1:1
Canli Garden · two garden boundaries
11.79 m : 12.53 m
Lengths along the inner faces of the upper and right boundary walls; approximately 5.9% from 1:1. The entrance recess is excluded.
01
Begin with Sanfen Sunyi
Sanfen Sunyi · One-Third Subtraction and Addition
Divide the original length L into three equal parts. Removing one leaves 2L/3; adding one gives 4L/3. Click through the steps below to follow each operation.
Click through the steps to follow the change in string length.
Guqin schematic and interactive diagrams: Yanzhi Yang, 2026. Seven strings and thirteen hui follow the instrument’s form; a partial vibrating string length illustrates the arithmetic.
For an ideal string of constant linear density and tension, frequency varies inversely with effective length. This length relationship provides a mathematical comparison with garden proportions.
Two Northern Song systems of measure
Ruan Yi and Hu Yuan’s Huangyou xin yue tu ji records the musical reforms of the Huangyou era. In juan 1, “Huangyou lülü tu,” the authors cite the Shiji’s treatise on pitch pipes and give a length sequence beginning with subtraction:
81 → 54 → 72 → 48 → 64
Successive factors are 2/3, 4/3, 2/3 and 4/3. This Northern Song text records and applies an older method.
Music: generating pipe lengths
Huangyou xin yue tu ji treats pitch pipes alongside millet-based length, capacity and weight. Subtracting and adding thirds generates successive pipe lengths: 81 becomes 54, then 72.
54 : 81 = 2 : 3
Building: sizing members in cai-fen
Juan 4 of Yingzao fashi specifies eight grades of cai. Its breadth is divided into fifteen fen and its thickness into ten. The fen depends on the chosen cai grade.
10 : 15 = 2 : 3
Cai diagram and comparison: Yanzhi Yang, 2026. Facsimile sources are credited in the captions.
Both works connect measurement and calculation to making objects. In music, 2:3 relates successive pipe lengths; in building, it relates the thickness and breadth of a timber section.
02
A further operation: difference and sum
Let a and b be two lengths. Both 2:3 and 3:4 differ by one unit; their sums are five and seven. Thus 1:5 and 1:7 arise from difference and sum. This operation changes the quantities being compared.
Select a length ratio to compare its difference and sum.
(3 − 2) : (3 + 2) = 1 : 5
Sanfen Sunyi length operations use the prime factors two and three; the further difference-and-sum operation yields 1:5 and 1:7. At Qunxing Caotang, 1:5 compares two clear courtyard distances; here it compares a difference with a sum.
03
Tong Jun on proportion
Tong Jun’s 1930 manuscript “Proportion” states: “人眼见下列之比例，则觉适足” (the eye finds the following proportions satisfying), then lists 1:2, 2:3, 3:4 and 4:5. Below, these are drawn as short-side to long-side ratios: the shapes become progressively less elongated.
1:2
2:3
3:4
4:5
Interior length and width: plan shape
The table headed “室中之比例” (interior proportions) lists length and width. A brace beside 3:2 and 2:1 reads “巨室适用（大厅）” (suitable for large rooms / halls).
L denotes length and W width. Their relationship determines the plan’s orientation and degree of elongation.
Room height: the mean of length and width
Original: “高＝1/2（长＋宽）” — height equals half the sum of length and width.
H = (L + W) / 2
(5 + 3) / 2 = 4
For length five and width three, the height is four. The formula translates two plan dimensions into a vertical dimension, relating the room’s height to its overall volume.
The diagram uses proportional values in a common unit.
04
Ratios in gardens
One example for each ratio, distinguishing published dimensions, drawing measurements and mathematical comparisons.
1:2
Lingering Garden · Linquan Qishuo Hall
5.52 m : 11.15 m ≈ 1 : 2
Column-centre lines define one hall depth of about 5.52 m and a combined depth of 11.15 m. The approximate 1:2 ratio compares one hall with the two-hall whole.
2:3
Humble Administrator’s Garden · Yuanxiang Hall
1.53 m : 2.28 m
The west gallery span and adjacent side bay along one column line, measured between column centres. Approximately +0.6% from 2:3.
3:4
Master of the Nets Garden · Xiaoshan Conggui Xuan
4.86 m : 6.53 m
Opposite sides are averaged between four wall-centre intersections. The ratio compares hall depth and width, approximately −0.8% from 3:4.
4:5
Garden of Cultivation · main pond
22.20 m : 27.13 m
Perpendicular extents of the main pond’s water edge, including the southern inlet. Approximately +2.3% from 4:5.
1:5
Qunxing Caotang · either side of the pavilion
1.91 m : 9.53 m
Clear distances from opposite pavilion edges to the hall veranda and the rock-court wall. The pavilion itself is excluded. Calibrated against the original 5 m scale.
1:6
Yu Garden · rockwork and viewing distance
7.4 m : 45.8 m ≈ 1 : 6.19
From Sansui Hall, the main peak rises 7.4 m across a horizontal viewing distance of 45.8 m, giving an elevation angle of about 9°. The approximate 1:6 height-to-distance relationship creates a broad, distant view.
1:7
Three peaks · height and court-width comparison
h : 7h
Lingnan Tingyuan suggests a peak height of about one-sixth to one-eighth of court width. The analytical diagram uses 1:7 as an illustrative value within that range.
3:5
Canglang Pavilion Garden · Mingdao Hall
9.86 m : 16.86 m ≈ 3 : 5
Mingdao Hall is approximately 9.86 m deep and 16.86 m wide, giving a depth-to-width ratio close to 3:5. Its broad front faces the courtyard to the north.
5:7
Fengyuan North Street · the two lower peaks
200 : 280 = 5 : 7
The heights of the Baihu and Qinglong peaks. The number matches Tong Jun’s 5:7 entry, but the objects being compared are different.
7:10
Fengyuan North Street · Qinglong and main peak
280 : 400 = 7 : 10
Qinglong and Xuanwu peak heights, reduced directly from the original figure’s numbers.
5:10:7
Fengyuan North Street · three peak heights
200 : 400 : 280 = 5 : 10 : 7
The heights of Baihu, Xuanwu and Qinglong in that order: a comparison of the three peak heights.
05
Further ratios in the literature
Area, length and viewing-distance ratios concern different quantities. The following records retain the published numbers and their scope.
| Ratio | Example | Meaning and source |
|---|---|---|
| 1:2.5 = 2:5 | Yipu historical planting grid | Multiples of the Ruyuting grid reported in the article’s abstract. |
| 1:7.5 = 2:15 | Yipu historical planting grid | The abstract gives a grid ratio of 1:7.5, equivalent to the integer ratio 2:15. |
| 1:1 / 2:1 | Jichang / Huishan / Kuoran Dagong | Water area / facing rockwork footprint: 1:1, 2:1 and 2:1 respectively. |
| 3:1 / 2:1 | Jichang and Huishan / Kuoran Dagong | Main pond length / width: 3:1 for the first two gardens and 2:1 for the last. |
| 1:5 / ≈1:10 | Borrowed views in the three gardens | Target height difference / horizontal distance; each viewing relation is distinct. |
| 4.4:1 / 1:1 | Jichang looking towards Huishan / Xishan | Mountain breadth / viewing distance, following table 1. |
| 3.5:1 / 1:2 / 1.1:1 | Huishan Garden: rockwork / Wanshou / northern hill | Mountain breadth / viewing distance, in the stated order. |
| 1.3:1 | Kuoran Dagong looking towards the northern hill | Northern hill breadth / viewing distance, table 1. |
| 32:24:28:16 | Mountain Resort: present-condition model | Model area percentages: trees and shrubs, lawn, water and buildings. |
| 53:22:15:10 | Rehe imperial-palace model | Model area percentages for trees and shrubs, lawn, water and buildings. |
| 3:5:8 / 1:2:3 / 2:3:5 | Candidate planting ratios | Proposed planting ratios in the literature, corresponding to height, volume and area respectively. |
06 Bibliography
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- 李忠達。〈聲數同原：《周易時論合編》律學中的象數思維〉。《清華學報》52，第1期（2022）：41–77。 Link
- 童寯。〈比例〉。载《童寯文集》，第1卷，23。北京：中国建筑工业出版社，2000。原稿1930年。
- 刘敦桢。《刘敦桢全集》。第8卷，苏州古典园林。北京：中国建筑工业出版社，2007。Figures: Canli Garden · two garden boundaries · 389, fig. 811; Lingering Garden · Linquan Qishuo Hall · 126, fig. 166; Humble Administrator’s Garden · Yuanxiang Hall · 123, fig. 158(3); Master of the Nets Garden · Xiaoshan Conggui Xuan · 331, detail of fig. 687; Garden of Cultivation · main pond · 360, fig. 745; Canglang Pavilion Garden · Mingdao Hall · 323, fig. 664
- 夏昌世、莫伯治。《岭南庭园》。曾昭奋整理。北京：中国建筑工业出版社，2008。Figures: Qunxing Caotang · either side of the pavilion · 74, fig. 4-20; Three peaks · height and court-width comparison · 212–13, fig. 8-28; Fengyuan North Street · the two lower peaks · 213, fig. 8-28; Fengyuan North Street · Qinglong and main peak · 212–13, fig. 8-28; Fengyuan North Street · three peak heights · 212–13, fig. 8-28
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- Yu Hui and Chen Yingshi. “Theorizing ‘Natural Sound’: Ancient Chinese Music Theory and Its Contemporary Applications in the Study of Guqin Intonation.” In The Oxford Handbook of Music in China and the Chinese Diaspora, edited by Jonathan P. J. Stock and Yu Hui, 30–46. Oxford: Oxford University Press, 2023. Link
- UNESCO. “Guqin and Its Music.” Intangible Cultural Heritage. Accessed October 12, 2026. Link