The Harmonic System

The Harmonic System is one of variations of the Universal Unit System.

1 The Universal Unit System

By surprising coincidences, if the dozenal number system is used to express the speed of light in vacuum and the quantum of action as the defining constants such that these constants are strictly multiples of integer powers of twelve of the unit quantities, it is possible to construct a coherent unit system in which not only the constant that was used in the definition but also the Rydberg constant, the Bohr radius, the unified atomic mass unit, and half the value of the Planck length can be approximated to about or within an error of 2 per gross (1_1/2%) by a multiple of integer powers of twelve of the unit quantities.

In that case, many other physical constants, including the charge and mass of an electron, the fine structure constant, the molar volume of an ideal gas under standard conditions, the black-body radiation at the ice point, the density and surface tension of water, and others, can be approximated by multiples of integer powers of twelve of the unit quantities. Moreover, by adding the Boltzmann constant and using it in the definition of thermodynamic temperature, the gas constant of an ideal gas can be approximated by a multiple of an integer power of twelve of the unit quantity and the Stefan-Boltzmann constant, and the specific heat of water can be approximated by multiples of integer powers of twelve of the unit quantities with a factor 2 remaining.

For putting these coincidences to use, the dozenal number system is the only choice.

We define the Universal Unit System as “the unit system that is constructed by using the dozenal number system and the speed of light in vacuum, the quantum of action, and the Boltzmann constant as the defining constants in such a way that these constants become strict multiples of integer powers of twelve of the unit quantities, and the Rydberg constant, the unified atomic mass unit, the Bohr radius, and half the value of the Planck length can be approximated by multiples of integer powers of twelve of the unit quantities”.

(Please refer to dozenal.com for the overall Universal Unit System. The notation of this article is in principle according to footnote 1 of revised.pdf.)

2 Definition of the Harmonic System

The Harmonic System is conceptually designed and strictly defined as following:

2.1 Conceptual design

'Conceptual design' is the system design that prescribes the rough quantity of the units to constitute a unit system conceptually.

The Harmonic System concept is '35G=c₀=ħ=k_B=Z_P=1' .(See also here(Wikipedia Talk:Planck_units).)

The Harmonic System is conceptually designed to express the following quantities as multiples of integer powers of twelve of the corresponding unit quantities.

- 2E;(35.) times Newtonian gravitational constant (35G)

- speed of light in a vacuum (c₀)

- quantum of action (ħ)

- Boltzmann constant (k_B)

- Planck impedance (Z_P)

2.2 Strict definition

'Strict definition' is the system definition that corresponds strictly to the theoretical equations that a unit system is based on.

The Harmonic System is strictly defined to express the following quantities as the corresponding unit quantities.

- 100,1700; cycle over Rydberg constant

- 10;^-8 times speed of light in a vacuum

- 10;^+26; times quantum of action

- 10;^+20; times Boltzmann constant

- Planck impedance

3 Units

All the units which have special names peculiar to the Harmonic System are listed as following:

3.1 Length

harmon [hm] - 272.352206 mm, 10.7225278 inches (8 / 9 feet, difference 0.52%)

This is slightly shorter than 1/1000,0000; of the quadrant meridian length of the Earth(=279.136507 mm, difference 2.5%).

Volume of a cube whose side is half harmon (i.e. cubic half harmon) is 2/3 U.S. liquid gallon.(difference 0.06%) Water in this volume includes about 100;^10; H₂O molecules.(see section 3.3)

100. yards equal 240; harmons.(difference 0.08%)

1 megalithic yard equals about 3 harmons.

3.2 Physical time

nic [nc] - 390.625115 ms (25./64. SI seconds, almost exact)

This is almost 1/1000; of the difference of the length of one Julian year and one mean tropical year.(difference 0.14% at the beginning of year 1900.)

100,000. SI seconds(difference 2.8%), 10,0000; nics(almost exact), and 1 ⅛ days are nearly equal.

1 harmon / nic (=atol [al]) equals 2.51 km / hour. (almost exact)

See also chapter 4.

3.3 Mass

looloh [ℓℓ] - 131.829287 g, 4.65014126 ounces

This is slightly lighter than 1/100; of the mass of water in one cubic harmon capacity.(difference 6.4%)

On the other hand, this is slightly heavier than 1/100; of the mass of ice in the same capacity.(difference 2.4%)

'looloh' is named for the fact that it is 100;^10; times unified atomic mass unit.(almost exact) It means that Avogadro constant is expressed by 100;^10; times unit quantity(=reciprocal universal mol) of the Harmonic System.

The universal gravitational force between 2 mass points 5 loolohs and 7 loolohs put unit length apart is 10;^-8 times force unit.

When we use the Harmonic System, temperature rises twice the unit temperature if we give water of the unit mass 'looloh' heat of the unit energy.(almost exact)

3.4 Impedance

nohm [nh, Ω_n] - 29.9792458 Ω(exact)

This is the Planck impedance itself. The unit name 'nohm' is an abbreviation of 'natural ohm'.

The coulomb force between unit electric charges put unit length apart is 10;⁸ times force unit.

Relations of the electric and magnetic quantities in the Universal Unit System are shown as follows, where Ω₂ is total solid angle of the spherical surface(see this ref):

Supplemental comments

These difference percentages show us that the Harmonic System is a kind of dozenalized metric system.

(See also '[Addition]' part of this article.)

Using the Harmonic System, many constants can be approximated by 2ⁿ×10;^m times the unit quantity, where n = 0 or ±1 and m ∈ integer. The quadrant meridian length and gravitational acceleration of the Earth, the density and specific heat of water, which are used in the past for definition of units, are also included in these constants.

Please refer to this table and online converter for other units.

4 Time unit concepts

The Harmonic System distinguishes 'calendar time' from 'physical time' as a different concept.

4.1 Physical time

The natural time scale ladder of 'powers of twelve'

This quantity is time which flows uniformly from a viewpoint of physics.

'nic' is the physical time unit of the Harmonic System, and is on the natural time scale ladder of 'powers of twelve'.

The greatest common divisor of one tropical year and one day is 2^-7 days, which is also equivalent to about 1000; nics.

4.2 Calendar time

This quantity is the coordinated mean rotation angle of the Earth derived by using the direction of the Sun as a coordinate origin, and its dimension is not time but plain angle.

The dimension of ratio of physical time and calendar time is [time / plain angle], which means the period length of one mean solar day. Because of tidal friction, this length is not constant and becomes longer little by little.

The calendar time unit structure, 1 span = 10_0362,8000; nic-angles

The unit that should be dozenal divisions of mean solar day is calendar time rather than physical time.

Because physical length of mean solar day is not constant but variable, the relation of mean solar day and any unit of physical time will collapse by all means in the future.

'day', as a unit, is equivalent to plain angle of total circumference.

The Harmonic System adopts 'day' as a unit of calendar time corresponding to one mean solar day, and uses the calendar time unit structure shown on the right figure.

5 Number counting

The Universal Unit System recommends the dozenal myriad system.

When units of the Harmonic System are combined with Systematic Dozenal Nomenclature, only powers divisible with 4 or -1, -2, and -3 should be used. In this situation, dozen(10;), gross(100;) and doz gross(1000;) are used in substitution for powers 1, 2, and 3.

Also, the dozenal short scale system (dozenal thousand system):

(n) (n-illion) (number of zeros: 3n+3) (value: 10;³ⁿ⁺³)

1 million 6 (1,000,000;)

2 billion 9 (1,000,000,000;)

3 trillion 10; (1,000,000,000,000;)

4 quadrillion 13; (1,000,000,000,000,000;)

5 quintillion 16; (1,000,000,000,000,000,000;)

6 sextillion 19; (1,000,000,000,000,000,000,000;)

7 septillion 20; (1,000,000,000,000,000,000,000,000;)

8 octillion 23; (1,000,000,000,000,000,000,000,000,000;)

9 nonillion 26; (1,000,000,000,000,000,000,000,000,000,000;)

X dekrillion 29; (1,000,000,000,000,000,000,000,000,000,000,000;)

E elpillion 30; (1,000,000,000,000,000,000,000,000,000,000,000,000;)

10 dozillion 33; (1,000,000,000,000,000,000,000,000,000,000,000,000,000;)

11 undozillion 36; (etc...)

12 duodozillion 39

13 tredozillion 40

14 quattuordozillion 43

15 quindozillion 46

16 sexdozillion 49

17 septendozillion 50

18 octodozillion 53

19 novemdozillion 56

1X dekradozillion 59

1E elpindozillion 60

20 vigintillion 63

21 unvigintillion 66

22 duovigintillion 69

23 trevigintillion 70

24 quattuorvigintillion 73

25 quinvigintillion 76

26 sexvigintillion 79

27 septenvigintillion 80

28 octovigintillion 83

29 novemvigintillion 86

2X dekravigintillion 89

2E elpinvigintillion 90

30 trigintillion 93

31 untrigintillion 96

32 duotrigintillion 99

33 tretrigintillion X0

34 quattuortrigintillion X3

35 quintrigintillion X6

36 sextrigintillion X9

37 septentrigintillion E0

38 octotrigintillion E3

39 novemtrigintillion E6

3X dekratrigintillion E9

3E elpintrigintillion 100

40 quadragintillion 103

41 unquadragintillion 106

42 duoquadragintillion 109

43 trequadragintillion 110

44 quattuorquadragintillion 113

45 quinquadragintillion 116

46 sexquadragintillion 119

47 septenquadragintillion 120

48 octoquadragintillion 123

49 novemquadragintillion 126

4X dekraquadragintillion 129

4E elpinquadragintillion 130

50 quinquagintillion 133

51 unquinquagintillion 136

52 duoquinquagintillion 139

53 trequinquagintillion 140

54 quattuorquinquagintillion 143

55 quinquinquagintillion 146

56 sexquinquagintillion 149

57 septenquinquagintillion 150

58 octoquinquagintillion 153

59 novemquinquagintillion 156

5X dekraquinquagintillion 159

5E elpinquinquagintillion 160

60 sexagintillion 163

61 unsexagintillion 166

62 duosexagintillion 169

63 tresexagintillion 170

64 quattuorsexagintillion 173

65 quinsexagintillion 176

66 sexsexagintillion 179

67 septensexagintillion 180

68 octosexagintillion 183

69 novemsexagintillion 186

6X dekrasexagintillion 189

6E elpinsexagintillion 190

70 septuagintillion 193

71 unseptuagintillion 196

72 duoseptuagintillion 199

73 treseptuagintillion 1X0

74 quattuorseptuagintillion 1X3

75 quinseptuagintillion 1X6

76 sexseptuagintillion 1X9

77 septenseptuagintillion 1E0

78 octoseptuagintillion 1E3

79 novemseptuagintillion 1E6

7X dekraseptuagintillion 1E9

7E elpinseptuagintillion 200

80 octogintillion 203

81 unoctogintillion 206

82 duooctogintillion 209

83 treoctogintillion 210

84 quattuoroctogintillion 213

85 quinoctogintillion 216

86 sexoctogintillion 219

87 septenoctogintillion 220

88 octooctogintillion 223

89 novemoctogintillion 226

8X dekraoctogintillion 229

8E elpinoctogintillion 230

90 nonagintillion 233

91 unnonagintillion 236

92 duononagintillion 239

93 trenonagintillion 240

94 quattuornonagintillion 243

95 quinnonagintillion 246

96 sexnonagintillion 249

97 septennonagintillion 250

98 octononagintillion 253

99 novemnonagintillion 256

9X dekranonagintillion 259

9E elpinnonagintillion 260

X0 dekragintillion 263

X1 undekragintillion 266

X2 duodekragintillion 269

X3 tredekragintillion 270

X4 quattuordekragintillion 273

X5 quindekragintillion 276

X6 sexdekragintillion 279

X7 septendekragintillion 280

X8 octodekragintillion 283

X9 novemdekragintillion 286

XX dekradekragintillion 289

XE elpindekragintillion 290

E0 elpagintillion 293

E1 unelpagintillion 296

E2 duoelpagintillion 299

E3 treelpagintillion 2X0

E4 quattuorelpagintillion 2X3

E5 quinelpagintillion 2X6

E6 sexelpagintillion 2X9

E7 septenelpagintillion 2E0

E8 octoelpagintillion 2E3

E9 novemelpagintillion 2E6

EX dekraelpagintillion 2E9

EE elpinelpagintillion 300

100 centillion 303

(there is no consistent and widely accepted way to extend cardinals beyond centillion)

For the dozenal long scale system (dozenal million system), "n" should be corresponding to 6n (10⁶ⁿ) instead of 3n+3 (10³ⁿ⁺³), and 6n+3 can be either "thousand n-illion" or "n-illiard".