Prefatory note:This paper is dedicated to musicians who, like me, don’t know the first thing about mathematics. Some notions are therefore inaccurate to physicists and mathematicians. Pythagorean references are not either historical since they actually cover a large period of about 15 centuries involving many authors and viewed with a nowadays mathematical conception.
From some Physics basis this paper explores the Four mains acoustics system
 Pythagorean system
 Zarlino’ system
 Physician’s system(Holder)
 Equal temperament
then introduces temperaments and modes intricacy
Resonance, intonation, temperament and so on, seem to be very confused notions to many musicians. Through well known notions, I’ll try to clarify things, within my own understanding(mental representation I should say) of phenomena since I am impervious to mathematics and physics.
Pitch: The pitch is the relative position of a sound in the scale, i.e. the height of the sound. It is related to frequency. Low frequency sounds are written with the Bass clef, high frequency notes are notated with treble clef and ledger lines.
Sound speed and frequency seems often mixed up, so I will try to develop those concepts with rather unorthodox means.
A simple approach to the notion of frequency is to imagine sound as vibrations that regularly knock at your ear drum. Between two knocks, the distance is named Wavelength and the duration is called Period.
Physics teaches us that the ratio Distance/Time is the definition of Speed.
Higher frequency,that means more frequent knocks (i.e. gaining speed), needs to shorten the distance (wavelength) and therefore the duration (period) between knocks; so we immediately see the relation wavelength, period and frequency.
Physically speaking, the sound travels in a regular wave motion called cycle.
Figure 1 shows a cycle made of a convex then a concave waves. The amplitude (A) from the rest line to the top of the crest (or to the bottom of the trough) determines the intensity of the sound (loudness). The whole distance is the wavelength (λ) and the duration is the period (P).
From figure 1, you will notice that the wavelength is a kind of short cut; so the speed mentioned above is not the speed of sound which have a longer trip to do, but the speed of the occurrence of the knocks which is another way to express frequency.
With the period equal to one second time, we have the frequency unit call Hertz (Hz) [1 Hz = 1 Cycle/s]
Figure 2
In figure 2, we see the possibility to put more cycles in our « one second time box », provide the wavelength, measured from crest to crest, and its linked period be narrowed. That shows how frequency (number of cycles per time unit) is related to sound speed.
Sound speed or Celerity (C) (term used for propagation of waves) is almost a constant in a given medium such air.
Its value in air is about 340 meters (1115.49 feet) per second with a slight difference according to temperature and pressure.
For more accuracy the value is 331 meters (1085.96 feet) per second at 0 degree Celsius plus 0.6 meter per degree (in a normal atmospheric pressure). Therefore at 15 degrees Celsius C=331+ (0.6*15) =331+9 =340 meters/s.
Our one second time box is therefore a 340 meters wide box containing f cycles that are λ meters long. That is mathematically expressed: C=f*λ
Where C=Celerity (sound speed)
f= Frequency =Wavelength
Caution: C=340 m/s is only true in air as it is from the instrument to the hearer’s ear but the musician controls the speed or the wavelength to produces a given frequency by changing some parameters.
Sound speed will be discussed later on
Summary Frequency says how often a vibration occurs per time unit while sound speed tells how fast it passes through the medium. Period (P), that refers to the time it takes to complete a cycle, is the reciprocal of frequency (P=1/f)
The frequency (number of cycles per second) is inversely proportional to the length of the vibrating string or air column.
The shorter is the string, the higher is the frequency and its related pitch.
Table 1 computes frequencies of a virtual kind of sitar composed of seven similar, equally taut strings which are different only by their length; the first string, 20 centimeters (0.65 feet) long, yields a hypothetical frequency of 420 Hertz. The length of the second string is twice the length of the first one, the third string is 3 times longer than the first producing a frequency thrice smaller and so on.
You will notice that, with only seven strings, our one hand wide sitar (20 centimeters or (0.65 feet)is nearly the size of a Piano (1.4 Meter or 4.59 Feet) at the other extremity.
To get a handier sitar, we can obtain the same frequency by reducing the length of some strings and increasing their diameter.
The general idea is that frequency in not specifically proportional to length of the string or the air column but is proportional to the quantity of matter put into resonance.
Actually, sound speed(C) depends upon String Tension (T) String Masse (M) and String length (L) The relation is written [ C=T/(M/L)]
Frequency is related to speed and string length F=C/2L
From the basic formula C=F*λ we can extract F=C/λ and note that λ = 2 L and see the relation with wavelength
Resonance: A string or an air column put into vibrations emits a predominant sound named fundamental (that corresponds to the frequency computerized above).
Many other sounds generally accompany this fundamental. Those added sounds, called overtones or more precisely harmonics, which the frequency is a multiple of the fundamental, have a lower intensity and are therefore less perceived.
The number of overtones is theoretically unlimited but actually the only six first harmonics are clearly perceived.
Our 20 centimetres long string would ring at a frequency of 420 Hertz which is the fundamental that gives the pitch, plus, according to the resonant material qualities, 840 HZ (420*2), 1260 Hz (420*3), 1680 HZ (420*4), and so on
Resonance accounts for Timbre and Harmony
Resonance accounts for Timbre and Harmony
Timbre or sound quality
 Rare instruments emit only the fundamental harmonic, called pure sound.
 Unfortunately, purity may appear boring.
 Quality of sound depends upon richness in harmonics.
 According to the number of harmonics and their relative amplitude, the sound takes a specific quality (Timbre) that makes different a clarinet from a flute or an oboe.
Harmony
If you play a long C on the piano with the Forte pedal on you can perceive part of the following series.
In this example the fundamental is C, considered as first harmonic (H1) .
Notice
 all powers of 2 (H2, H4, H8, H16) are octaves that reinforce the fundamental
 paired harmonics (H2, H4, H6, H8, and H10) are replica of a former sound
 impaired harmonics (H3, H5, H7, H9, H11 and H13) add new sounds to the fundamental, creating a chord.
 the octave contains more and more divisions while the series increases
H2, the octave, is common to all civilizations and has to be pure.
H3 is the fifth of the fundamental and corresponds to the first accompanied melodies called organum or diaphony;
then comes the major triad chord H4 H5 H6 (tonic chord CEG), then the seventh /ninth (dominant) chord of the inferior fifth (F).
Interval ratios are given by reading two successive sound from right to left (example fifth=3/2, major third =5/4 )
Here are written by the nature the historical bases of tonal music,
with progressive climbing of the resonance series, and a summary of the tonal concept: a steady tonic chord that begins and ends the musical talk, flanked by an inferior and a superior fifth that create motion.
Remark: All above discussed notions on length, frequency and harmonics refer to a single note
Now let’s consider a one stringed instrument named Unicorde
The vibration part (L) of the string goes from the nut to the bridge.If we press the string with a finger, the string (L) is divided into two parts:
 a non vibrating part (A) from the nut to the finger
 a vibrating one (B) from the finger to the bridge.
With the finger placed in the middle
of the string, the string is divided into two equal parts A=1/2 and B=1/2; the frequency, which is the reciprocal of the length, is 2/1 that yields the octave.
Because 4=2*2 (or2 ^{2})would produce a double octave, placing our finger to the third of the string is the only other possibility to divide the string with
a whole number.
However, we encounter the alternative A=2/3+ B=1/3 and A=1/3+B=2/3.
In the first choice, the vibrating part =1/3 would multiply the frequency by 3 therefore beyond the octave.
So, we have to adopt
a 2/3 vibrating part which multiply the frequency by 3/2 or 1.5 and produce the fifth of the original note.
Keeping the first finger in the upper third of the string, it becomes a new nut and we can, like violinists or guitarists do, add a second finger to stop
the string at the upper third of remaining vibrating part.
That will yield the fifth of the fifth (3/2*3/2)
We can also measure the distance between the finger and the bridge and turn our violin into a lyre with strings 1.5 times shorter than the preceding one
Table 2

Open String 
String 1 
String 2 
String 3 
String 4 
Length(Ratio) 
1 
2/3 
4/9=(2/3)^{2} 
8/27=(2/3)^{3} 
16/81=2/3)^{4} 
Length(in Cm) 
81 
54 
36 
24 
16 
Frequency(Ratio) 
1 
3/2 
9/4 =(3/2)^{2} 
27/8=(3/2)^{3} 
81/16=(3/2)^{4} 
Frequency(Hertz) 
87 
130.5 
195.75 
293,62 
440,43 
Let’s call the first (open) string F; the following strings are C G D A E B.
That is the seven stringed Orpheus’ lyre
.
All the notes form the C major scale but
it needs length adjustments, listed in table 3, to remain within an octave range.
Table 3
Notice the symmetry
Note 
Modification of length 
C 
L= 
D 
L*2 
E 
L*4 
F 
L/2 
G 
L= 
A 
L*2 
B 
L*4 
Two strings keep their length C (tonic) and G (dominant), some lengths are doubled (L*2) or multiplied by 4.
The only sub dominant is divided.
Remark: The tonic is not the first string but the second string.
Now if we want to add an eighth string to our lyre, we get F# then, keeping on, C# G# D# A# E# B# that is another
lyre half a tone higher, with B# higher than C and therefore out of the range of an octave
.
Table 4
Notice the symmetry
Note 
Modification of length 
C# 
L/16 
D# 
L/32 
E# 
L/64 
F# 
L/8 
G# 
L/16 
A# 
L/32 
B# 
L/64 
So, to carry on we would need a hypothetical F## then F ### and so on.an endless spiral.
We therefore possess only two lyres.
To put our two lyres together in a single octave, we again have to adjust the length of the strings.
Summary : Division of string into 2 equal parts (1/2)produce an Octave while division into 3 equal parts (1/3)yields a fifth plus an octave .
To remain in the span of the octave we have to divide the ratio (1/3):(1/2)= (1/3)x (2/1)=2/3=1.5(2/3)=1.5
Succession of strings 1.5 time shorter or longer than the original produce an endless spiral chain of F C G D A E B always half a tone higher than the preceding one
Fb Cb Gd Db Ab Eb Bb
F C G D A E B
F# C# G# D# A# E# B#
F## C## G## D## A## E## B##
Since B# is higher than C only two sequences are available
Now, let’s compute the frequencies of our 14 stringed lyre.
Starting with a hypothetical string called F and an arbitrary frequency of 100 Hertz
 our first note C is 100*1.5 =150 Hz
 the second is G (150*1.5=225) or 100*1.5^{2}
 the third is D (225*1.5=337.5) or 100*1.5^{3}
 and so on
 following the cycles of the fifths up to B# (100*1.5^{14})
However, this calculation results in a rather steep sound since the 150 Hz initial
frequency reaches 29192.93 Hz (about 7 Octaves) in 14 Steps. To smooth the scale, we therefore have to contain all frequencies in one octave range
(from 150 Hertz to 150*2=300 Hertz in our example). To do so, each frequency superior to 300 has to be divided by 2 as many times as necessary. Table 5 shows
the two lyres put together but sorted separately into increasing frequency.
Table 5
Note 
Frequency 
Note 
Frequency 

C 
150 




C# 
160.18 
D 
168.75 




D# 
180.20 
E 
189.845 




E# 
202.72 
F 
200 




F# 
213.57 
G 
225 




G# 
240.27 

A 
253.12 




A# 
270.30 

B 
284.76 




B# 
304.09 

C 
300(150*2) 







If you play all the frequencies in zigzag from top to bottom, you will notice that the progressive rising scale is broken with two falling (E# F) and B#C.
Therefore, we have to get rid of two notes to keep a smooth rising scale.
B# overtakes the octave, so we can easily rule it out since octave, which is a repetition, reinforces the fundamental and awards it the tonic function,
furthermore octave must be pure i.e. in a simple ratio.
Choice between E# and F is more difficult because, whatever the choice, one fifth E# C or A# F will be too short; that is the Wolf fifth.
In a tonal point of view we would eliminate E# since the tonal system rests on a tonic flanked by a lower and an upper fifth.
So, this Pythagorean scale, based on fifths, is made of 11 pure fifths and 1 smaller fifth.
This descriptive scale is an upgraded Pythagorean’s scale (the Ptolemy’ scale) which doesn’t reflect the historical reality.
While we use raising frequencies, Pythagoras would have increased the length of the strings which would have resulted in a decreasing frequency series like below.
Table 6
NOTE 
C 
F 
Bb 
Eb 
Ab 
LENGTH 
1 
3 
9 
27 
51 
FREQUENCY 
4500 
1500 
500 
166,66 
55,55 
That is why all antic scales are descending.
Actually, Pythagoras used a general philosophical concept based on number and ratio.
Besides the tetraktys (1+2+3+4=10),
music has been built on one of his favorite special series called The universal proportion: a group of 4 numbers such 12, 9, 8 and 6,
lying on the combination of pair and impair
Table 7 Pair & Impair
Pair 
1 
2 

4 

6 

8 

10 

12 
Impair 
1 

3 


6 


9 


12 
that yields 3 kinds of ratio:
 Arithmetic proportion: 129=96=3
 Harmonic proportion 128/86=4/2=2
 Geometric proportion 12/8=9/6=3/2
 6 and 12 are common to both series
 9 is the Arithmetic mean (6+12)/2
 8 is the Harmonic moiety 2*(6*12)/12+6)
Summary Pythagorean music has been built on The universal proportion: a group of 4 numbers such 12, 9, 8 and 6, lying on the combination of pair and impair
With 4 homogeneous strings which are respectively 12, 9, 8, 6 units long, such
Exploring each combination, he could deduct
 CF: 12/9=4/3=the fourth
 CG: 12/8=3/2= the fifth
 CC: 12/6=2/1= the octave
 FG: 9/8=the tone
 FC: 9/6=3/2=the fifth
 GC: 8/6=4/3=the fourth
Before Pythagoras the lyre, based on the 3/2 ratio as formerly described, had only 5 strings.
In order to frame all the notes within an octave range, lets apply the length factor to get the following table
Table 8
String
Number 
Note 
Original string length 
Length adaptation factor 
New
String length 
Frequency 
1 
F 
3/2 
1/2 
3/4 
4/3 
2 
C 
1 
1 
1 
1 
3 
G 
2/3 
1 
2/3 
3/2 
4 
D 
(2/3)^{2} 
2 
8/9 
9/8 
5 
A 
(2/3)^{3} 
2 
16/27 
27/16 
Table 9:Sorted by increasing frequency
String number 
2 

4 

1 

3 

5 


Note 
C 

D 

F 

G 

A 

C 
Frequency 
1 

9/8 

4/3 

3/2 

27/16 

2 
Interval 

9/8 

32/27 

9/8 

9/8 

32/27 


T 

Minor third 

T 

T 

Minor third 
We have the asymmetric, pentatonic scale
Only two intervals are involved:
 The tone (T) (9/8) which is the difference between the fourth (3/4) and the fifth (2/3).
 The minor third (32/27) which is the difference between C and A: 32/27=2/1*16/27
Caution: difference means division (or multiplication by the inverted ratio) in term of proportion so 9/8 =3/4 *3/2
Table 10:With 7 Strings
String Number 
Note 
Original string length 
Length adaptation factor 
New String length 
Frequency 
1 
F 
3/2 
1/2 
3/4 
4/3 
2 
C 
1 
1 
1 
1 
3 
G 
2/3 
1 
2/3 
3/2 
4 
D 
(2/3)2 
2 
8/9 
9/8 
5 
A 
(2/3)3 
2 
16/27 
27/16 
6 
E 
(2/3)4 
4 
64/81 
81/64 
7 
B 
(2/3)5 
4 
128/243 
243/128 
Table 11:Sorted by increasing frequency
String number 
2 

4 

6 

1 

2 

5 

7 

1 
Note 
C 

D 

E 

F 

G 

A 

B 

C 
Frequency 
1 

9/8 

81/64 

4/3 

3/2 

27/16 

243/128 

2 
Interval 

9/8 

9/8 

256/243 

9/8 

9/8 

9/8 

256/243 


T 

T 

ST 

T 

T 

T 

ST 

We obtain the Pythagorean scale with only two kinds of intervals.
The tone (T) and the Semitone (ST) that is the difference between
F and E 256/243=64 /81*4/3 or C and B 256/243= 2/1*128/243
Two notes in succession form an interval which is the difference between the lower and the higher note.
Each note of the scale is related to the tonic. For example, Interval EF is CFCE.
Since notes are ratio, difference is actually quotient
The seven notes scale satisfied musicians for centuries. To add variety to melodies, some composers thought to divide the scale into twelve parts while keeping the assymmetry by adding the fifths on either direction.
Increasing direction introduces sharp notes while the decreasing way creates flat notes. The ratio is then inverted
Bb←F←C→G→D 2/3 2/3 3/2 3/2
Starting from C=150 Hertz, let’s multiply length string, or divide frequency by 1.5 to get decreasing fifths (Ratio 2/3). Using adaptation factor to remain within the octave span, we can compute the following table.
Table 12 Decreasing frequencies>
Note 
Frequency 
Multiplying Factor 
Corrected frequency 
C 
150 
1 
150 
F 
100 
2 
200 
Bb 
66.67 
4 
266.68 
Eb 
44.45 
4 
177.8 
Ab 
29.63 
8 
237.04 
Db 
19.75 
8 
158 
Gb 
13 
16 
210.72 
Cb 
17 
32 
280.96 
Fb 
5.85 
32 
187.2 
Compared to table 5 we remark that sharp notes frequency is superior to their enharmonic flat note
The slight difference between chromatic semitone (called Apotome) and diatonic semitone (or Limma) is the Pythagorean comma 2^{7}/_{312/212}=2^{7}*2^{12}/3^{12}=2^{19}1/3^{12} =0.987 (or 312/219) = 1,013.
Again :The difference between C and C# is a ratio: 160.18/150=1.067 (Apotome)
The difference between C and Db is the ratio: 158/150=1.053 (Limma)
Pythagorean comma =Apotome/limma=1.067/1.053=1.013 
12 fifths approximately travel 7 octaves (2^{7}) =128 whilst 12 pure fifths (3^{12}/2^{12} = 129.74) so the difference (quotient) is the Pythagorean comma:
128/129, 74=0.987 
Apotome is the difference (quotient) between the tone (9/8) and semitone (256/243) thus 9/8/256/243=9/8*243/256=2187/ 2048=37/211
Limma is the difference (quotient) between the fourth (4/3) and the ditone or Pythagorean third 9/8*9/8=81/64 thus 4/3/81/64=4/3*64/81= 256/243=28/35
Since 12 consecutive pure fifths lead to B# instead of C, one has to reduce a fifth to keep the octave pure. That is the wolf fifth.
Any fifth can be the wolf fifth but we generally choose the interval G#D# which is unused most of the time. However this method doesn’t allow modulation and need to retune instrument for each tonality.
Furthermore, Pythagoras/ Ptolemy ‘ scale possesses two different kinds of semitone. All this was quiet acceptable as long as music was melodic, played with a not fixed pitched instrument such violin but it became difficult with keyboard since extra keys were required.
Let’s now examine what happened when two strings are plucked simultaneously.
Going back to frequency cycle (figure 5), we notice the sine curve crosses the rest line three times per cycle.
The locations of the crossing the rest line, which means no sound, is called node. The parts between two nodes or antinodes are the vibrating parts.
It must be pointed out that this sinusoidal curve belongs to pure sound, i.e. having a sole frequency.
Sound is generally a mixture of pure sounds (sinusoidal curves) with different frequencies, named Harmonics that change the shape of the resulting wave by adding or subtracting their intensity according to the following rule:
Crest +Crest or Trough +Trough add each other making the sound louder
Crest + Through subtract each other lowering the sound.
Two strings played together will change the shape of the resultant wave according to the same rule.
Frequencies which have whole number ratios are musical since the smaller cycles divide exactly the larger cycle so nodes coincide with the node of the largest wave (the fundamental) and create a regular pattern.
Nodes can be compared to strings length division .
Figure 5 ,which represents the fundamental frequency (F1) (called First harmonics H1),requires a special attention. The wavelength λ is twice the string length because the wave goes back and forth ; a direct wave and a reflected one
As harmonic representation, only the first part of the cycle is concerned .String length division is addition of nodes as shown on figure 6
It must be pointed out that addition of node implies addition of half a wave
Since the fundamental frequency (F1) (called First harmonics H1) has a wavelength that is twice the length of the string (L), λ1=2/1*L,
second harmonics wavelength is λ2=2/2*L
,
third harmonics wavelength is λ3=2/3*L
nth harmonics= λn=2/n*L
Frequency of nth harmonics F(n) is F1 (fundamental frequency)* n and λn=1/n* L1
Fundamental note is accompanied by several higherfrequency tones called overtones having frequencies of 2,3,4,5 ..etc times the fundamental.
Many musicians use Harmonics,overtones and partials as synonyms
Others consider Fundamental as the first harmonics, second harmonics being the first overtone; fundamental may also be named first partial
 Some authors reserve the term « partials to overtones that are not an integer multiple of the fundamental
A very confusing situation
Musical doesn’t mean consonant;
The smaller ratio (nearer the fundamental sound), the more consonant
so 2/1(octave) is more consonant than 3/2 (fifth) which is more consonant than 4/3 (fourth) because it shares less harmonics than does an octave.
 In Octave interval every two harmonics are common
 in fifth interval every three harmonics correspond to each other and so forth
Here again, we meet the resonant series.
Now, two very near frequencies (< 7 cycles) interfere with one another and yield a beat pattern characterized by regular change in amplitude from 0 (no sound) to an increasing then decreasing wave.
The frequency of the beat pattern, which is a kind of swelling of the sound, is the difference (not the ratio) between the two frequencies.
For example 258 Hz â€“ 256 Hz = 2 Hz
Here is the trouble
Looking back upon the resonance series, major third (H5) is 5 times the fundamental;
with C= 150
E is 150*5 =750 divided by 4 to enter in the octave range 750/4=187.5
while our Pythagorean scale (table 4) reads E =189.84.
Because of the beat, this difference, named syntonic ( see below) is unacceptable for polyphony and chord;
that is why ornamentation and temperaments had been evolved.
Summary
Two very near frequencies (< 7 cycles) interfere with one another and yield a beat pattern characterized by regular change in amplitude creating a kind of swelling of the sound which was is unacceptable for polyphony and chord; that is why ornamentation and temperaments have been evolved.
Ornamentation
Especially in baroque music for harpsichord, beats prominent in long chords were masked with addition of extra notes such trill, mordent and so forth named ornaments
Temperaments
The aim of musicians is to overtake a challenge: built a pure octave scale with regular intervals, using simple ratio.Unfortunately, the musicians’ curse being that 2 (the octave) cannot be exactly divided by 3 (the begetter of notes),he has to make a choice to overcome this impossibility
As seen previously,
 one can tune eleven pure fifths and disregard the twelfth one (the wolf fifth)
 or evenly distribute (principle of the temperament) the reduction on several fifths (unequal temperament, as Werckmeister’s ones, used in Bach&’s Time) that smooths the impurity progress allowing some modulations
 or on the twelve fifths (equal temperament) which are no longer pure but some thirds and the wolf fifth are then enhanced.Two systems are particularly important:
 Meantone or Mesotonique temperament(XVIIth Century):Thirds are pure and fifths are almost pure for a specific tonality
 Our modern Twelvetone equal temperament where all intervals but octave are false
To summarize Pythagorean scale:
 all fifths but one are pure
 all major thirds are larger that the pure thirds of the resonance series
 all minor thirds are therefore smaller
 Furthermore, it contains two kinds of semitone which do not yield a tone.
Despite all these disadvantages, this scale cannot be ignored since only fifths are able to generate the twelve tones.
For a test, using the resonance series, we see the consonance next to the fifth is the major third (H5). Let’s try to generate a scale using this interval: C E G# B#; in only three skips we encounter the imperfect octave.
Since Pythagorean third is larger than the pure one, the triad is out of tune,
Zarlino, introduced the number 5, to try to adjust other intervals in order to keep the third its purity.
The general idea was to promote the three tonal chords I .IV.V. .
To do so he kept Pythagorean fifths FC, CG, and GD (for the C Major scale) and adapted DA / AE / EB.
He obtained the following ratios:
 CD=9/8 the Pythagorean tone
 CE= 5/4 the pure Major third
 CF=4/3 the pure fourth
 CG=3/2 the pure fifth
 CA= 5/3 the third (5/4) of the fourth (4/3) â†’ (5/4*4/3=20/12=5/3)
 CB=15/8 the third (5/4) of the fifth (3/2) â†’ (5/4*3/2=15/8)
Unfortunately we see that this scale leads to two kinds of tone 9/8 (Pythagorean tone) and 10/9, one sort of semi tone (16/15) and a wolf fifth (DA)
(Original A is the pure third of F, not the fifth of D).
Table 14 A :Zarlino’s C major scale
Notes 
Initial frequency 
Ratio 
Resultant frequency 
C 
150 
1/1 
150 
D 
150 
9/8 
168.75 
E 
150 
5/4 
187.5 
F 
150 
4/3 
200 
G 
150 
3/2 
225 
A 
150 
5/3 
250 
B 
150 
15/8 
281.25 
C 
150 
2 
300 
Table 14 B :Zarlino ‘s D major scale
Notes 
Initial frequency 
Ratio 
Resultant frequency 
D 
168.75 
1/1 
168.75 
E 
168.75 
9/8 
189.84 
F# 
168.75 
5/4 
210.93 
G 
168.75 
4/3 
225 
A 
168.75 
3/2 
253.12 
B 
168.75 
5/3 
281.25 
C# 
168.75 
15/8 
316.40 
D 
168.75 
2 
337.5 
.
Comparing C major and D Major Scales, you may notice that some notes (D G B) have the same frequency in both tonalities but A is different since it is now the fifth of the tonic instead of the third of the subtonic in C major.
Furthermore extension to chromatic scale is very complicated since two cycles (fifths and thirds) are involved allowing several ways to achieve it but leading to different results with many kinds of commas
The flat notes and sharp notes have then variable value according to tones succession.
However this system is the most convenient to harmony and chords
because it uses the resonance series elements (4:5:6 for the Major triad and 10:12:15 for the minor triad) with not beats
Since Pythagoras factors (2 and 3) generate too large thirds and Zarlino’s addition of the factor 5 creates more complications than it solves problem, Holder used a quiet different method of computation using irrationals numbers (roots) to divide the octave into 53 equal parts to find a compromise between the syntonic comma of the Zarlino « system and the Pythagorean comma.
53 is related to Pythagorean octave which is made of 5 apotomes of 5 commas and 7 limmas of 4 commas.
.53= (5*5) + (7*4)
Table 15 :Holderian C Major scale
NOTES 
Power of 2 
Initial frequency 
Multiplied by 
Frequency 
C 
0 
150 
1 
150 
D 
9/53 
150 
1,12491136 
168,736703 
E 
18/53 
150 
1,26542556 
189,813834 
F 
22/53 
150 
1,33338587 
200,00788 
G 
31/53 
150 
1,4999409 
224,991135 
A 
40/53 
150 
1,68730056 
253,095083 
B 
49/53 
150 
1,89806356 
284,709534 
C 
1 
150 
2 
300 
Each holderian comma worth 2 ^{1/53} (53^{th }root of 2)
Holderian scale (the one described in our modern theory books), like Pythagorean scale, has one sort of tone and two kinds of semitone, the diatonic semitone and the chromatic semitone.
 Tone has 9 holderian commas
 diatonic semitone worth 4 commas
 chromatic semitone 5 commas
.
Therefore
 a tone is 2^{9/53} (=1,1249)times original the frequency
 a diatonic semitone is 2^{4/53} (=1,0537)times original the frequency
 a chromatic semitone is 2^{5/53} (=1,0675)times original the frequency
If C=150 Hz then
 D is 150* 2^{9/53}=168.73
,
and
So D flat is lower than C sharp in that system
Computing the chromatic Holderian scale is rather complicated with the various combinations of the numbers 1, 4, 5 and 9 that lead to a 31 notes scales
Table 16: Numbers combinations of the Holder system: First column is the numerator n of the root
0123456789CB# DbC# D
Frequencies of notes are not much different from the Pythagorean system
but their nature is different and will open the way to the equal tempered scale.
Because the two kinds of semitones allow expressive intonation this scale is the singers’ scale.
Table 17: The 31 ordered notes of the Holder system: First column is the numerator n of the root
: Example G=2^{31/53}
In the Equal Tempered scale,the octave (2) is divided into 12 equal (so with no comma ) parts or semitones that worth 2^{1/12} or 12^{th } root of 2 =1.05946
 C =150
 C#=Db= 150*2^{1/12} =150 *1.05946=158,919
 D=150 (C)*2^{2/12} =150*1.12246 = 150*1.05946*1.05946=168,369
or 158,919 (C#)*1.05946
Each frequency is 1.05946 (2 ^{1/12})times the preceding frequency or the original frequence* 2^{rank/12}
The original rank is 0
C=150 =150* (2^{0/12})=150*1
Equal tempered scale is the best compromise since the enharmonic notes allow modulations in all tones.
However all intervals are false and scales seem to be the same in all tonalities.
Remark:There are many other acoustics systems of little usage.
UNITS used to compare frequencies
 1ratio used hitherto
 2 Savart Use of the Logarithm (base10) to transform ratio into difference. The octave (2/1) is 301.030 (=Log_{10}2Log_{10}1)*1000. it is the smallest perceptible interval
 3Cent: It is the smallest unit .Cent is the hundredth part of the semitone of the equal tempered scale 2^{1/1200} =1.0005778 Octave =1200 Cents
To transform ratio M/N into cents the formula is
1200/Log_{10}2*Log_{10}(M/N) 
Different kinds of commas
 Pythagorean comma: difference CB# 3^{12}/2^{12}2^{7}=1.0136 =5.885 = 23.4 Cents (approximately since cent is dedicated to equal tempered system)
 Syntonic or Zarlinian comma :Difference between the Pythagorean Major third (81/64)and Pure Major Third (5/4=80/64) =81/80= 5,395= log_{10}81log_{10}80=(1,9084850191,903089987)*1000=5,39503 or =21.5 cents
Shisma is the difference between Pythagorean and syntonic comma ˆ1.96 cents
 Theoretic Holder’s Comma: The ninth part of a tone=5.680 = 22.64 Cents
Pure intonation is indirectly related to temperament since it is a way of playing that uses only pure intervals involving the 2 3 and 5 numbers like Zarlino’ system. This method shares the same advantages and disadvantages of Zarlino ‘ scale
Modes is a rather complicated matter.
Dorian, Lydian and so forth are Greek names that refer to antic descending Greek scales but they have been assigned to more recent ascending Gregorian scales.
A given name doesn’t necessarily apply to the same scale according to the epoch or the author, furthermore Gregorian scales had two forms (authentic and plagal which is a fifth lower than the authentic one) so we encounter a very confusing situation.
This is easily resolved by ignoring the nomenclature. From the C major scale, modes start with each degree of the scale. So we have D mode scale (Dorian), E mode scale (Phrygian) and so on.
Each mode may of course be transposed and we say D mode transposed in G for example.
Descending series doesn’t have resonant properties in normal conditions.( electroacoustic devices are necessary to display them)
It is symmetric to the ascendant one but leads to minor chord and only higher harmonics can be perceived; that is why harmony always refers to the bass note.
The descending series produces the minor chord a fifth lower than the fundamental therefore C yields the ascending C major chord and the F minor descending chord.
Caution:Descending series must not be mixed up with Tartini’s third sounds
Symmetry in ratio does not necessarily mean symmetry in scale.
Semi tones in ascending (C maj) and descending (F min) scales do not match together
C 
D 
E 
F 
G 
A 
B 
C 
F 
Eb 
Db 
C 
Bb 
Ab 
G 
F 
However any ascending scale has its exact mirror that starts at the upper major third so the mirror of C major is E minor
C Maj ascending series and its mirror
C 
D 
E 
F 
G 
A 
B 
C 
E 
D 
C 
B 
A 
G 
F 
E 
The reverse is true
So the true minor relative scale of C major is E minor( Phrygien mode) and not Am
A min or is not the mirror of C Major
However E scale (Lydien scale) has a relationship with am :From E, as fundamental, the descending series is A minor (Aeolian mode) chord(a fifth below E).
Futhermore Em descending scale shares chords on IV and V scale degree with Am ascending scale creating either an authentic cadence or a plagal cadence.
Caution: the reverse is not valid
From A C E A E (decreasing harmonics fro E) Considering A the lower note as fundamental the descending series starts on a wrong note and is not the perfect mirror of the ascending C scale but the mirror of .E MaJ
Trying to look like the ascending major scale, ascending minor scale had its seventh degree raised and then went out of the original tonality. The scale became an hybrid
Confusion stems from the baroque period with its figured bass (basso continue) where the lower note of a chord prevailed so chord is said minor (meaning small) when the lower third is minor. It must be pointed out that minor chord is not smaller than a major one since both cover a pure fifth made of a major and a minor third in the reverse order
Minor chord is just a mirror of the Major chord but not Â a mirror of the scale.
Modality and temperament
Although antic Pythagorean scales, made of five then seven notes,content tones and semitones, there was no hierarchy since any notes is 1.5 times higher than the next so there is no imposed tonic.
Since descending scales they were not based upon resonance and were dedicated to melodies.
With ascending motion of scales and the advent of the polyphony, modality became an ambiguous notion because resonance concepts were added to a non resonant system.
Mode is no longer a change in scale direction but a change in semitones distribution which grant the scales a specific taste.
Unfortunately the equal temperament deleted the spice of those scales
New concepts let out defects that were more or less solved but, ignoring the original cause, everything got confused.
Concept of modality and temperament can be compared to an handyman that lays wallpaper crookedly; he would cut the parts that stick out at the top and the bottom, add a ribbon to cover the missing part, try to adjust and so on but the result is never satisfactory and forgetting the reason why he had to make adjustments,had a new device that creates a new problem.
To conclude: It seems to me that confusion lies in the fact that the single ratio principle reflects several intricate musical fields.
The musician, who is a frequencies maker, can play differently a single note(a given frequency) by applying different sort of ratios:
 Frequency is proportional to the length of a string
; an affirmation that refers to melody (successive notes). However, since nodes and antinodes are equivalent to string length, it also accounts for resonance (simultaneous notes or harmony)
Note: In many situations the differences between different acoustic systems are so little that we may think there is much ado for nothing but it must be pointed out that the differences in concepts lead to new ways of research such rules of counterpoint and harmony, new instruments factory with the introduction of temperament or new timbre with micro intervals.