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# 1-2-5 Series

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### 1-2-5 Series

In industrial design, preferred numbers (also called preferred values) are standard guidelines for choosing exact product dimensions within a given set of constraints. Product developers must choose numerous lengths, distances, diameters, volumes, and other characteristic quantities. While all of these choices are constrained by considerations of functionality, usability, compatibility, safety or cost, there usually remains considerable leeway in the exact choice for many dimensions.

Preferred numbers serve two purposes:

1. Using them increases the probability of compatibility between objects designed at different times by different people. In other words, it is one tactic among many in standardization, whether within a company or within an industry, and it is usually desirable in industrial contexts. (The opposite motive can also apply, if it is in a manufacturer's financial interest: for example, manufacturers of consumer products often have a financial interest in lack of compatibility, in planned obsolescence, and in selling name-brand and model-specific replacement parts.)
2. They are chosen such that when a product is manufactured in many different sizes, these will end up roughly equally spaced on a logarithmic scale. They therefore help to minimize the number of different sizes that need to be manufactured or kept in stock.

## Renard numbers

The French army engineer Col. Charles Renard proposed in the 1870s a set of preferred numbers for use with the metric system. His system was adopted in 1952 as international standard ISO 3. Renard's system of preferred numbers divides the interval from 1 to 10 into 5, 10, 20, or 40 steps. The factor between two consecutive numbers in a Renard series is constant (before rounding), namely the 5th, 10th, 20th, or 40th root of 10 (1.58, 1.26, 1.12, and 1.06, respectively), which leads to a geometric sequence. This way, the maximum relative error is minimized if an arbitrary number is replaced by the nearest Renard number multiplied by the appropriate power of 10.

The most basic R5 series consists of these five rounded numbers:

 R5:  1.00        1.60        2.50        4.00        6.30


Example: If our design constraints tell us that the two screws in our gadget should be placed between 32 mm and 55 mm apart, we make it 40 mm, because 4 is in the R5 series of preferred numbers.

Example: If you want to produce a set of nails with lengths between roughly 15 and 300 mm, then the application of the R5 series would lead to a product repertoire of 16 mm, 25 mm, 40 mm, 63 mm, 100 mm, 160 mm, and 250 mm long nails.

If a finer resolution is needed, another five numbers are added to the series, one after each of the original R5 numbers, and we end up with the R10 series:

 R10: 1.00  1.25  1.60  2.00  2.50  3.15  4.00  5.00  6.30  8.00


Where an even finer grading is needed, the R20, R40, and R80 series can be applied:

 R20: 1.00  1.25  1.60  2.00  2.50  3.15  4.00  5.00  6.30  8.00
1.12  1.40  1.80  2.24  2.80  3.55  4.50  5.60  7.10  9.00

 R40: 1.00  1.25  1.60  2.00  2.50  3.15  4.00  5.00  6.30  8.00
1.06  1.32  1.70  2.12  2.65  3.35  4.25  5.30  6.70  8.50
1.12  1.40  1.80  2.24  2.80  3.55  4.50  5.60  7.10  9.00
1.18  1.50  1.90  2.36  3.00  3.75  4.75  6.00  7.50  9.50

 R80: 1.00  1.25  1.60  2.00  2.50  3.15  4.00  5.00  6.30  8.00
1.03  1.28  1.65  2.06  2.58  3.25  4.12  5.15  6.50  8.25
1.06  1.32  1.70  2.12  2.65  3.35  4.25  5.30  6.70  8.50
1.09  1.36  1.75  2.18  2.72  3.45  4.37  5.45  6.90  8.75
1.12  1.40  1.80  2.24  2.80  3.55  4.50  5.60  7.10  9.00
1.15  1.45  1.85  2.30  2.90  3.65  4.62  5.80  7.30  9.25
1.18  1.50  1.90  2.36  3.00  3.75  4.75  6.00  7.50  9.50
1.22  1.55  1.95  2.43  3.07  3.87  4.87  6.15  7.75  9.75


In some applications more rounded values are desirable, either because the numbers from the normal series would imply an unrealistically high accuracy, or because an integer value is needed (e.g., the number of teeth in a gear). For these needs, more rounded versions of the Renard series have been defined in ISO 3:

 R5″: 1           1.5         2.5         4           6

R10′: 1     1.25  1.6   2     2.5   3.2   4     5     6.3   8

R10″: 1     1.2   1.5   2     2.5   3     4     5     6     8

R20′: 1     1.25  1.6   2     2.5   3.2   4     5     6.3   8
1.1   1.4   1.8   2.2   2.8   3.6   4.5   5.6   7.1    9

R20″: 1     1.2   1.5   2     2.5   3     4     5     6     8
1.1   1.4   1.8   2.2   2.8   3.5   4.5   5.5   7      9

R40′: 1     1.25  1.6   2     2.5   3.2   4     5     6.3   8
1.05  1.3   1.7   2.1   2.6   3.4   4.2   5.3   6.7    8.5
1.1   1.4   1.8   2.2   2.8   3.6   4.5   5.6   7.1    9
1.2   1.5   1.9   2.4   3     3.8   4.8   6     7.5    9.5


As the Renard numbers repeat after every 10-fold change of the scale, they are particularly well-suited for use with SI units. It makes no difference whether the Renard numbers are used with metres or kilometres. But one would end up with two incompatible sets of nicely spaced dimensions if they were applied, for instance, with both yards and miles.

Renard numbers are rounded results of the formula

$R\left(i,b\right) = 10^\left\{\frac\left\{i\right\}\left\{b\right\}\right\}$,

where b is the selected series value (for example b = 40 for the R40 series), and i is the i-th element of this series (with i = 0 through i = b).

## Rail gauges

Only two rail gauges are preferred numbers, and these are likely accidental, but are remarkable in that they are in the R10 series whether expressed in inches or millimetres.

The more common gauge is the Irish gauge, 63 inches, which rounds to 1600 mm, both numbers in the R10 series. It is also used in Australia and Brazil. The other gauge is just half this, 800 mm or 31.5 inches, and is used by the Wengernalpbahn in Switzerland, between Lauterbrunnen and Grindelwald by way of Kleine Scheidegg.

## 1-2-5 series

In applications for which the R5 series provides a too fine graduation, the 1-2-5 series is sometimes used as a cruder alternative. It is effectively an R3 series rounded to one significant figure:

... 0.1 0.2 0.5 1 2 5 10 20 50 100 200 500 1000 ...

This series covers a decade (1:10 ratio) in three steps. Adjacent values differ by factors 2 or 2.5. Unlike the Renard series, the 1-2-5 series has not been formally adopted as an international standard. However, the Renard series R10 can be used to extend the 1-2-5 series to a finer graduation.

This series is used to define the scales for graphs and for instruments that display in a two-dimensional form with a graticule, such as oscilloscopes.

The denominations of most modern currencies follow a 1-2-5 series. An exception are some quarter-value coins, such as the Canadian quarter and the United States quarter (the latter denominated as "quarter dollar" rather than 25 cents). A ¼-½-1 series (... 0.1 0.25 0.5 1 2.5 5 10 ...) is used by currencies derived from the former Dutch gulden (Aruban florin, Netherlands Antillean gulden, Surinamese dollar), some Middle Eastern currencies (Iraqi and Jordanian dinars, Lebanese pound, Syrian pound), and the Seychellois rupee. However, newer notes introduced in Lebanon and Syria due to inflation follow the standard 1-2-5 series instead.

## E series

In electronics, international standard IEC 60063 defines another preferred number series for resistors, capacitors, inductors and Zener diodes. It works similarly to the Renard series, except that it subdivides the interval from 1 to 10 into 6, 12, 24, etc. steps. These subdivisions ensure that when some arbitrary value is replaced with the nearest preferred number, the maximum relative error will be on the order of 20%, 10%, 5%, etc.

Use of the E series is mostly restricted to resistors, capacitors, inductors and zener diodes. Commonly produced dimensions for other types of electrical components are either chosen from the Renard series instead (for example fuses) or are defined in relevant product standards (for example wires).

The IEC 60063 numbers are as follows. The E6 series is every other element of the E12 series, which is in turn every other element of the E24 series:

E6  ( 20%): 10      15      22      33      47      68

E12 ( 10%): 10  12  15  18  22  27  33  39  47  56  68  82

E24 (  5%): 10  12  15  18  22  27  33  39  47  56  68  82
11  13  16  20  24  30  36  43  51  62  75  91


With the E48 series, a third decimal place is added, and the values are slightly adjusted. Again, the E48 series is every other value of the E96 series, which is every other value of the E192 series:

E48  ( 2%): 100  121  147  178  215  261  316  383  464  562  681  825
105  127  154  187  226  274  332  402  487  590  715  866
110  133  162  196  237  287  348  422  511  619  750  909
115  140  169  205  249  301  365  442  536  649  787  953


E96 (  1%): 100  121  147  178  215  261  316  383  464  562  681  825
102  124  150  182  221  267  324  392  475  576  698  845
105  127  154  187  226  274  332  402  487  590  715  866
107  130  158  191  232  280  340  412  499  604  732  887
110  133  162  196  237  287  348  422  511  619  750  909
113  137  165  200  243  294  357  432  523  634  768  931
115  140  169  205  249  301  365  442  536  649  787  953
118  143  174  210  255  309  374  453  549  665  806  976

E192 (0.5%) 100  121  147  178  215  261  316  383  464  562  681  825
101  123  149  180  218  264  320  388  470  569  690  835
102  124  150  182  221  267  324  392  475  576  698  845
104  126  152  184  223  271  328  397  481  583  706  856
105  127  154  187  226  274  332  402  487  590  715  866
106  129  156  189  229  277  336  407  493  597  723  876
107  130  158  191  232  280  340  412  499  604  732  887
109  132  160  193  234  284  344  417  505  612  741  898
110  133  162  196  237  287  348  422  511  619  750  909
111  135  164  198  240  291  352  427  517  626  759  920
113  137  165  200  243  294  357  432  523  634  768  931
114  138  167  203  246  298  361  437  530  642  777  942
115  140  169  205  249  301  365  442  536  649  787  953
117  142  172  208  252  305  370  448  542  657  796  965
118  143  174  210  255  309  374  453  549  665  806  976
120  145  176  213  258  312  379  459  556  673  816  988


The E192 series is also used for 0.25% and 0.1% tolerance resistors.

1% resistors are available in both the E24 values and the E96 values.

## Buildings

In the construction industry, it was felt that typical dimensions must be easy to use in mental arithmetic. Therefore, rather than using elements of a geometric series, a different system of preferred dimensions has evolved in this area, known as "modular coordination".

Major dimensions (e.g., grid lines on drawings, distances between wall centres or surfaces, widths of shelves and kitchen components) are multiples of 100 mm, i.e. one decimetre. This size is called the "basic module" (and represented in the standards by the letter M). Preference is given to the multiples of 300 mm (3 M) and 600 mm (6 M) of the basic module (see also "metric foot"). For larger dimensions, preference is given to multiples of the modules 12 M (= 1.2 m), 15 M (= 1.5 m), 30 M (= 3 m), and 60 M (= 6 m). For smaller dimensions, the submodular increments 50 mm or 25 mm are used. (ISO 2848, BS 6750)

Dimensions chosen this way can easily be divided by a large number of factors without ending up with millimetre fractions. For example, a multiple of 600 mm (6 M) can always be divided into 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, etc. parts, each of which is again an integral number of millimetres.

## Paper documents, envelopes, and drawing pens

Main article: Paper size

Standard metric paper sizes use the square root of two and related numbers (√√√2, √√2, √2, 2, or 2√2) as factors between neighbour dimensions (Lichtenberg series, ISO 216). The √2 factor also appears between the standard pen thicknesses for technical drawings (0.13, 0.18, 0.25, 0.35, 0.50, 0.70, 1.00, 1.40, and 2.00 mm). This way, the right pen size is available to continue a drawing that has been magnified to a different standard paper size.

## Computer engineering

When dimensioning computer components, the powers of two are frequently used as preferred numbers:

 1    2    4    8   16   32   64  128  256  512 1024 ...


Where a finer grading is needed, additional preferred numbers are obtained by multiplying a power of two with a small odd integer:

(×3)  6   12   24   48   96  192  384  768 1536 ...
(×5) 10   20   40   80  160  320  640 1280 2560 ...
(×7) 14   28   56  112  224  448  896 1792 3584 ...

Preferred aspect ratios
16: 15: 12:
:8 2:1 3:2
:9 16:9 5:3 4:3
:10 8:5 3:2
:12 4:3 5:4 1:1

In computer graphics, widths and heights of raster images are preferred to be multiples of 16, as many compression algorithms (JPEG, MPEG) divide color images into square blocks of that size. Black-and-white JPEG images are divided into 8x8 blocks. Screen resolutions often follow the same principle. Preferred aspect ratios have also an important influence here, e.g. 2:1, 3:2, 4:3, 5:3, 5:4, 8:5, 16:9.

## Retail packaging

In some countries, consumer-protection laws restrict the number of different prepackaged sizes in which certain products can be sold, in order to make it easier for consumers to compare prices.

An example of such a regulation is the European Union directive on the volume of certain prepackaged liquids (75/106/EEC [1]). It restricts the list of allowed wine-bottle sizes to 0.1, 0.25 (1/4), 0.375 (3/8), 0.5 (1/2), 0.75 (3/4), 1, 1.5, 2, 3, and 5 litres. Similar lists exist for several other types of products. They vary and often deviate significantly from any geometric series in order to accommodate traditional sizes when feasible. Adjacent package sizes in these lists differ typically by factors 2/3 or 3/4, in some cases even 1/2, 4/5, or some other ratio of two small integers.

## Music

Main article: tuning system

While some instruments (trombone, theremin, etc.) can play a tone at any arbitrary frequency, other instruments (such as pianos) can only play a limited set of tones. The very popular "twelve-tone equal temperament" selects tones from the geometric sequence

$f\left(n\right) = k \times 2^\left\{\frac\left\{n\right\}\left\{12\right\}\right\}.$

where k is typically 440 Hz, though other standards have been used. However, other less common tuning systems have also been historically important as preferred audio frequencies.

Since 210≈103, 21/12≈103/120=101/40, and the resultant frequency spacing is very similar to the R40 series.

## Photography

In photography, aperture, exposure, and film speed generally follow powers of 2:

The aperture size controls how much light enters the camera. It's measured in f-stops: f/1.4, f/2, f/2.8, f/4, etc. Full f-stops are a square root of 2 apart. Digital cameras often subdivide these into thirds, so each f-stop is a sixth root of 2, rounded to two significant digits: 1.0, 1.1, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.5, 2.8, 3.2, 3.5, 4.0.

The film speed is a measure of the film’s sensitivity to light. It's expressed as ISO values such as “ISO 100”. Measured film speeds are rounded to the nearest preferred number from a modified Renard series including 100, 125, 160, 200, 250, 320, 400, 500, 640, 800... This is the same as the R10′ rounded Renard series, except for the use of 6.4 instead of 6.3, and for having more aggressive rounding below ISO 16. Film marketed to amateurs, however, uses a restricted series including only powers of two multiples of ISO 100: 25, 50, 100, 200, 400, 800, 1600 and 3200. Some low-end cameras can only reliably read these values from DX encoded film cartridges because they lack the extra electrical contacts that would be needed to read the complete series. Some digital cameras extend this binary series to values like 12800, 25600, etc. instead of the modified Renard values 12500, 25000, etc.

The shutter speed controls how long the camera records light. These are expressed as fractions of a second, roughly but not exactly based on powers of 2: 1 second, 1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/250, 1/500, 1/1000 of a second.

## References

• ISO 3, Preferred numbers — Series of preferred numbers. International Organization for Standardization, 1973.
• ISO 17, Guide to the use of preferred numbers and of series of preferred numbers. 1973.
• ISO 497, Guide to the choice of series of preferred numbers and of series containing more rounded values of preferred numbers. 1973.
• ISO 2848, Building construction — Modular coordination — Principles and rules. 1984.
• ISO/TR 8389, Building construction — Modular coordination — System of preferred numbers defining multimodular sizes. International Organization for Standardization, 1984.
• IEC 60063, Preferred number series for resistors and capacitors. International Electrotechnical Commission, 1963
• 75/106/EEC [2], European Union Directive on volume of liquids. 1975.
• BS 2045, Preferred numbers. British Standards Institution, 1965.
• BS 2488, Schedule of preferred numbers for the resistance of resistors and the capacitance of capacitors for telecommunication equipment. 1966.
• ANSI Z17.1, American National Standard for Preferred Numbers. 1973
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