferred a1 = 432 and c = 128 Hz and it’s octave

256 Hz. Some statements for C were:

c1 = 256 Hz belongs to the human being, gives space, sounds peaceful, pleasant and

full and sounds as prime within the whole human being.

To c1 = 261.656 Hz over 90 % said it sounded jabbing, irritating, unpleasant, heady,intellectual, makes one nervous…

Further experiments were made with the correspondingly lower C, c1 = 252 Hz.

Those tested said that this tone gives rise to a feeling of bodily comfort, made one drowsy, but

at the same time had a calculating, merciless quality…

Thus Renold showed that a sense for the ethical quality of a tone described by the

Greeks, still exists today, and that it was important to take this fact into account in the

development of a new method of tuning. Further results of these experiments can be found in

her book (Renold, 2004, pp. 76-79).

The mathematical-cosmic origin of the frequencies C = 128 Hz and A = 432Hz

Everyone discovers soon enough that numbers have a lot to doing with music. They

are connected with all kinds of rhythms and are thereby capable of showing the correlation

between macrocosmic rhythms, microcosmic rhythms and musical harmonies. Numbers are

not allegations but are pure phenomena, and as such enable us to gain the required objective

basis.

Music has a very interesting and important connection to time. If something (e.g. a

string) moves 16 times, and the speed at which it does so is fast enough for us to hear it, we

can say with certainty: ‘that is a second’. We are unable to hear one cycle per second, but one

11

5 In this connection it is interesting to note that Rudolf Steiner originally gave thisspecification to Kathleen Schlesinger who had noticed that her newly rediscovered modes (1939) haddifferent effects on the listeners, depending on what pitch they were tuned to.

6 For a while the story has been upheld that Rudolf Steiner suggested the concert pitch ofa=432 Hz be used on the flutes built by Mr and Mrs Ziemann-Molitor for the second Waldorf School.However recent research made by Michael Kurtz (responsible for the music department at the Section

for Eurythmy, Speech, Music, Puppetry and Drama at the Goetheanum, Switzerland) has proven thisto be a myth. However, as Maria’s Temperament gives rise to this concert pitch as a matter of coursewhen taken from c=128 Hz, and because tuning usually begins with A the merits of this pitch are alsogone into here.



cycle times 24 (2 x 2 x 2 x 2) is 16 Hz, which is the lowest tone we are able to hear. 164 = 128

Hz, i.e. 128 Hz7 is the 4th octave of 16 Hz.

Table 1 The 10 x 7 factors of the Platonic cosmic year

Herewith the number 128 finds its musicalsubstantiation, and at the same time its

importance for music and the human being. This

tone is also known as the philosophic pitch.

The number 432 is found in the harmony of the

rhythms contained within the so-called platonic

cosmic year, which takes 25920 years. This

number represents the progression through the

whole zodiac of the point where the sun rises at

the spring equinox.

The number 25920 has 70 factors, which areconnected to each other in a wonderful,

harmonic rhythm. E.g. 1 occurs 25920 times,

while 2 occurs simultaneously 12960 times, 3

8640 times etc. One of these rhythms is 432 : 60;

432 divides 25920 60 times. The number 60 is to

be found in the rhythm of the Saturn and Jupiter

conjunctions: the conjunction between Saturn

and Jupiter takes place every 20 years. Every 60

years they meet again at the point of departure.

Microcosmically, the cosmic number 25920 hasa connection to the average breaths a human

being takes in a day: 18 breaths a minute

multiplied by 1440 (60 minutes x 24 hours)

makes 25920.

“This number corresponds also exactly with the

number of days, which–in accordance with the

old Babylonian year of 360 days–make 72 years.

The statement of Rudolf Steiner’s, that

esoterically the human life spans 72 years, isbased on this. After this time the sun ‘covers’

another point of the heavens and ‘releases’ the

star governing ones destiny .8 Thus we breathe

an average of 25920 times a day and our life

contains 25920 days” (Glöckler & Glöckler,

2007, p. 186).

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7 Also 27 = 128.

8 See Rudolf Steiner, GA 237, lecture of 5th July, 1924

1 x 25 920

2 x 12 960

3 x 8 640

4 x 6 480

5 x 5 184

6 x 4 320

8 x 3 240

9 x 2 880

10 x 2 592

12 x 2 160

15 x 1 728

16 x 1 620

18 x 1 440

20 x 1 296

24 x 1 080

27 x 960

30 x 864

32 x 81036 x 720

40 x 648

45 x 576

48 x 540

54 x 480

60 x 432

64 x 405

72 x 360

80 x 324

81 x 320

90 x 288

96 x 270108 x 240

120 x 216

135 x 192

144 x 180

160 x 162



If we look more closely at the 24 th to 48th factors we find the intervals of the major

scale: 24:27:30:32:36.40:45:48.

“The intervals of the major scale begin with t he prime on the factor 24. The

proportion 24:24, completely reduced, is the same as the proportion 1:1, i.e. the prime… The

proportion 24:27, when completely reduced, corresponds with the proportion 8:9, i.e. the

second. And so on till 24:48 (1:2), the octave…

Table 2 Intervals of the major scale

24 : 24 = 1 : 1 Prime

24 : 27 = 8 : 9 Second

24 : 30 = 4 : 5 Major third

24 : 32 = 3 : 4 Fourth

24 : 36 = 2 : 3 Fifth

24 : 40 = 3 : 5 Major Sixth

24 : 45 = 8 : 15 Major Seventh

24 : 48 = 1 : 2 Octave

…The Classical intervals of the minor scale arise when we depart from the factor

360

Table 3 Intervals of the minor scale

360 : 360 = 1 : 1 Prime

360 : 405 = 8 : 9 Second

360 : 432 = 5 : 6 Minor third

360 : 480 = 3 : 4 Fourth

360 : 540 = 2 : 3 Fifth

360 : 576 = 5 : 8 Minor sixth

360 : 648 = 8 : 15 Minor seventh

360 : 720 = 1 : 2 Octave

So we find, hidden within the two factor rows, as a continuous proportion, the

intervals of the major and minor scales. Therein included is the number 432 [27]–the

vibration frequency [Hz] of the archetypal concert pitch” (Glöckler & Glöckler, 2007)

If we include the lower and higher octaves of 432, we discover that this concert pitchappears 7 times within the 70 factors (27, 54, 108, 216, 432, 864 and 1728)

Qualitative characteristics

Because the size of the various types of intervals in the Middle Tuning varies, each

chord and consequently each key gains its own definite character. A modulation is thereby

experienced more strongly. Because of this an organ builder and piano tuner once said that the

Middle Tuning in comparison to equal tempered tuning is like a relief in contrast to a flat

map. Besides this enrichment in the keys, a greater sonority is achieved through the difference

tones, which arise through the size of the tuned fifths. 9 Because of this even upright pianos

sound a lot better as a result of the tuning.

13

9 For more about the difference tones, see ‘Tuning open fourths and fifths’ below



In contrast to the bright, even tense-sounding equal temperament on concert pitch

440 Hz, the Middle Tuning is more relaxed and warmer. Many people experience that the

music speaks more directly to the heart and that the barrier between audience and music is

broken down.10

But enough has been said! Let us proceed to a discussion of the Middle Tuning itself,

before proceeding to the tuning instruction, so that you will be able to make your ownexperiences!

14

10 See the Appendix for a report on doing eurythmy with the Middle Tuning.



15



New discoveries

16



17



‘Open intervals’

The second method of tuning, which is introduced here, was made possible by the

discovery of open fourths and fifths, as well as the discovery that a genuine sounding octave

is bigger than the ‘perfect’ proportion 2 : 1.Even though the latter is a surprise, ‘stretching’ theoctaves has long become established tuning practice.

As a string player, Maria Renold was aware of the problem posed by the double

octave flageolet. In the lower regions it sounds flat because of the inharmonicity of the

strings; in higher regions in sounds low because of so-called psycho-acoustic reasons. She

made several measurements of the sizes of the double octave:

Table 4 Measurements of the double octave–Interval sizes in cents

Grand piano 2404.6 2406.8 2411.4 2418.6 Bechstein ENCello,flageolet von c 2398.7 2399.2 2399.8

c1 sounded toolow

Cello,stopped 2401,7 2404,4 2406,7 c1 sounded right

“Note: If one multiplies the widest stopped double octave on the cello by the factor

3.5 ( = seven octaves), the result is the same interval as that of twelve perfect fifths; the

Pythagorean comma does not apply: 2406.7 x 3.5 = 8423.45; 701.955 [= cents of perfect

fifth] x 12 = 8423.46” (excerpt from Renold, 2004: Table 27)

The secret of the fifths

Maria Renold also discovered that different perfect fifths exist:

“Further observations can be made with fourths and fifths on resonant instruments.

Grand pianos, harpsichords and pipe organs are best for first attempts because of the

required precision. Let us begin with the beat-free perfect fifth. It sounds peaceful, almost

solid. The lower octave of the fundamental hums along, totally absorbed in the sound, giving

a silvery colouring to the whole. If one minimally enlarges the fifth–by either raising the pitch

of the higher, or lowering that of the bottom tone–the whole texture is set in motion. First of

all the interval vibrates violently, then it opens up and the fifth sounds peaceful and clear

once again but also wide. The difference tone emerges sonorously and sounds stable. Some

people recognise the openness of the interval by a relaxation in the area of the diaphragm. If

the fifth is enlarged further, the difference tone begins to beat, the sound loses its coherence

and the beats dominate. If one makes the fifth smaller than perfect, it immediately loses its

lustre and strength and grows milder and more inward. When the interval is smaller than a

perfect fifth, the difference tone does not stabilise, being weak at first then suddenly breaking

off. Where the beating sets in, the interval is roughly the same size as the equal-tempered fifth.

If one makes it still smaller, the fifth will be utterly calm again.

Much the same applies for the fourth…”(Renold, 2004, p. 129)

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The fourth and fifths were also measured:

Table 5 Fifths and Fourths–interval sizes in cents

Instrument Interva Comments

Open fifths, sonorous difference tone

Grand piano 703.5 702.9 704.2 Bechstein ENPiano 703.4 702.7 704.2 Sabel

Violin 703.5 703.7 704.2 open strings, steel

Viola 703.6 704.0 704.8 open strings, steel

Cello 702.9 703.8 706.5 open strings, steel

Organ pipes 703.0 704.3 705.5 stopped

Perfect fifths. silvery difference tone

Cello 701.0 702.2 open strings, steel

Organ pipes 702.0 Stopped

Small fifth. no ifference tone

Cello 698.5 699.0 700.4 open strings, steel

Open fourths. s norous difference tone

Grand piano 499.7 499.9 500.8 502.2 Bechstein EN

Piano 499.6 499.9 501.3 502.3 Sabel

Chord zither 499.5 499.7

Organ pipes 499.7 499.9 500.9 502.5 Stopped

(Renold, 2004, p. 27)

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Oscilloscope pictures of the different fifths were made:

Table 6 Oscilloscope pictures of differently sized fifths

Open fifth

Transition

Perfect fifth

Transition

Small fifth

Table 7 (Renold, 2004, p. 136, fig. 4)

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21



Tuning instructions

22



23



The structure of the scale of twelve fifths

The octave proportion used here is: 2.003873819 which amounts to 1203.35 cents

Maria Renold specifies three pitches:

1. c = 128 Hz

2. a1 = 432 Hz3. gelis1 = 362.4 Hz

The first one originates from Rudolf Steiner; the first was given to the musicologist

and rediscoverer of the Aulos modes, Kathleen Schlesinger, the second was given to Mr and

Mrs Ziemann-Molitor, the builders of the flutes for the second Waldorf School. Both tones

originate in the duodecimo row of c.

The scale of twelve fifths consists of two groups of duodecimo rows: the first of

seven diatonic open-fifth tones (F, C, G, D, A, E, B) which belong to the Pythagorean-Dorian

octachord, and the second from five geometric mean tones (Gelis, Alis, Belis, Delis, Elis).

The latter is built, beginning from the geometric mean of the minimally enlarged octave

(1203.350 cents) c = 256 Hz –c1 = 512.99 Hz, gelis1 = 362.4 Hz and progressing in open

duodecimos.

The audible difference tones are an important part of the scale of twelve fifths. They

add resonance and fullness to the sound. Therefore Maria Renold recommends that instead of

tuning to octaves one tunes to fourths and fifths. But in the extreme regions this is impractical,

where octave tuning is necessary.

Tuning instructions–Maria Renold

The method of tuning “is made up of three fundamental parts:

1. gelis1 = 362.04 Hz and Rudolf Steiner's pitch indications c = 128 Hz and a1 = 432

Hz (for these pitches it is necessary to have three tuning forks that must be

calibrated to within ± 0.5 Hz);

2. the open (minimally enlarged) octave, fifth and fourth intervals which, despite

background beating, sound totally calm and open and have sonorous difference

tones;

3. tuning in contrary direction of the Apollonian Sun scale: the pre-Christian

descending direction of the true Dorian octachord and the Christian ascending

direction of the true-tone C major resurrection scale(Renold, 2004, p. chapter 19

and Table 11)...”(Renold, 2004, p. chapter 24)

As can be seen through the measurements of the fourths and fifths above, it isevident how exactly one has to listen in order to attain the required intervals.

“The results of the interval measurements as presented in [ Table 4 and Table 5

above] therefore showed a surprising variation in the size of the named intervals. This means

that the cents can only be a rough guide and that very exact hearing and the ability to

recognize these intervals is vital for successful tuning of the scale of twelve fifths. Every tone

must always be fine-tuned by ear.”

Tuning open fourths and fifths

First tune a perfect fourth or fifth, making sure that the interval has absolutely no

beats. “If it is perfect it sounds stable and the difference tone, though not strong, is clearlyaudible and sounds silver.” (Renold, 2004, p. 140)

24



Fourths are easier to tune as the difference tone is lower than that of the fifth and is therefore

easier to hear. The difference tone is the difference in frequency between the lower and upper

tone, e.g. 341.3 Hz (F) minus 256 Hz (C) = 85.3 Hz (F). Written musically this is as follows:

Table 8 Difference tone of the fourth

Fourth:

Differenzton:

An example of the fifth is: 384 Hz (G) minus 256 Hz (C) = 128 Hz (C), in notes:

Table 9 Difference tone of the fifth

Fifth:

Difference tone:

Next, enlarge the interval slowly. “The interval will begin to beat intolerably: this

has often shocked piano tuners so much that they hardly dared to continue. Do not give in to

‘fear of beats’, but calmly increase the interval in minute steps. The beats will gradually get

slower until they suddenly disappear altogether, the sound of the interval opens up and the

differential begins to sound strongly and sonorously… these open fourths and fifths, together

with their sonorous difference tones, lend their richness of colour to the scale of twelve

fifths…

[Table 9] gives a musically notated tuning guide. The open note head of each

interval indicates the tone from which one tunes, and the solid head the one to be tuned. The

diamond-shaped heads on the base stave show the differential of the interval directly above…

The three tones on the extreme left indicate the tones c1 = 256 Hz, gelis1 = 36 2.40 Hz and a1 =

432 Hz. First tune these three tones as exactly as possible. They must equal the tone of the

tuning fork so exactly that they are beat-free. This is best achieved when one inwardly

experiences the tuning fork's tone and then recreates the experience on the instrument.

Tuning proceeds in the two directions of movement belonging to the Apollonian

scales: the pre-Christian descending direction of the true Dorian octachord and the ascending

direction of the true-tone C major resurrection scale. Starting with c1 = 256 Hz, tune an

ascending open fourth to f 1 and open fifth to g 1. Then tune a descending open fifth from a1 =

432 Hz to d 1 and an open fourth to e1. From this e1 , tune a descending open fourth to b and an

25



ascending open fifth to b1. The seven non–altered diatonic fifth tones of the scale of twelve

fifths have now been tuned. The first open octave sounds between b and b1…

As the two directions of movement meet in the fourth d 1 –g 1 , the first steps must be

carefully tuned. The first formed fifth appears between b and gelis1. It sounds serious, but it

must be calm and totally harmonically acceptable. If this is not the case, look first for the

cause of the problem in the b which has probably been tuned too sharp, and not in the gelis1

which has been tuned from the tuning fork. Then go back to the descending open fourth a1 –e1

and check this thoroughly. Pay attention to tuning the e1 really flat enough. This fourth is wide

and open and is correctly tuned when its differential sounds really strong and sonorous like

the sound of an organ. The same applies to the descending fourth e1 –b. When these two

fourths are tuned correctly, then the formed fifth b–gelis1 also sounds correct.”

My experience has been that the descending fourths a-e and e-b need to be particularly wide,

otherwise the b is too high, resulting in an intolerable 5 th b-gelis. A beginning help is to tune

the 5th b-gelis first. (it should have about 2.5 – 3 beats) and then tune the e to fit between the b

and the a.

“Now proceed from gelis1 and tune a descending open fourth of the same size to

delis1 , and from there an ascending open fifth to alis1; from alis1 a descending open fourth to

elis1 and from elis1 a descending fourth to belis1. The second formed fifth sounds between

belis1 and f 1. What applied above is also applicable here: if the formed fifth belis1 –f 1 sounds

wrong, go back to gelis1 and tune the tone sequence delis1 , alis1 , elis1 , belis1 once again until

they are correct and the formed fifth belis–f 1 sounds right.”

As above, the fifth belis-f may be tuned first, beating about the same as the fifth b-gelis, andthen progression back to gelis. The fourths need to be wide and the fifth delis-alis relatively

small, i.e. perfect rather than open.

“Finally, tune the two ascending open fifths elis1 –belis1 and f 1 –c2 and check the

latter with the open fourth g 1 = c2. The 9th belis–c2 has thus been tuned to the scale of twelve

fifths and the three minimally enlarged octaves b–b1 , belis–belis1 and c1 –c2 have been gained.

Success in tuning the lower octaves downwards is again dependent on tuning the

open fourths and fifths large enough, i.e., tuning the lower tone flat enough. To make sure that

tuning has been successful, the following check is imperative. Having completed the loweroctaves between c2 and subcontra 2 A, play all 24 major and minor arpeggios in ascending

order one after the other. If they sound pleasant to the ear they have been correctly tuned. If

each higher octave sounds too flat, then the open fourths and fifths have been tuned too small,

with each new tone then tuned too sharp. In this case, return to e1 and tune again until the

correct size has been achieved and all chords sound pleasing.

To tune the higher octaves of the instrument, proceed from gelis1 and tune in

ascending direction to open fourths and fifths… A similar problem exists here as above, but

the opposite. Take care that each new tone is tuned sharp enough, in other words that the

open fourths and fifths are really tuned open. The sonorous resonating of the differential can

be a helpful guide. When tuning is complete, play all 24 major and minor arpeggios again in

ascending direction through all registers. The incorrectly tuned tones will be immediately and

uncomfortably noticeable. When they have been corrected and all chords sound pleasing, the

26



instrument is ready to be played; you can play in all styles and rejoice in a true and beautiful

sound.

When testing the 19ths of the base tones, it will be noticed that some beat more than others.

E.g. the 19th f-a beats (this interval is Pythagorean) very fast while the 19th e - g hardly beats

(this interval is almost just)

27



Table 10 Tuning to the scale of twelve fifths - Renold

Temperament octave

28

O p e nn o t e h e a d : t on e f r om wh i c h t uni n g pr o c e e d s .

Bl a c k n o t e h e a d : t on e t o b e

t un e d .

Di a m on d n o t e h e a d : d i f f e r e

n c e t on e .

N o t e s pr e c e d e d b y a c r o s s a r e g e om e t r i c m e a n t on e s .

T uni n gf or k s a t t h e f ol l o wi n gf r e q u e n c i e s a r e r e q ui r e d f or

t h e t uni n g : c 1 =2 5 6 Hz , g e l i s 1 =

3 6 2 .4 Hz a n d a 1 =4 3 2 Hz .



The Ideal Mathematical Representation

The word ideal is used because such a temperament is neither obtainable nor

desirable. A temperament will always be adjusted by ear to suite a particular instrument.

However, for the sake of an accurate representation of this temperament this is vital and is a

milestone in the development of this tuning. This is shown first in the style of Jorgenson

(1991) for tuning by ear and then a table is given showing the offset for setting a tuner.

Table 11 Tuning instructions – Davis

29

œ

œ

œ

Tune totuning forks

˙

–1.32

¿

Listen fordifferencetones

?

˙

+0.50

¿

œ

+0.56

¿ etc.

œ

Test

–1.49

œ

–1.67

œ

–1.25

œ

Test

–2.34

˙

+0.63

œ

œ

+0.94Test

˙

+0.66

œ

œ

+0.99Test

œ

–2.21

œ

–2.34Test: beats faster

œ

–1.40

˙

+0.53

œ

–1.58

˙

+0.59

œ

œ

+0.89Test

˙

+8.33

Major thirds

˙

+8.83

˙

+14.33

˙

+9.89

˙

+10.48

˙

+11.12

˙

+11.79

˙

+19.13

˙

+20.28

˙

+21.51

˙

+14.84

˙

–14.31

Minor thirds

˙

–15.17

˙

–10.13

˙

–10.74

˙

–18.02

˙

–19.10

˙

–20.25

˙

–13.52

˙

–14.33

˙

–15.20

˙

–16.12

˙

–27.04

Tune to Tuning forksc1 = 256 Hza1 = 432 Hzgelis1 = 362.40 Hz

+ before the beats indicates beats faster – beats that are slower than equal temperamentblack note heads: ntoes arleady tunedopen note heads: notes to be tuned

Beats:



Table 12 Offset for Renold 2 for the Tuner

30

-1 2 0 3 . 3 5 D ownw ar d s

1 2 0 3 . 3 5 U pw ar d s

o c t av e

c o c t av e s

un s t r e t c h e d

R en ol d c -

o c t av e s

s t r e t c h e d

e. t . c f r om A

=

4 3 2 HZ

D

i f f er en c e

o c t av e s of

A 4 4 0

e. t . c f r om A

=

4 4 0

d i f f

er en c e t o

R en

ol d c - o c t av e s

s t r e t c h e d

C 4

- 5 . 8 7

C 0 t oB 0

1 6

1 5 . 8 7 7

1 6 . 0 5

-1 9 .2 7 4

2 7 . 5

1 6 . 3 5

- 5 1 . 0 3 2

- 4 5 .1 6

C 0

d el i s 4

-4 .4 7

C 1 t oB 1

3 2

3 1 . 8 1 5

3 2 .1 1

-1 5 . 9 2 4

5 5

3 2 .7 0

-4 7 . 6 8 2

- 4 1 . 8 1

C 1

D4

- 3 . 0 7

C 2 t oB 2

6 4

6 3 .7 5 3

6 4 .2 2

-1 2 . 5 7 4

1 1 0

6 5 .4 1

-4 4 . 3 3 2

- 3 8 . 4 6

C 2

el i s 4

-1 . 6 8

C 3 e t c

1 2 8

1 2 7 .7 5 3

1 2 8 .4 4

- 9 .2 2 4

2 2 0

1 3 0 . 8 1

-4 0 . 9 8 2

- 3 5 .1 1

C 3

E 4

- 0 .2 8

C 4

2 5 6

2 5 6 . 0 0 0

2 5 6 . 8 7

- 5 . 8 7 4

4 4 0

2 6 1 . 6 3

- 3 7 . 6 3 2

- 3 1 .7 6

C 4

F 4

- 5 . 5 9

C 5

5 1 2

5 1 2 . 9 9 2

5 1 3 .7 4

-2 . 5 2 4

8 8 0

5 2 3 .2 5

- 3 4 .2 8 2

-2 8 . 4 1

C 5

g el i s 4

-4 .1 9

C 6

1 0 2 4

1 0 2 7 . 9 7 1

1 0 2 7 .4 8

0 . 8 2 6

1 7 6 0

1 0 4 6 . 5 0

- 3 0 . 9 3 2

-2 5 . 0 6

C 6

G4

-2 .7 9

C 7

2 0 4 8

2 0 5 9 . 9 2 3

2 0 5 4 . 9 6

4 .1 7 6

3 5 2 0

2 0 9 3 . 0 0

-2 7 . 5 8 2

-2 1 .7 1

C 7

al i s 4

-1 .4 0

C 8

4 0 9 6

4 1 2 7 . 8 2 7

4 1 0 9 . 9 2

7 . 5 2 6

7 0 4 0

4 1 8 6 . 0 1

-2 4 .2 3 2

-1 8 . 3 6

C 8

A 4

0 . 0 0

b el i s 4

1 .4 0

B 4

2 .7 9

Of f s e t f or R en ol d 2 –r e g u

l ar i s e d a c c or d i n g t oP a ul D avi s

o

c t av e s t r e t c h i n c en t s

U s e t h el a s t t w o c ol umn s of

t h ef ol l owi n g t a b l e ( r e d ) t o s e t t h e s t r e t c h

i n d i vi d u al l yf or e a c h o c t av ei f an ov er al l s t r e t c h

c ann o t b e s e t .

s e t t h e of f s e t f or t h e

n o t e s of t h e ch r om a t i c

o c t av e a sf ol l ow s:

of f s e t i nr el a t i on t o C 4 . S e t

t h e c or r e s p oni d n g o c t av

e s t o

t h i s of f s e t



Appendix

31



32



Experiences in doing Eurythmy with the Middle Tuning

In repeated experiments, the concert pitches c = 128 Hz and c = 130.828 Hz and the

equal-tempered and twelve fifth-tones tuning have been eurythmically compared with one

another, singly and in groups.The tone c = 128 Hz as prime streams evenly and harmoniously through the whole

stature. Even the feet are self-evidently 'there' and can be reached without exertion.

With the tone c = 130.828 Hz the eurythmist experiences himself to be drawn

upwards, away from the feet, because the tone does not sound through but outside of the

body.

The same fundamental tendencies appear with the 30° tone movements. The tones

relating to c = 128 Hz rest within the area of the muscle and bone structure of the human

being. There exists a wonderful correspondence between gesture and tone. With c = 130.828

Hz the eurythmist can imagine these movements to lie within the body but they never become

real and direct experience there. If the eurythmist tries to find where they lie, the place, and/or angle cannot be found; he experiences only that they lie ‘outside’.

By eurythmical improvisation to the C major prelude from J. S. Bach's Well-

Tempered Clavier, the following experiences arose. With c = 130.828 and equal-tempered

tuning one experienced a world of most beautiful radiance. Nothing more beautiful exists. In

striving after this ‘world’ the eurythmist initially feels bigger, like a ‘wonderful soloist’. It is a

great feeling! But it deludes one; this world’ of most beautiful radiance is outside of him and

his true being.11 It proves to be an unattainable illusion, and the human being loses his

humanity, his freedom. If the attempt is made to bring this world of beautiful radiance to

expression–and this is only possible in that the eurythmist strives after this world, because it

does not come to him–then he tires very quickly. For the observer, the eurythmist looks tense

and small and the movement angular. This is due to the strong muscle tension which is

induced. A group of eurythmists move very inharmoniously together.

With the twelve fifth-tones tuning on c = 128 Hz the eurythmist feels as if he is being

invited to move. He moves his arms and the world streams in. The eurythmist feels himself in

balance between centre and periphery. The space between the arms is also filled, bringing

about relationship and conversation between the arms. The arms find a connection to one

another. The eurythmist does not need to strive after something. Rather he is left free to

approach the music with questioning interest. He then receives an answer. He experiences a

grace-given peace, peace as joy and quiet at the same time. Seen by the viewer, the movement

is peaceful, but big, and the periphery is filled. A group of eurythmists move together

harmoniously. In eurythmy terms one would say that the etheric surrounding and streaming is

full and evident.

Both pianos were in the same practice room, so comparisons could easily be made.

Sometimes a period of adjustment from one piano to the next was necessary-after the

brilliance and tension of equal-tempered tuning and c = 130.828 Hz (a = 440 Hz), the twelve

33

11 This feeling of outside of one, is not just caused by the pitch but also arises through equaltempered tuning. This feeling is possibly the reason many concert goers experience the music to bedistant, disconnected to one. Only accomplished musicians are able to overcome the quality of the

pitch and the tuning through their inner activity.Equal tempered tuning may leave one free because the archetypal music does not immediately

sound. But the eurythmist needs tones and intervals which bring to expression what he or she shows inthe gesture.



fifth-tones tuning and c = 128 Hz could occasionally feel empty. Here the above-mentioned

feeling of questioning interest was the key to finding the new quality. On other occasions the

change was experienced as a relief. A change the other way around was mostly experienced as

being unpleasant. On one day we would begin with the one piano and then we progressed to

the other; the next day we started with the other one. Sometimes we stayed with one piano for

the whole session.On one occasion the pianist told how the one piano required to be played very

differently to the other and suggested that she could try to swap the difference around, playing

on the one as if for the other. The effect was immediate. Playing on twelve fifth-tones tuning

as if for equal-tempered tuning, the falseness was clearly observed. The feeling was like

having caught a thief red-handed. The other way round gave the feeling that what was

wanting to be musically created was being attacked and destroyed by the equal-tempered

tuning, or, seen from the other angle, just did not speak–like an unsalted meal.

A piano tuned to a pitch correspondingly lower than c = 128 Hz as c = 130.828 Hz is

higher (c = 126 Hz) was not available. but comparisons were made with tuning forks and

tones played on a monochord. With the tone c = 126 Hz as prime, the eurythmist feels as if heis pressed so far into his body that the air is squeezed out of him. The head area feels very

small and the inner movement weighs downwards. One is drawn through the feet into weight.

These experiences show that a relatively small difference in pitch and method of

tuning have a big effect on our experience. The Middle Tuning is a tremendous discovery and

contributes to the expansion of musical expression and is an enrichment to musical sensation.

34



Explanation of Terms

• What is the duodecimo row ? This is an independent row within the overtone row ofC, in which the tones appear in the proportion of 3:1 (a duodecimo), e.g. C, g, d2,

a3… or the 1st, 3rd, 9th, 27th… overtones. Tones, which belong to this row, are alsocalled fifth tones . Two of the pitch specifications belong to this row: the tone a1 =432 Hz is the 4th tone of the duodecimo row beginning on 2C = 16 Hz (c = 128 Hz isthe 3rd octave of this C).

• What is Pythagorean tuning ? This is the tuning based on perfect fifths. Acharacteristic of this tuning is the relatively large Major third, which sounds bright,and to unaccustomed ears almost sharp.

• What are geometric mean tones? There are three means that build musicalintervals: the harmonic , arithmetic , and geometric mean. In the octave C–c the fifth,G, is arrived at through the arithmetic, the fourth, F, through the harmonic and thegeometric mean gives the middle between g and f12, to the equal tempered tritone f

sharp or g flat. However this tone is the middle of the octave and is thereforeneither the sharpened tone f sharp nor the flattened tone g flat, but the middle ofthe two. Therefore Maria Renold gave these geometric mean tones new names:Gelis (for the equal tempered tritone g flat/f sharp), Alis, Belis, Delis and Elis.13 Thethird tone specification gelis1 = 362.4 Hz is therefore a geometric mean tone.

• What are formed intervals? This is the name given by Maria Renold to the intervals,which lie between a diatonic tone and a geometric mean tone. E.g. b–gelis (formedfifth).

35

12 Musically, the middle of an interval is arrived at through the geometric mean. I.e. theresulting intervals are equally large. (Purely numerically, the middle is built through the arithmetic

mean.)

13 In German a flattened tone is called Ges and a sharpened tone Giss. The ending (e)lis isderived from these two terms.



Bibliography

Glöckler, G., & Glöckler, M. (2007). Das Musikalrische geheimnis des PlatonischenWeltenjahres. In A. Husemann (Ed.), Menschenwissenschaft durch Kunst

Verlag Freies Geistesleben.Jorgensen, O. (1991). Tuning: containing the perfection of eighteenth-century

temperament, the lost art of nineteenth-century temperament, and the scienceof equal temperament, complete with instructions for aural and electonictuning East Lansing, Mich.: Michigan State University Press.

Renold, M. (2004). Intervals, scales, tones and the concert pitch c = 128 Hz (B. M.Stevens, Trans.). Forest Row: Temple Lodge.

Schlesinger, K. (1939). The Greek aulos: a study of its mechanism and of its relationto the modal system of ancient Greek music . London: Methuen.

Documents

A Handbook on the Middle Tuning