Reassessing the Proportional System of Joseon Era Wooden Architecture: The Bracket Arm Length and Width as a Standard Modular Method

Abstract

Previous research has agreed that standard modular methods in Song Yingzao Fashi and Qing Gongcheng Zuofa were not applied to Korean wooden architecture. This study notes the size of bracket arms as a standard modular method by investigating the proportion systems of Sungnyemun, Paldalmun, and Heunginjimun Gates, the official government buildings of the Joseon Dynasty. The purlin direction bracket arms in the intercolumnar bracket sets apply a proportional system in the ratio of a regular integer relationship to the front and side facades and building height. Challenging current assumptions, the application of the bracket arm width as a modular rule is divided into more subdivided values than the measurement units. A particularly important finding is that, unlike the height of the bracket arms, the width and length of the brackets are standard members that determine the height of the side facades. This is very similar to the official government building styles in the Song and Qing Dynasties, premodern China. Therefore, this study is meaningful in reassessing wooden frame structures of the Joseon era, deriving parametric measuring rules universally applied in East Asia to provide basic data useful for heritage conservation.

1. Introduction

1.1. Overview

In East Asia, the concept of measure is widely adopted in all construction styles and types when constructing wooden structures. This concept originates from the human body and differs from the widely used meter (m), based on 1 qi, where 1 qi can be divided into 10 cun and 1 cun into 10 fen. One cun corresponds to the width of a thumb at the knuckle. Units of measurement (yeongjo-cheok) are employed for the design of East Asian wooden architecture or the manufacture of objects. In the Joseon Dynasty, it was standardized to 31.22 cm in the 12th regime by King Sejo (1466), and after a gradual reduction, it was set at 30.3 cm in the 6th regime by Emperor Gwangmu (1902) [1] (p. 455). Despite standard measurement units, the construction site depended on several factors with no fixed rules, such as the style of the times, artisans’ techniques, and regional characteristics.

However, in most cases, the lengths of all bays could not be divided into one cheok or two cheoks (Korean pronunciation of the qi unit in Chinese). The length unit cannot be expressed as an integer ratio, and the units of measurement themselves are not uniform as they depend on the type and size of the building. Since a lack of other indicators or criteria of reference points have been proposed for most buildings, there have been issues of authenticity to the restoration works.

In China, instead, the Song Yingzao Fashi (State Building Standards, 1103, hereafter YZFS) [2] uses the basic module system (mushu), which divides the building grades into eight levels and indicates the size of the building according to the flower-shaped bracket arm [3] (pp. 6–11). The Qing Gongcheng Zuofa Zeli (Building Code of the Construction Bureau, 1734, hereafter GCZF) [4] has the size of the cap block divided into eleven stages, indicating the shape and grade of the building and the scope of buildings [5] (pp. 10–12). The stage is determined based on the size of the cap block by the purlin direction of the cap block placed on the intercolumnar brackets. Depending on the width of the cap blocks placed on the bracket arms, the first grade has a width of 6 cun, and the eleventh grade has a width of 1 cun (Figure 1 and Figure 2).

In the YZFS and GCZF, the parametric proportional system is determined by the size of the bracket arm. To procure the reference value, the Song YZFS is based on the width of the flower-shaped bracket arms in the direction of the beams, whereas the Qing GCZF considers the width of the bracket arms in the direction of the purlins. They all jointly include buildings with bracket sets and use the width of the bracket arms as a unifying module for all members [6] (p. 8), [7] (p. 67) (Figure 3).

Figure 1. Grades of Cai (a standard module for the design of all parts of a building) and Zhi (a secondary modular timber) module in the YZFS [8]. The eight grades of a cai module of the YZFS: left to right: first grade (9 × 6 cun (a measure of length, 3.03 cm), second (8.25 × 5.5 cun), third (7.5 × 5 cun), fourth (7.2 × 4.8 cun), fifth (6.6 × 4.4 cun), sixth (6 × 4 cun), seventh (5.25 × 3.5 cun), eighth (4.5 × 3 cun) (featuring a smaller modular unit zhi on top of a cai timber of each grade) [9] (p. 240).

Figure 1. Grades of Cai (a standard module for the design of all parts of a building) and Zhi (a secondary modular timber) module in the YZFS [8]. The eight grades of a cai module of the YZFS: left to right: first grade (9 × 6 cun (a measure of length, 3.03 cm), second (8.25 × 5.5 cun), third (7.5 × 5 cun), fourth (7.2 × 4.8 cun), fifth (6.6 × 4.4 cun), sixth (6 × 4 cun), seventh (5.25 × 3.5 cun), eighth (4.5 × 3 cun) (featuring a smaller modular unit zhi on top of a cai timber of each grade) [9] (p. 240).

Figure 2. Grades in the GCZF [6] (p. 8). The doukou (the width of a bracket arm) has 11 grades, and the minimum dimension of doukou is 1 cun (3.5 cm), obtaining an approximate value of the eave column diameter (D) from 3.6 doukou to 5.5 doukou [10].

Figure 2. Grades in the GCZF [6] (p. 8). The doukou (the width of a bracket arm) has 11 grades, and the minimum dimension of doukou is 1 cun (3.5 cm), obtaining an approximate value of the eave column diameter (D) from 3.6 doukou to 5.5 doukou [10].

Figure 3. Conceptual definition of the reference value and direction (beam purlin) for setting the basic module between YZFS and GCZF. Line drawing of Jeong-Hee Lee.

Figure 3. Conceptual definition of the reference value and direction (beam purlin) for setting the basic module between YZFS and GCZF. Line drawing of Jeong-Hee Lee.

In other words, in China, restoration designs and buildings do not use a unit of measurement but estimations of the overall size of buildings depending on the size of the flower-shaped bracket arm or cap block. A standard model is presented of the most appropriate building structure to optimize the building use. Such a model reduces material loss and standardizes the structure of a building by adjusting the number of timber materials required for construction. The authenticity restoration according to the times is secured to some extent. A previous study attempted to investigate the presence of a standard modular system for wooden buildings in Korea, akin to the Chinese construction norms. This was achieved by analyzing the proportional systems and construction methods employed in traditional Korean wooden-framed buildings [11] (p. 29). The research presumed that a module system by the cheok measurement unit might be used as historical evidence for the repair and restoration of traditional architecture. However, the study concluded that there were no modular systems in Korea similar to those in China [11] (pp. 613–626). As a result, the restoration of buildings in the Goryeo or before, the Unified Silla period, the Three Kingdoms period, and the Joseon period faced many difficulties. Therefore, this study emphasizes the need to search for a standard modular system in the Korean peninsula, especially after previous studies have found the standard form of wood-frame constructions (Figure 4).

1.2. Related Studies

There is no study on the proportion system of the wooden architecture of the official style in Korea. Most studies are related to temple architecture and are analyzed without separating local and official-style buildings. There is a study on proportionality from the point of view of modular design [12]. The study discusses differences depending on the load-bearing capacity of the wooden structures, so the scale and proportions are determined by the floor plan. The appearance of a building is determined by the type of roof and the proportions of the facade [12]. The study that analyzed the proportions of 103 wooden buildings in the Joseon Dynasty concluded that the proportions were not uniform [13]. Most of the conclusions of these studies are that in Korea, unlike in China, the system of standard modules and specific elements is not applied to the standard modules [11]. The comparison with the Great East Hall of the Foguangsi Buddhist Monastery is noteworthy because a quantitative value in the form of a moshu module with the size of the bracket arm (nidaogong) connected to the capital block along the wall plane has attracted attention. The construction principle of the Great East Hall supports that the classes in YZFS are determined by the size of the flower-shaped protruding arm (salmi). As for the three Joseon Dynasty gate tower buildings in this study, the width of the bracket arm was found to be a reference point in mutual similarity with the Song YZFS, although the previous studies assumed that there was no such rule as the Moshu and Doukou modular system in ancient Korean architecture [11] (p. 616).

In all studies of East Asian architecture, the exact dimensions and shapes of the wooden members are not mentioned at all [11,12,13] because there are no concrete drawings or historical records from the time when these buildings were built. As these buildings survived for a long time, the wooden members were reinforced and replaced accordingly. Therefore, the major assumption of these studies is to focus on analyzing the current building construction method and the combination of wooden members to understand the new construction principle through the dimensions of the bracket arm.

1.3. Research Methods and Scope

In Chinese wooden-framed architecture, the moshu module system and doukou module system in each Song YZFS and Qing GCZF allow us to know the scale, shape, grade, and proportional system of wooden buildings. The units of measurement are a tool for a unit of standard size used in design and construction to facilitate the assembly of building components and reduce the waste of materials. There is no proportional system with a specific shape in timber frame buildings remaining in Korea [11] (pp. 613–620). As in China, there is a boundary that is not significantly different from the units of measurement but only a reference point based on certain components. To investigate the commonality of timber frame buildings in East Asia, the length, width, and height of bracket arms were measured based on those commonly used in YZFS and GCZF [6,14]. In this way, the relationship that determines the overall size of the building was analyzed to establish the qualities of the buildings. Because all remaining buildings of the Goryeo period were built by local temples, there are many difficulties in presenting the standard measuring units (yeongjo-cheok) because the local taste differs from the official-style buildings.

This study focused on the gate towers along the city walls of the Joseon Dynasty, which are among the official-style buildings still standing today. The reason why the gate towers were selected is that of the early buildings of the Joseon era, only a few official-style buildings have survived, and the Sungnyemun Gate is the only one that still exists. Among the Joseon era buildings, three two-storied gate tower buildings with few repairs and clear records have been studied: Sungnyemun Gate, Paldalmun Gate, and Heunginjimun Gate as early, middle-late, and late gates, respectively.

In addition, the size of the bracket arm in the intercolumnar bracket complexes was determined as the axis concept and not the size of the bracket arm in the bracket complexes placed directly on the column heads. Then, in the case of the bracket arm in the purlin direction, the x-axis (bracket arm length), y-axis (bracket arm height), and z-axis (bracket arm width) were applied to the XYJ coordinate system. Conversely, since the XYJ coordinate system differs when placed in the beam direction, the x-axis (length), y-axis (height), and z-axis (width) were defined for the following reason. First, the x-axis and z-axis change the axial direction from the front to the side facades, as viewed from the front facade, and then the y-axis (height of the bracket arm) is set. In addition, in a plan view of a wooden structure, the extension of the building size in the longitudinal direction of the bracket arm can be easily carried out when viewed from the front of the wooden-framed building. Second, it is not easy to expand in the lateral direction of the building, which utilizes the width of the bracket arm when viewed from the front of the building.

Hence, the lack of detailed drawings and historical records from the foundation time requires the significant assumption that should grasp the construction principle in consideration of historic states through reinforcement and replacement over time, resting on the current size of timber-framed components, in particular the length and width of bracket arms. To show that the same construction principle as the modular method for determining the scale of buildings mentioned in Song YZFS and Qing GCZF in ancient China was used through multi-storied official-style buildings in the late Joseon Dynasty, focusing on the current historical states of buildings, this study proves that the buildings are constructed using the size of the bracket arms as a standard module by identifying the size and dimensions of the wooden members and analyzing them accordingly.

2. Overview of Sungnyemun, Paldalmun, and Heunginjimun Gates

2.1. Historical Context of Sungnyemun, Paldalmun, and Heunginjimun

Sungnyemun and Heunginjimun, in the Seoul Hanyangdoesong Wall, and Paldalmun, in the Suwon Hwaseong Fortress, have similarities as they are gate tower buildings in a capital city, use a two-storied hipped roof, and have the same number of bracket sets on the front and side facades. Sungnyemun Gate is the oldest surviving gate tower in the walled city, while Heunginjimun is the largest gate tower. Sungnyemun Gate was built in the 15th century (1479), Hwaseong Paldalmun Gate in the 18th century (1794), and Heunginjimun Gate in the 19th century (1869), representing the early, middle, and late Joseon eras, respectively [15,16,17].

Sungnyemun Gate is a two-storied hipped roof with intercolumnar bracket complexes and a ground plan with five bays in the front and two bays on the side facades. The southern main gate of Hanyangdoesong Wall is commonly called Namdaemun. There were several repair and reconstruction works, and the recent dismantling work in 1961–1963 revealed various historical facts [18] (p. 82). On 10 February 2008, in the fire at Sungnyemun Gate, the roof of the second floor of the Gate Tower collapsed and part of the roof on the second floor was destroyed. It was severely damaged. After five years and two months, the restoration work was completed and opened to the public on May 4, 2013. The construction of Sungnyemun Gate began in the 5th year of King Taejo (1396) and was completed in the 7th year of King Taejo (1398). The building was rebuilt in the 30th year of King Sejong (1448) after the decision of reconstruction (1443), and had repair work during the 10th of King Seongjong (1479). During the dismantling and repair from 1961 to 1963, it was found that there was major construction in the 10th year of King Seongjong (1479) [17,18]. After an arsonist set fire to the Sungnyemun Gate in 2008, the Gate was restored in 2013.

Paldalmun Gate in Hwaseong Fortress is a two-storied building with five bays on the front and two bays on the side facades, with intercolumnar bracket complexes and a hipped roof. It is a large-scale building built in 1794 and represents the half-timbered buildings of the Hwaseong Fortress. Hwaseong has more than 20 buildings, but except for Paldalmun Gate (1794), Hwaseomun Gate (1796), Dongjangdae Command Post (1795), Banghwasuryujeong Pavilion (1794), and Seobukgongsimdon Observation Tower (1796), the rest of the buildings in Hwaseong were constructed in 1975 to 1979, after the 20th century. Paldalmun had a partial restoration of Ongseong (semi-circular bastion wall) in 1975 after the demolition of its right and left walls in 1923. Paldalmun Gate resembles Sungnyemun in length and overall proportions of the front and side facades. It is believed that Sungnyemun was used as a model for constructing the building [15] (pp. 63–65). However, there is a time difference of about 300 years between the Paldalmun Gate and the Sungnyemun Gate, and the Paldalmun Gate is about 1.5 m higher than the Sungnyemun Gate [19]. Paldalmun Gate was better developed in the composition of the brackets, and the height of the beams connected to the brackets was relatively higher than that of the Sungnyemun Gate. In addition, the inner wooden frame structure used high pass-through columns near the lower corner columns. The only difference is that Sungnyemun is placed using corner columns on the lower and the upper floors and Paldalmun is placed using off-corner tie beams on the lower floor without continuous tall columns at each corner (

Table 1. Chronology of target buildings during the Joseon Dynasty (Park, Kim, and Han 2018).

Era (Year)	Period	Year	Key Content	Remarks
Joseon Dynasty (1392–1910)	Pre-modern period	1392	Founding of Joseon
		1394	The capital was moved from Gaeseong, the former capital of Goryeo (918–1392), to Hanyang, the new capital. Construction of Gyeongbokgung Palace
		1398	Completion of Sungnyemun Gate	In 1443, Joseon decided to rebuild Sungnyemun Gate
		1479	Reconstruction of Sungnyemun Gate
		1592	Imjin War (Japanese invasions of Korea (1592–1598))	Gyeongbokgung Palace was burnt down
		1794	Construction of Suwon Hwaseong Construction of Paldalmun Gate
		1865–1872	Restoration of Gyeongbokgung Palace
		1868	Repair of Sungnyemun Gate	(Park, Kim, and Han 2018, p. 196)
		1869	Reconstruction of Heunginjimun Gate
	Modern period	1888	Construction of Daebul Hotel	The first modern building
		1907	Right and left walls of Heunginjimun Gate were damaged
Japanese occupation (1910–1945)		1908	Right and left walls around Sungnyemun were damaged
		1923	Right and left walls around Paldalmun were damaged
Korea (1945-present)	Contemporary period	1961	Dismatlement and Repair of Sungnyemun Gate
		1975	Restoration of Ongseong (semi-circular bastion wall) at Paldalmun Gate
		2008–2013	Sungnyemun Gate was restored after being destroyed by fire	Description from the Precise Survey Report (Junggu Office Seoul 2006)

and

Table 2. Overview of Sungnyemun, Paldalmun, and Heunginjimun Gates (Park, Kim, and Han 2018).

	Sungnyemun	Paldalmun	Heunginjimun
Period	1479	1794	1869
Roof shape	Two-storied, hip roof	Two-storied, hip roof	Two-storied, hip roof
Scale of a ground plan	Upper floor: 5 × 2 Lower floor: 5 × 2	Upper floor: 5 × 2 Lower floor: 5 × 2	Upper floor: 5 × 2 Lower floor: 5 × 2
Structural characteristics	Use of pass-through high columns at each corner Use of inner high columns	Non-use of pass-through high columns at each corner Use of off-corner beams Use of inner high columns	Use of pass-through high columns at each corner Use of inner high columns
Ground plan (location of columns)

).

Heunginjimun Gate in Seoul was first built in the 5th year of King Taejo of the Joseon Dynasty (1396) and then was rebuilt in the 1st year of King Danjong (1453). The building was rebuilt to its present form in the 6th year of King Gojong (1869). The right and left walls of Heunginjimun Gate were damaged in 1907. It is a two-storied building with a hipped roof with intercolumnar bracket complexes, with five bays (72.5 cheok) on the front and two bays (24.1 cheok) on the side facades. The original terrain was low when first built, and there was the Yigansumun Water Gate (a double-arched flood gate) through which the city’s water to the south collected and flowed downward. Judging from the contents of the Sangyangmun (a written record informing about the date of construction), Heunginjimun was reconstructed during the repair work in the 6th year of King Gojong by increasing the height of the existing stone foundation by 8 cheok. In this sense, the old gate tower was demolished and rebuilt [16] (p. 63). Structurally, it shows an internal lattice structure like the Sungnyemun using a continuous high column.

The application of pass-through high columns at each corner, subsuming inner pass-through high columns, is confirmed only in official buildings, which are actively used in palace and fortress buildings in the Joseon era: Sungnyemun Gate (1398); Heunginjimun Gate (1869); Gwanghwamun Gate (1865) at Gyeongbok Palace; Injeongjeon Hall (1803) at Changdeokgung Palace; Gunjeongjeon Hall (1867) at Gyeongbok Palace; Junghwajeon Hall (1906) at Deoksugung Palace (Park, Kim, and Han 2018, p.198).

2.2. Bracket Arms at Sungnyemun, Paldalmun, and Heunginjimun Gates

The timber frame construction is generally divided into three parts, namely, columns, brackets, and roof. Bracket sets form bracket complexes by placing the cap block at the bottom, and purlin-directed bracket arms and beam-directed bracket arms cross each other. Generally, since the Joseon Dynasty, beam-direction bracket complexes consist of transversely downward projecting bracket arms (salmi in Korea, outwardly projecting bracket arm to beam direction), while purlin-directed bracket complexes consist of upwardly projecting arms. Korean wooden buildings are divided into two styles, the Jusimpo style, with bracket complexes placed directly on the column heads before the Joseon Dynasty, and the Dapo style, with bracket complexes between columns after the Joseon Dynasty. One has no bracket sets between columns, while the other has bracket sets between columns. Sungnyemun, Paldalmun, and Heunginjimun belong to Dapo-style buildings with intercolumnar bracket complexes.

2.2.1. Characteristics of Sungnyemun Bracket Arms

The upper story of the Sungnyemun Gate has ten upper brackets, four corner brackets, and twenty four intercolumnar bracket sets on the columns, and the upper story has ten upper brackets on the pillars, four corner brackets, and thirty two intercolumnar bracket sets. The bracket sets of the upper floor consist of inner two-layer projecting arms in the transverse direction and outer three-layer projecting arms in the transverse direction, while the lower floor consists of inner and outer two-layer projecting arms. The brackets on the upper floor are relatively larger than those on the lower floor. The size of bracket arms on the lower floor varies within a bracket set, as there are large bracket arms and small bracket arms, but in the middle and side bays. The sizes (length of purlin direction) of large and small bracket arms in the middle and side bays differ. The central bay is relatively wider than the side bays, and the distance between the centers of the sets of brackets is also greater than that of the side bays.

The size of the bracket arm is not the same depending on each bay length, but it is adjusted to an appropriate size. There is a continual change in the x-axis (purlin direction). The bracket arm has a regular size on the z-axis and y-axis, and a variable is set on the x-axis. The width (z-axis) of the intercolumnar bracket complexes and the column top bracket complexes is 105 mm front and back. The height (y-axis) is 220 mm and includes 150 mm front and back, excluding the large bracket arm. The size of the bracket arm includes the middle and side bays and the top and bottom floors, and the width (z-axis) has a fixed size value. Similar values are found in the projecting bracket arm of the intercolumnar bracket complexes, except for the projecting bracket arm at the column heads. The width is 105 mm front and rear, and the height is 225 mm. It follows that the width of Sungnyemun’s bracket arm has a fixed value of 105 mm. This does not apply to the highest five-layer bracket set. This value is independent of whether the bracket arm is placed in the purlin or beam direction. A uniform bracket arm width (105 mm) and height (150 mm) have a ratio of 1/3 and 1/2 in Sungnyemun’s dimension units (308 mm) [17] (p. 204). For the others, the value resulting from the change in the length of the bracket arm (x-axis) in the purlin direction is expanded to a fixed value of the bracket arm width (z-axis) and the length of the bracket arm (y-axis). The width of the bracket (z-axis) changes little compared to the height of the y-axis.

This is consistent with the fact that wooden architecture is easy to expand in the direction of the torii, while it is very difficult to expand the building on the side. The bracket arms are considered expanding by adjusting the value of the x-axis by adjusting the values of the z-axis and the y-axis, which are uniform. Then the rules for the scalability of the x-axis are as follows. The bracket arms of the Sungnyemun bracket complexes have large and small arms. The x-axis length of the small bracket arm is about 775 mm, and that of the large bracket arm is about 1165 mm. The length of the small bracket arms is about 1.5 times that of the large bracket arms. The distance between the intercolumnar bracket sets is more important than the length of the bracket arm. Except for the front and rear central bays, the spacings between the bracket sets are uniformly set at 185–1295 mm front and back. In the case of the center bay, the spacings are 1400–1405 mm wide. This difference shows that the length of the x-axis bracket arms is not uniformly distributed. The distance between the bracket complexes of the center and side bays is approximately 110 mm. The measurement has a size value of 105 mm, the width of the bracket arm. In other words, the width of the bracket arm (z-axis) is an important standard unit for the length and height of wooden components of a building and the distance between the bracket complexes (Figure 5).

2.2.2. Paldalmun’s Bracket Arm Analysis

The number of bracket sets for Paldalmun is almost the same as for Sungnyemun. There are four intercolumnar bracket sets in the middle bay on the lower level. There are two in the remaining outermost bays, and on the upper level, there are four intercolumnar bracket sets in the central bay, two in the side bays, and one in the outermost bays. The bracket arm width (z-axis) on the upper floor and lower floor of Paldalmun Gate have a numerical value of about 105 mm and a height (y-axis) of 180 mm. This height is 0.34 cheok when calculated as 308.5 mm, the measuring units of Paldalmun. The height is calculated to be 0.58 cheok. The number of bracket arms in the middle bay in the purlin direction is slightly higher than that in the side bay. It is assumed that the standard shape was chosen based on the specific shape of the bracket arms. This means that there are no constant changes in the size of any bracket arms. The changes in the length, height, and width of the bracket arms are small. This is more similar to the YZFS moshu module system based on the width and thickness of bracket arms. The length of the central bay is slightly larger than the other bays. The length of the central bay is 6848 mm (22.2 cheok), and the length of the other side bays is 3705 mm (12 cheok). The spacings between each bracket set in the central bay average 1390 mm (4.5 cheok), and the spacings to the other side bays average 1052 mm (3.42 cheok). In addition, the distance between the bracket sets in the central bay is 1390 mm, and in the other bays, such as the side bays, is 1230 mm.

Assuming that the moshu module system is configured based on the size of the bracket arm in the purlin direction on the XYZ coordinate system, it is configured to some extent except for the size of the central bay. If this is the case, the width of the center bay is greater than that of the other bays, which is common in Joseon Dynasty wood frame construction. As for the gate tower, since the gate is formed as an arch below the timber-framed tower, columns should not be placed on the upper part of the arch gate. Thus, it is possible that the middle bay of the gate tower part was set wider to retain a proper width. The bracket sets in the purlin direction consist of projecting and non-projecting bracket arms placed on the column heads, and the length is properly adjusted according to the size of protruded arms and their position. In general, the closer the bracket arms are to the column heads, the smaller the bracket arms of the second, third, and fourth layers of bracket sets, all of which have the same length. The length of the bracket arms is also important, but the length based on the distance between one bracket arm and the other, the arms protruding across, is uniform. Thus, the length of the bracket arm in Paldalmun is 309 mm, which is almost equal to the units of measurement. The length of the side facade can be adjusted proportionally to the width of the bracket arm (z-axis) in the multi-layer bracket sets (Figure 6).

2.2.3. Heunginjimun’s Bracket Arm Analysis

The bracket complexes of Heunginjimun have the same format as Sungnyemun and Paldalmun. The lower floor consists of four sets of brackets in the central bay, two sets of brackets in each side bay, and two intercolumnar sets of brackets on both side facades. In the two outermost bays on the upper floor, there is only one set of brackets for a clerestory, and the rest of the substructure is the same as that on the lower floor.

The width of the bracket arm (z-axis) is 104–107 mm (average 105 mm), and the height is 235 mm. The length (x-axis) of the small bracket arm and the large bracket arm is 735 mm and 1045 mm, respectively, and the upper and lower layers of brackets have the same composition. However, the size is slightly larger in the middle bay. This is because the length of the middle bay is relatively long, so they do not employ the same bracket arm, and the distance between the bracket complexes is not the same accordingly. The unit of measurement used on the ground plan of Heunginjimun is 303.2 mm [16] (p. 306). If this is expressed as the proportional size of the bracket arm, the width (z-axis) of the bracket arm is about 1/3 (0.34), and the height (y-axis) is 0.77 times. The spacing of each bracket set is very similar to that of Sungnyemun and Heunginjimun Gates. In the central bay, the spacings between each bracket set are greater than elsewhere. The spacings are about 1420–1435 mm (average 1425 mm) for the central bay and 1210 mm for the side bays. The length of the bracket arm (in the x-axis direction) is 735 mm for the small bracket arm and 1045 mm for the large in most bays except for the central bay section. The difference in length between the small bracket arm and the large bracket arm is almost equal to that of Heunginjimun’s measuring unit, which is about 304 mm. The length of the outwardly transverse projecting arms of bracket sets is longer than that of the Sungnyemun and Paldalmun. The protruding arms are 365 mm long and about 1.2 times wider than a unit of measurement (Figure 7).

2.3. Comparative Analysis of the Bracket Arms of Sungnyemun, Paldalmun, and Heunginjimun Gates

The configuration of the size of the bracket arms used at Sungnyemun, Paldalmun, and Heunginjimun is as follows (Table 3). A wooden frame structure built in the Joseon Dynasty has a dimension unit of about 300 mm. It is slightly larger than that of Goryeo-era buildings. The length of standard units of measurement for construction is also increasing over time in the Korean peninsula [20] (pp. 8–10). A closer look at the length (x-axis), height (y-axis), and width (z-axis) of the bracket arms reveals a very consistent trend of about 105 mm. Although the construction period between Sungnyemun and Heunginjimun shows a difference of about 400 years, the width of the bracket arm (z-axis) is regular. A reason why the width of the bracket arm has a fixed size is that the building was constructed by fixing the width of the bracket arm itself and adjusting the size of the other brackets on the x and y axes. The height of the bracket arm (y-axis) increases gradually over time. This is related to the increase in the overall height of the building from the early and middle to the late Joseon Dynasty. In Chinese wooden architecture, when YZFS and GCZF modular methods were used on the same floor plan, the height of the ridge purlin was in fact, the highest for buildings constructed with GCZF. This shows that the height of buildings generally increases as the modern period progresses [8] (p. 225).

In addition, the height increases with the number of projecting stratum of bracket arms. The number of inward projecting layered bracket arms is one higher at Paldalmun and Heunginjimun than at Sungnyemun Gates. From them, the height of the beam is set slightly higher than the top floor. The length of the bracket arm is slightly different from that of the large and the small bracket arms. The length of the small bracket arms of Sungnyemun, an early Joseon building, is slightly longer than that of Paldalmun and Heunginjimun. The size of the central bay and the side bays of the three buildings is almost the same as in the ground floor plan. The spacings between the bracket sets are narrower at Sungnyemun. Generally, the small bracket arms are at the bottom, and the large bracket arms are at the top. The greater the number of protruding layer arms, the shorter the length of the bracket arms. The distance of each protruding multilayer arm was set to 1.2 times the measurement unit, or the width of the bracket arm was set to the same length as the measuring unit. This shows that this bracket composition is separate from the length and height of the bracket arms, and rather the bracket arm width (z-axis), as seen from the front and rear facades, is related to the scalability decision in the cross-section. The transversely projecting length of the bracket sets placed through the cantilever was within 1.2 cheok. The projecting length of the brackets of buildings after the Joseon Dynasty is smaller than that of the Goryeo Dynasty.

Table 3. Comparison of Sungnyemun, Paldalmun, and Heunginjimun (unit: mm).

	Sungnyemun (1479)	Paldalmun (1794)	Heunginjimun (1869)
Measuring unit	308	308.5	303.2
Bracket arm width	105	105	105
Bracket arm height	150	180	235
Bracket arm length (large protruding bracket arm)	1150	1006	1045
Bracket arm length (small protruding bracket arm)	770 (Average)	733	735
Projecting length of bracket arms	370 (1.2 cheok)	308 (1 cheok)	365 (1.2 cheok)
The number of protruding bracket arms	Lower floor: interior-exterior two-layer protruding arms Upper floor: interior three-layer and exterior two-layer protruding arms	Lower floor: interior three-layer and exterior two-layer protruding arms Upper floor: interior-exterior three-layer protruding arms	Lower floor: interior three-layer and exterior two-layer protruding arms Upper floor: interior-exterior three-layer protruding arms
Spacing between the one bracket arm and the other (x-axis)	Not evenly distributed	Not evenly distributed	Not evenly distributed

Table 3. Comparison of Sungnyemun, Paldalmun, and Heunginjimun (unit: mm).

	Sungnyemun (1479)	Paldalmun (1794)	Heunginjimun (1869)
Measuring unit	308	308.5	303.2
Bracket arm width	105	105	105
Bracket arm height	150	180	235
Bracket arm length (large protruding bracket arm)	1150	1006	1045
Bracket arm length (small protruding bracket arm)	770 (Average)	733	735
Projecting length of bracket arms	370 (1.2 cheok)	308 (1 cheok)	365 (1.2 cheok)
The number of protruding bracket arms	Lower floor: interior-exterior two-layer protruding arms Upper floor: interior three-layer and exterior two-layer protruding arms	Lower floor: interior three-layer and exterior two-layer protruding arms Upper floor: interior-exterior three-layer protruding arms	Lower floor: interior three-layer and exterior two-layer protruding arms Upper floor: interior-exterior three-layer protruding arms
Spacing between the one bracket arm and the other (x-axis)	Not evenly distributed	Not evenly distributed	Not evenly distributed

In consideration of the length (x-axis), height (y-axis), and width (z-axis) of the bracket arms, a fixed quantity is the value corresponding to the width of the protruding bracket arm. If the length (x-axis) of the protruding bracket arm has a regular or proportional value, the parts could be the reference value. The three buildings show fixed size values in the width of the bracket arm (z-axis). If the x-axis is the reference value, the size of the bracket arm should have a regular value, or the distance values for the intervals of each bracket set should be evenly distributed. For all three substructures, the spacings between the bracket arms are larger from the central bay than from the side bays. Since they are not uniformly distributed, these cannot be reference points with a coherent value. Moreover, for the y-axis, as the number of protruding arms increases, the height of the bracket arm also gradually increases simultaneously. It would, therefore, be difficult to establish a reference value with the height (y-axis).

Consequently, the width (z-axis) of the bracket arm indicates the absolute value concerning its size (intercolumnar bracket set), which is considered a common feature in buildings representing the two-storied gate tower in the Joseon Dynasty [11] (p. 616).

3. Cross-Section Analysis Using Wooden Bracket Arm Members and XYZ Coordinates

The relationship between the length (x-axis), height (y-axis), and width (z-axis) of the bracket arm affects the length of the front and side facades of the building. This indicates the relationship between the extent of the bracket arms, the ratio of the bracket sets, and the length extent of the side facade. Moreover, since they are related to the overall size composition of the buildings on the floor plan, it is possible to guess the size of the wooden parts and positions with the reference points.

3.1. Cross-Sectional Relationship with the Bracket Arm of Sungnyemun Gate

Sungnyemun has different lengths for each set of brackets located on the side facades of the lower and upper floors, and the number of sets of brackets is also different (Figure 8). The lower floor is longer. The number of bracket sets complexes is one on the upper floor and two on the lower floor. The brackets in each bay of the upper side facade are relatively long. They do not appear proportional to the standard form, given the configuration of the bracket sets and arms of the front and side facades.

The bracket arm width (z-axis) is 105 mm. The length of the interior–exterior two-layer protruding arm of the column top bracket sets from the front facade of the building is 1470 mm, which is about 14 times different [16] (p. 242), while the length of each bay of the side facade (about 3852 mm) or the total length of the side facade (about 7705 mm) is 36.685 times and 73.3809 times, respectively. In other words, since the integer relationship is not fixed, the cross-sectional length of bracket complexes is relatively independent of the length of the side facade. Conversely, there is a correlation between the cross-sectional length of bracket complexes and the z-axis length of bracket arms. However, the x-axis length of the bracket arm is converted to the z-axis when it is attached to the side facade.

The length of the x-axis of the small bracket arm is 770 mm, and the length of the bay of the side facade is 3852 mm. This is a five-to-one integer ratio with a correlation. On the other hand, if the entire side façade is considered, ten times the x-axis length of the small bracket arm is the total length of the side facade. This correlates proportionally with the length of the bracket arms and the total side facade. For the front length, the proportional relationship between the x-axis length of the bracket arm and the front bay or total length cannot be expressed as an integer relationship. This is because the wooden buildings are relatively free in the x-axis extension. In Sungnyemun, the central bay is relatively long, and the number of bracket sets is also different in the central and side bays. This results in the spacing of the bracket sets to appear uneven. However, in a lateral arrangement, the length of the x-axis of the bracket arms shows a very constant correlation with the length of a bay of the side facade or the total length of the side facade. It can be assumed that the length of the bracket arm is proportional to the lateral length. When the bracket arm is attached to the side facade of the building, it is important to change from the direction of the x-axis, which is the length of the bracket arm, to the direction of the z-axis. That is, the total extension length of the side facade is strongly related to the length of the x-axis of the bracket arm. The cross-sectional length of the bracket sets has a similar relationship since they are in integer ratios.

The height of the bracket arm is 150 mm, and it is very uniformly designed. Since the height of the building is divided by an integer ratio of 105 mm (z-axis), which is equal to the width of the bracket arm, the width of the bracket arm is closely related to the height. It is approximately 92 times the width of the z-axis of the bracket arm and 47 times the height of the lower floor, or 4936 mm. The z-axis width of the bracket arm becomes a constant reference length for the entire building since a fixed value appears as an integer ratio in the height of the upper and lower floors. When the coordinate points of the bracket arm are set to the XYZ coordinates of Sungnyemun, the width of the bracket arm (z-axis) becomes the reference point for the size of the building.

Putting the side width of the building on the side of the width of the bracket arm (z-axis) and converting the x-axis length of the bracket arm to the z-axis width, the proportional relationship of the whole building is determined according to the z-axis direction of the bracket arm. Although the unit of measurement of Sungnyemun is 308 mm, the z-axis width of the bracket arm, 105 mm, can be considered the smallest unit that determines the size of the building. This shows that the length of the bracket arm determined in the z-axis direction plays an important role as a mediator in determining the total size of all parts of the building (

Table 4. Proportional relationship between the side facade and building height, and bracket arm (Sungnyemun).

	Side Façade Length	Building Height	Remarks
Bracket arm length (x-axis)	Related (based on the small bracket arm)	Non-related
Bracket arm width (z-axis)	Non-related	Related (based on the small bracket arm)	However, even if the direction of the axis is changed from the side facade of the building, the width of the bracket arm (in the z-axis direction) becomes the criterion for determining the size of all members.
Bracket arm height (y-axis)	Non-related	Non-related

).

3.2. Cross-Sectional Relationship with the Bracket Arm of Paldalmun Gate

The distribution of the bracket arms of Paldalmun almost resembles that of Sungnyemun (Figure 9). However, as with Sungnyemun, the arms with one clasp are on the upper side, and the two bracket sets are on the lower side. The difference between Paldalmun and the upper floors of Sungnyemun is that the length of the bracket arms is not unusually long. This was constructed by connecting cut-off columns in the upper layer using off-corner tie beams (gwijabibo) instead of high pass-through columns. The column structure connecting the first and second floors has the off-corner tie beams used in diagonal directions at 45 degree angles above bracket sets on the four corners. The length of each bay of the upper side facade is uniformly configured. The bracket arm width in the cross-section (z-axis) is 105 mm. The cross-sectional length of one bracket set is 1701 mm, calculated accurately by 16.2 times, and appears close to the integer ratio. The length of the side facade of the building is 3712 mm from the distance of one lower floor. The length of the upper story is 2773 mm, which is not close to the integer ratio. Thus, there appears to be a small proportional relationship between the lateral distance of the bracket arm and the lateral length of the building.

The length (x-axis) of the bracket arm on the front facade of the Paldalmun is 733 mm for the small bracket arm and 1006 mm for the large one. The length of the bracket arm is 1/5 or 5 times the ratio of each bay on the side facade. This ratio is almost similar to that of Sungnyemun when comparing the length of the bracket arm. In other words, the lateral distance of the building is shifted to the z-axis when the length of the bracket arm is placed at the side, and it is applied when calculating the lateral distance. Considering the total height of the building, the length from the upper part of the foundation to the lower part of the ridge purlin is 11,015 mm (260 mm (main column height under the floorboards) + 4708 mm (upper part of the second floor) + 6047 mm (lower part of the ridge purlin) [15] (p. 421). In these quantitative data, there is no integer relationship between the height of the bracket arm and the width of the side facade. It is related to the length of the small bracket arm and appears in the form of an integer ratio at a ratio of about 15 times. Compared to the small bracket arm, the shape of the integer ratio is also small, but if the length of the large bracket arm of 1006 mm is used, the ratio is about 11 times (

Table 5. Proportional relationship between the side facade and building height, and bracket arm (Paldalmun).

	Side Façade Length	Building Height	Remarks
Bracket arm length (x-axis)	Related (based on the small bracket arm)	Related (based on the small bracket arm)
Bracket arm width (z-axis)	Non-related	Non-related	Related to the cross-sectional length of the bracket sets.
Bracket arm height (y-axis)	Non-related	Non-related

).

3.3. The Relationship between Bracket Arms and the Cross-Section of Heunginjimun Gate

The number of bracket sets in Heunginjimun Gate (Dongdaemun) is similar to that in Paldalmun (Figure 10). Unlike Sungnyemun, the upper bracket arms resemble the bracket complexes on the front and side facades of Paldalmun. However, Heunginjimun engages the pass-through tall columns on the upper floor, as in Sungnyemun. The length of one bay seen from the side facades of the upper floor appears stable.

The width of the small Heunginjimun bracket arms is 105 mm, the height is 235 mm, and the length is 735 mm. Overall, the width of the bracket arms is equivalent to two buildings. The length is smaller than Sungnyemun and similar to Paldalmun. The height and size of the other two buildings are relatively large.

The one-bay length (lower floor) of the side facade of Dongdaemun is 3680 mm. It is very well divided in an integer ratio of 35 times the width of the bracket arms of 105 mm. From the side facade, it has a length equal to 70 times the width of the bracket arm. In addition, five times the proportionality of length is established with the length of the bracket arm of 735 mm. In this way, the integer ratio between the width of the bracket arm and the length of the side facade is established.

The total height of the building is 11,447 mm from the foundation to the lower edge of the ridge purlin, with the first and second floors added together. Using quantitative numbers for the bracket arms, the width is 105 mm divided by 109, which is close to an integer ratio. However, the length of the bracket arms and the height of the bracket arms are not divided by the integer ratio of the width. The ratio in height with the small bracket arm and the side facade of the building located in the intercolumnar bracket complexes of Heunginjimun is similar to that of Paldalmun (

Table 6. Proportional relationship between the side facade and building height, and bracket arm (Heunginjimun).

	Side Façade Length	Building Height	Remarks
Bracket arm length (x-axis)	Related (based on the small bracket arm)	Non-related
Bracket arm width (z-axis)	Related (based on the small bracket arm)	Related (based on the small bracket arm)
Bracket arm height (y-axis)	Non-related	Non-related

).

3.4. Discussion

By comparing the width, length, and height of the small bracket arms located on the top column bracket sets of the above three buildings and the top column bracket sets of each building, the following conclusions can be drawn.

First, the length of the bracket arms is divided by the integer ratio of the length of the side facade for all three buildings. In this context, the bracket arms attached to the side facade change the direction of the XYZ coordinate system from the x-axis to the z-axis. There is a very close relationship between the length of the side facade and the bracket arms. This shows that the length of the bracket arms is an important measure for determining the total length of the side facade of a building. Second, the width of the bracket arms also plays an important role in determining the overall height of the building. The width is the smallest part of the bracket arms in the XYZ coordinate system, and the building height at Sungnyemun and Heunginjimun Gate is divided by an integer ratio. At Heunginjimun, the width of the bracket arms is divided by the integer ratio of the length of the side facade and the building height. Third, the height of the bracket arms is not divided by the integer ratio of the length of the side facade and the height of the three buildings. The height of the bracket arms themselves has little to do with the height of the buildings.

Considering Paldalmun Gate as an example, Figure 11 and Figure 12 show that the length of the bracket arms on the front facade is 733 mm for the small bracket arms and 1006 mm for the large ones, respectively. The bracket arm width in the cross-section is approximately 105 mm. The bracket arm length is one-fifth the ratio of each bay on the side facade. In other words, the length of one bay at the side façade is five times the bracket arm length. Conversely, the dimension decision of the length of the side facade of the building is 3712 mm (a total of 7424 mm) from a bay distance of one upper floor. The whole height of the building, from the upper part of the foundation to the lower part of the ridge purlin, is 11,015 mm. The dimension of the entire building is decided on an integer ratio of about 15 times the length of the small bracket arms.

The dimensions of most wooden buildings change over time; thus, the current dimensions are not those of the original plan. The original dimensions would have been based on the bracket arm dimensions, and the building would have been constructed based on the measurement unit; therefore, these current variations can be used to estimate the planned dimensions of the building at the time of construction.

First, in the case of Paldalmun Gate, in comparison to the initial module (red) and the current transformed (black) dimensions between Figure 11, Figure 12 and Figure 13, the planned length of the bracket arms at the front facade is 740 mm for the small bracket arms and 1010 mm for the large bracket arms in base modules, with a bracket arm width of 105 mm in cross-section. The small bracket arm length (740 mm) is 1/5 times the length ratio of each bay on the side façade (3700 mm). Instead, the dimension decision for the side facade of the building is 3700 mm (7400 mm total). In this case, the measurement unit (1 cheok) can be calculated as 310 mm long.

Similarly, regarding Sungnyemun Gate, the planned length of the bracket arms at the front facade is 750 mm for the small bracket arms and 1150 mm for the large bracket arms in base modules, with a bracket arm width of 105 mm in cross-section. The length of each bay on the side façade (3850 mm) is five times the ratio of the small bracket arm length (770 mm). The measurement unit (1 cheok) can be calculated as 310 mm (Figure 14).

As for Heunginjimun Gate, the planned length of the bracket arms at the front facade is 760 mm for the small bracket arms and 1050 mm for the large bracket arms in base modules, with a bracket arm width of 105 mm in cross-section. The length of the side façade (2630 mm) is 3.5 times the ratio of the small bracket arm length (760 mm). The measurement unit (1 cheok) can be calculated as 310 mm (Figure 15).

These three buildings have in common that the planning module (1 cheok) can be assumed to be 310 mm. The length of the side façade is five times that of the small bracket arms, except for Heunginjimun Gate. As mentioned above, the bracket arms are all 105 mm in width. Another similarity is that the length and width of large bracket arms and small bracket arms are important criteria for determining the whole length and height of front and side facades.

In the Joseon era, similar to the GCZF, the width of the cap block is proportional to that of the bracket arms in the purlin-center direction. In official buildings in Korea and China, the size (length and width) of the bracket arm is closely related to the overall building scale (the length of the front and side facades and the building height). This is similar to the specifications in the building manuals of QCZF in the Joseon era. The width of the cap block is proportional to that of the bracket arms in the purlin-center direction. The overall scale of the building and the common factors that shape the proportional system are likely to provide fundamental data for the study of the construction principles and systems of East Asian wooden architecture.

4. Conclusions

Given the front and side façade ratios and the height of the buildings, Sungnyemun (Namdaemun), Paldalmun, and Heunginjimun (Dongdaemun), the two-storied gate tower buildings of the Joseon Dynasty conserve the size of the intercolumnar bracket arms through the application of modular coordination, which points to commonalities between official buildings in Korea and China. The results of this study are as follows.

First, the comparative analysis of the bracket arms of three buildings revealed that they are subdivided units rather than a unit of measure. Instead of defining the size of the building components based on the unit of measurement, the bracket arms are the minimum basic unit of the building and play a role in the production of the modular system of the building.

Second, the length and width of the bracket arms become the basic unit that determines the length of the front and side facades. This became an important factor in influencing the length decision of the front and side facades since the size of the bracket arms is the same on the front and side. It was found that the width and length of the bracket arms of the three buildings are in a regular integer relationship. Interestingly, this is a numerical value, which is independent of the measurement units (yeongjo-cheok).

Third, the width of the bracket arms of the three buildings has the same value (105 mm), although Sungnyemun and Heunginjimun have been constructed at a distance of almost 400 years. Thus, the width of the bracket arms represents a basic module, which is not divisible by an integer ratio with measurement units. The width of the bracket arms is different from that of the measuring unit.

Fourth, Heunginjimun Gate is the newest of the three buildings, as it was built after the mid-19th century. The width and length of bracket arms at Heunginjimun were more accurately applied in determining the total length of the side facade and the building height than those at Sungnyemun and Paldalmun. The basic unit of measurement based on the bracket arm was adjusted more closely to the integer ratio when the length and width of the bracket arms were superimposed, which is a new attempt based on the combination of the YZFS and GCZF construction methods to determine the dimension decision.

Fifth, the height of the bracket arms of the three buildings does not have an integer relationship with the lateral facade length and the building height. The numerical values of the direction of the y-axis of the bracket arms are unsuitable as standard unit values.

The results of this study challenge the existing theory of Korean architectural history, according to which Korean wooden architecture does not have a proportional system. The construction principle of wooden-framed architecture in Korea and China, which uses bracket arm length and width as a standard modular method, has been continuously used to construct wooden-framed buildings. The scale of buildings can be set through the ground plan, the arrangement of columns, and the spacing and size (width and length) of each bracket set from the size of the bracket arm, a small unit member of a wooden building. Combining the standard wooden members with a proportional system analysis would be significant in the restoration of wooden buildings employing parametric tools and structuring the BIM system for heritage monuments.

This study is significant as it derives parametric measuring rules universally applied in East Asia, creating wooden structures that preserve the construction characteristics of the late Joseon era and providing basic data useful for the conservation and restoration works of built cultural heritage.