{"id":16913,"date":"2026-08-12T14:49:00","date_gmt":"2026-08-12T04:49:00","guid":{"rendered":"https:\/\/geoffreychen.com\/?p=16913"},"modified":"2026-08-12T23:18:11","modified_gmt":"2026-08-12T13:18:11","slug":"why-does-doubling-a-frequency-still-sound-like-the-same-note","status":"publish","type":"post","link":"https:\/\/geoffreychen.com\/zh\/why-does-doubling-a-frequency-still-sound-like-the-same-note\/","title":{"rendered":"\u9891\u7387\u589e\u52a0\u4e00\u500d\uff0c\u4e3a\u4ec0\u4e48\u4ecd\u7136\u50cf\u201c\u540c\u4e00\u4e2a\u97f3\u201d\uff1f"},"content":{"rendered":"<\/p>\n<p><em>\u300a\u542c\u89c1\u6570\u5b66\u300b\u00b7\u201c\u58f0\u97f3\u600e\u6837\u8fdb\u5165\u8eab\u4f53\u4e0e\u5fc3\u7075\u201d\u00b7 \u7b2c\u4e8c\u7bc7<\/em><\/p>\n<p>\u94a2\u7434\u4e0a\u4e2d\u592e C \u9644\u8fd1\u7684\u4e00\u4e2a\u97f3\u4e0e\u9ad8\u516b\u5ea6\u7684 C \u663e\u7136\u4e0d\u4e00\u6837\u3002\u4e00\u4e2a\u66f4\u9ad8\u3001\u66f4\u8f7b\u3001\u66f4\u660e\u4eae\uff1b\u53e6\u4e00\u4e2a\u66f4\u4f4e\u3001\u66f4\u539a\u3001\u66f4\u6709\u91cd\u91cf\u3002\u53ef\u662f\u53d7\u8fc7\u897f\u65b9\u97f3\u4e50\u8bad\u7ec3\u7684\u4eba\u901a\u5e38\u6beb\u4e0d\u72b9\u8c6b\u5730\u628a\u5b83\u4eec\u90fd\u53eb\u4f5c C\u3002\u5408\u5531\u4e2d\uff0c\u6210\u5e74\u4eba\u548c\u513f\u7ae5\u5373\u4f7f\u5531\u5728\u4e0d\u540c\u97f3\u533a\uff0c\u4e5f\u53ef\u80fd\u88ab\u8ba4\u4e3a\u5728\u5531\u201c\u540c\u4e00\u6761\u65cb\u5f8b\u201d\u3002\u5df4\u8d6b\u300a\u7b2c\u4e00\u65e0\u4f34\u594f\u5927\u63d0\u7434\u7ec4\u66f2\u300b\u524d\u594f\u66f2\u4e2d\u7684\u5206\u89e3\u548c\u5f26\u4e0d\u65ad\u8de8\u8d8a\u97f3\u533a\uff0cG \u5728\u4e0d\u540c\u516b\u5ea6\u51fa\u73b0\u65f6\u65e2\u6539\u53d8\u7ebf\u6761\u7684\u65b9\u5411\uff0c\u53c8\u4fdd\u6301\u8c03\u6027\u8eab\u4efd\u3002<\/p>\n<p>\u8fd9\u79cd\u7ecf\u9a8c\u5f88\u5bb9\u6613\u88ab\u4e00\u53e5\u8bdd\u89e3\u91ca\u5b8c\uff1a\u516b\u5ea6\u7684\u9891\u7387\u6bd4\u662f 2:1\uff0c\u6240\u4ee5\u4e24\u4e2a\u97f3\u5f53\u7136\u662f\u201c\u540c\u4e00\u4e2a\u97f3\u201d\u3002\u4f46\u201c\u5f53\u7136\u201d\u6b63\u662f\u9700\u8981\u8ffd\u95ee\u7684\u5730\u65b9\u3002\u9891\u7387\u7ffb\u500d\u53ea\u8bf4\u660e\u4e00\u79cd\u7269\u7406\u5173\u7cfb\uff1b\u628a\u4e24\u4e2a\u4e0d\u540c\u9ad8\u5ea6\u5f52\u5165\u540c\u4e00\u97f3\u7ea7\uff0c\u662f\u542c\u89c9\u4e0e\u97f3\u4e50\u4f53\u7cfb\u5b8c\u6210\u7684\u5206\u7c7b\u3002\u4e3a\u4ec0\u4e48 220\u3001440 \u548c 880 \u8d6b\u5179\u4e4b\u95f4\u7684\u5dee\u5f02\u53ef\u4ee5\u540c\u65f6\u88ab\u542c\u6210\u5de8\u5927\u53d8\u5316\u548c\u67d0\u79cd\u540c\u4e00\u6027\uff1f<\/p>\n<h2>\u516b\u5ea6\u4e0d\u662f\u76f8\u5dee\u4e00\u4e2a\u56fa\u5b9a\u6570\u503c\uff0c\u800c\u662f\u76f8\u4e58\u4e24\u500d<\/h2>\n<p>\u5047\u8bbe\u4e00\u4e2a\u97f3\u7684\u9891\u7387\u662f 220 \u8d6b\u5179\u3002\u5b83\u4e0a\u65b9\u4e00\u4e2a\u516b\u5ea6\u662f 440 \u8d6b\u5179\uff0c\u518d\u4e0a\u4e00\u4e2a\u516b\u5ea6\u662f 880 \u8d6b\u5179\u3002\u4e24\u6b21\u8de8\u8d8a\u7684\u9891\u7387\u5dee\u5206\u522b\u662f 220 \u548c 440\uff0c\u7edd\u5bf9\u5dee\u5e76\u4e0d\u76f8\u7b49\uff1b\u76f8\u7b49\u7684\u662f\u6bd4\u4f8b\uff1a\u6bcf\u6b21\u90fd\u4e58\u4ee5 2\u3002<\/p>\n<p>\u8fd9\u63d0\u793a\u6211\u4eec\uff0c\u97f3\u9ad8\u5173\u7cfb\u66f4\u9002\u5408\u653e\u5728\u5bf9\u6570\u5c3a\u5ea6\u4e0a\u7406\u89e3\u3002\u5728\u8fd9\u6837\u7684\u5c3a\u5ea6\u4e0a\uff0c\u76f8\u540c\u7684\u9891\u7387\u6bd4\u88ab\u8868\u793a\u4e3a\u76f8\u540c\u8ddd\u79bb\u3002\u82e5\u4ece 220 \u5230 440 \u4e0e\u4ece 440 \u5230 880 \u90fd\u7b97\u4e00\u4e2a\u516b\u5ea6\uff0c\u90a3\u4e48\u97f3\u7a0b\u5c31\u4e0d\u662f\u7528 <code>f\u2082 - f\u2081<\/code> \u8861\u91cf\uff0c\u800c\u662f\u7531 <code>f\u2082 \/ f\u2081<\/code> \u51b3\u5b9a\u3002\u897f\u65b9\u97f3\u4e50\u5e38\u7528\u97f3\u5206\u8868\u793a\u8fd9\u4e00\u8ddd\u79bb\uff1a<\/p>\n<p><code>\u97f3\u5206 = 1200 \u00d7 log\u2082(f\u2082 \/ f\u2081)<\/code><\/p>\n<p>\u6bd4\u4f8b\u4e3a 2 \u65f6\u6b63\u597d\u5f97\u5230 1200 \u97f3\u5206\u3002\u516c\u5f0f\u7684\u91cd\u8981\u6027\u4e0d\u5728\u4e8e\u8ba9\u542c\u8005\u8ba1\u7b97\uff0c\u800c\u5728\u4e8e\u63ed\u793a\u4e00\u79cd\u7ed3\u6784\uff1a\u6211\u4eec\u542c\u89c1\u7684\u97f3\u4e50\u8ddd\u79bb\u66f4\u63a5\u8fd1\u4e58\u6cd5\u4e16\u754c\uff0c\u800c\u4e0d\u662f\u76f4\u5c3a\u4e0a\u7684\u7b49\u5dee\u4e16\u754c\u3002<\/p>\n<p>\u8fd9\u79cd\u5bf9\u6570\u5173\u7cfb\u4e5f\u4e0e\u4eba\u7c7b\u611f\u5b98\u7684\u5e7f\u6cdb\u7279\u70b9\u76f8\u547c\u5e94\u3002\u611f\u77e5\u7cfb\u7edf\u9700\u8981\u5904\u7406\u8de8\u8d8a\u5f88\u5927\u8303\u56f4\u7684\u5f3a\u5ea6\u548c\u9891\u7387\uff0c\u6309\u6bd4\u4f8b\u7f16\u7801\u80fd\u628a\u5de8\u5927\u7684\u7269\u7406\u8303\u56f4\u538b\u7f29\u4e3a\u53ef\u7528\u5dee\u5f02\u3002\u4e0d\u8fc7\uff0c\u201c\u97f3\u9ad8\u5927\u81f4\u6309\u5bf9\u6570\u5c3a\u5ea6\u7ec4\u7ec7\u201d\u4ecd\u4e0d\u7b49\u4e8e\u201c\u6240\u6709\u4eba\u90fd\u5fc5\u7136\u628a\u516b\u5ea6\u5f53\u4f5c\u76f8\u540c\u97f3\u7ea7\u201d\u3002\u524d\u8005\u5173\u4e4e\u8fde\u7eed\u9ad8\u5ea6\u600e\u6837\u8868\u793a\uff0c\u540e\u8005\u5173\u4e4e\u600e\u6837\u5206\u7c7b\u3002<\/p>\n<h2>\u6cdb\u97f3\u91cd\u53e0\u4e3a\u516b\u5ea6\u76f8\u4f3c\u63d0\u4f9b\u4e86\u5f3a\u58f0\u5b66\u7ebf\u7d22<\/h2>\n<p>\u8bbe\u4e00\u4e2a\u7406\u60f3\u7684\u8c10\u6ce2\u590d\u5408\u97f3\u57fa\u9891\u4e3a 220 \u8d6b\u5179\uff0c\u5b83\u7684\u6210\u5206\u53ef\u4ee5\u5305\u62ec 220\u3001440\u3001660\u3001880\u30011100\u2026\u2026\u8d6b\u5179\u3002\u9ad8\u516b\u5ea6\u7684 440 \u8d6b\u5179\u97f3\uff0c\u5176\u7406\u60f3\u8c10\u6ce2\u6210\u5206\u5305\u62ec 440\u3001880\u30011320\u30011760\u2026\u2026\u8d6b\u5179\u3002\u7b2c\u4e8c\u7ec4\u4e2d\u7684\u8bb8\u591a\u6210\u5206\u5df2\u7ecf\u51fa\u73b0\u5728\u7b2c\u4e00\u7ec4\u91cc\uff1a\u9ad8\u97f3\u7684\u57fa\u9891\u662f\u4f4e\u97f3\u7684\u7b2c\u4e8c\u8c10\u6ce2\uff0c\u9ad8\u97f3\u7684\u7b2c\u4e8c\u8c10\u6ce2\u53c8\u662f\u4f4e\u97f3\u7684\u7b2c\u56db\u8c10\u6ce2\u3002<\/p>\n<p>\u8fd9\u79cd\u5d4c\u5957\u8ba9\u4e24\u4e2a\u58f0\u97f3\u7684\u9891\u8c31\u5177\u6709\u7279\u6b8a\u4eb2\u7f18\u6027\u3002\u5b83\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u516b\u5ea6\u5408\u594f\u5bb9\u6613\u5728\u67d0\u4e9b\u6761\u4ef6\u4e0b\u53d1\u751f\u878d\u5408\uff1a\u9ad8\u97f3\u7684\u591a\u9879\u6210\u5206\u53ef\u4ee5\u4e0e\u4f4e\u97f3\u6cdb\u97f3\u5bf9\u9f50\uff0c\u6574\u4f53\u5468\u671f\u4e5f\u5177\u6709\u7b80\u660e\u5173\u7cfb\u3002\u58f0\u5b66\u5e76\u6ca1\u6709\u5ba3\u5e03\u4e24\u4e2a\u9891\u7387\u201c\u540d\u79f0\u76f8\u540c\u201d\uff0c\u5374\u7ed9\u542c\u89c9\u7cfb\u7edf\u63d0\u4f9b\u4e86\u6bd4\u8bb8\u591a\u5176\u4ed6\u97f3\u7a0b\u66f4\u5f3a\u7684\u5171\u540c\u5468\u671f\u548c\u8c31\u7ebf\u91cd\u53e0\u3002<\/p>\n<p>\u4f46\u771f\u5b9e\u4e50\u5668\u4e0d\u662f\u7406\u60f3\u516c\u5f0f\u3002\u5404\u6cdb\u97f3\u5f3a\u5ea6\u4e0d\u540c\uff0c\u94a2\u7434\u5f26\u8fd8\u4f1a\u51fa\u73b0\u975e\u8c10\u6027\uff0c\u949f\u58f0\u7b49\u58f0\u97f3\u7684\u5206\u97f3\u4e5f\u672a\u5fc5\u63a5\u8fd1\u6574\u6570\u500d\u3002\u5982\u679c\u516b\u5ea6\u8eab\u4efd\u53ea\u9760\u5b8c\u7f8e\u6cdb\u97f3\u5217\uff0c\u5b83\u5728\u73b0\u5b9e\u4e2d\u5e94\u5f53\u6bd4\u6211\u4eec\u7ecf\u9a8c\u5230\u7684\u8106\u5f31\u5f97\u591a\u3002\u542c\u89c9\u7cfb\u7edf\u4f1a\u5bb9\u5fcd\u504f\u5dee\u3001\u5229\u7528\u65f6\u95f4\u7ebf\u7d22\uff0c\u5e76\u628a\u58f0\u97f3\u653e\u5165\u65cb\u5f8b\u548c\u8c03\u6027\u8bed\u5883\u3002\u7269\u7406\u76f8\u4f3c\u662f\u6761\u4ef6\u4e4b\u4e00\uff0c\u4e0d\u662f\u5168\u90e8\u673a\u5236\u3002<\/p>\n<h2>\u201c\u66f4\u9ad8\u201d\u4e0e\u201c\u540c\u540d\u201d\u662f\u4e24\u4e2a\u53ef\u4ee5\u540c\u65f6\u6210\u7acb\u7684\u7ef4\u5ea6<\/h2>\n<p>\u97f3\u4e50\u5fc3\u7406\u5b66\u5e38\u628a\u97f3\u9ad8\u7ecf\u9a8c\u533a\u5206\u4e3a<strong>\u97f3\u9ad8\u9ad8\u5ea6<\/strong>\uff08pitch height\uff09\u4e0e<strong>\u97f3\u7ea7\u8272\u5f69<\/strong>\uff08pitch chroma\uff09\u3002\u4ece C4 \u5230 C5\uff0c\u9ad8\u5ea6\u660e\u663e\u4e0a\u5347\uff1b\u5728\u4ee5\u5341\u4e8c\u97f3\u7ea7\u7ec4\u7ec7\u7684\u97f3\u4e50\u4e2d\uff0cchroma \u53c8\u56de\u5230 C\u3002\u628a\u97f3\u9ad8\u60f3\u8c61\u6210\u87ba\u65cb\u6bd4\u76f4\u7ebf\u66f4\u5408\u9002\uff1a\u87ba\u65cb\u4e0d\u65ad\u4e0a\u5347\uff0c\u4f46\u6bcf\u7ed5\u4e00\u5708\u4f1a\u56de\u5230\u540c\u4e00\u65b9\u5411\u3002<\/p>\n<p>\u8fd9\u4e2a\u6bd4\u55bb\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u201c\u540c\u4e00\u4e2a\u97f3\u201d\u5fc5\u987b\u52a0\u5f15\u53f7\u3002\u4e24\u4e2a C 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href=\"https:\/\/imslp.org\/wiki\/Cello_Suite_No.1_in_G_major%2C_BWV_1007_%28Bach%2C_Johann_Sebastian%29\" target=\"_blank\" rel=\"noopener\">\u73b0\u5b58\u8d44\u6599\u4e0e\u4e50\u8c31<\/a>\u663e\u793a\u7684\u4e0d\u662f\u628a\u4e00\u4e2a G \u7b80\u5355\u590d\u5236\u5230\u66f4\u9ad8\u5904\uff0c\u800c\u662f\u5229\u7528\u97f3\u533a\u6539\u53d8\u7ebf\u6761\u7684\u91cd\u529b\uff1a\u4f4e\u97f3\u4e3a\u548c\u58f0\u5960\u57fa\uff0c\u9ad8\u97f3\u6253\u5f00\u7a7a\u95f4\uff0c\u8fd4\u56de\u7684\u97f3\u7ea7\u53c8\u8ba9\u542c\u8005\u4fdd\u6301\u65b9\u5411\u3002\u516b\u5ea6\u7b49\u4ef7\u4f7f\u6750\u6599\u80fd\u8de8\u97f3\u533a\u4fdd\u6301\u8eab\u4efd\uff1b\u516b\u5ea6\u4e0d\u7b49\u540c\u5219\u4f7f\u8fd9\u79cd\u8de8\u8d8a\u503c\u5f97\u542c\u3002<\/p>\n<h2>\u8de8\u6587\u5316\u7814\u7a76\u8ba9\u201c\u666e\u904d\u89c4\u5f8b\u201d\u53d8\u5f97\u66f4\u8c28\u614e<\/h2>\n<p>\u516b\u5ea6\u5728\u4e16\u754c\u8bb8\u591a\u97f3\u4e50\u4f20\u7edf\u4e2d\u6781\u5176\u5e38\u89c1\uff0c\u513f\u7ae5\u7684\u6b4c\u5531\u548c\u90e8\u5206\u52a8\u7269\u7814\u7a76\u4e5f\u88ab\u7528\u6765\u8ba8\u8bba\u5b83\u662f\u5426\u5177\u6709\u751f\u7269\u57fa\u7840\u3002\u7136\u800c\uff0c\u201c\u5e38\u89c1\u201d\u201c\u5bb9\u6613\u5f62\u6210\u201d\u4e0e\u201c\u6240\u6709\u6587\u5316\u3001\u6240\u6709\u4efb\u52a1\u4e2d\u5b8c\u5168\u76f8\u540c\u201d\u4e0d\u662f\u540c\u4e00\u4e2a\u547d\u9898\u3002<\/p>\n<p>\u4e00\u9879\u6bd4\u8f83\u7f8e\u56fd\u53c2\u4e0e\u8005\u4e0e\u4e9a\u9a6c\u5b59 Tsimane\u2019 \u53c2\u4e0e\u8005\u6b4c\u5531\u518d\u73b0\u7684\u7814\u7a76\uff0c\u53d1\u73b0\u4e24\u7ec4\u90fd\u663e\u793a\u51fa\u97f3\u9ad8\u5bf9\u6570\u5c3a\u5ea6\u4e0e\u9ad8\u9891\u97f3\u9ad8\u9650\u5236\u7684\u8bc1\u636e\uff0c\u5374\u6ca1\u6709\u5728 Tsimane\u2019 \u53c2\u4e0e\u8005\u4e2d\u89c2\u5bdf\u5230\u4e0e\u7f8e\u56fd\u53c2\u4e0e\u8005\u540c\u6837\u7684\u516b\u5ea6\u7b49\u4ef7\u8868\u73b0\u3002\u7814\u7a76\u8005\u56e0\u6b64\u63d0\u51fa\uff0c\u516b\u5ea6\u7b49\u4ef7\u53ef\u80fd\u4f9d\u8d56\u7279\u5b9a\u97f3\u4e50\u4f53\u7cfb\u4e2d\u7684\u7ecf\u9a8c\uff0c\u81f3\u5c11\u4e0d\u80fd\u7531\u73b0\u6709\u897f\u65b9\u5b9e\u9a8c\u76f4\u63a5\u5ba3\u5e03\u4e3a\u5b8c\u5168\u666e\u904d\u3002<a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC9907018\/\" target=\"_blank\" rel=\"noopener\">\u8fd9\u9879\u8de8\u6587\u5316\u7814\u7a76<\/a>\u7684\u4ef7\u503c\u4e0d\u5728\u4e8e\u8bc1\u660e\u516b\u5ea6\u201c\u53ea\u662f\u6587\u5316\u865a\u6784\u201d\uff0c\u800c\u5728\u4e8e\u62c6\u5f00\u51e0\u4e2a\u5bb9\u6613\u6df7\u5408\u7684\u5c42\u6b21\uff1a\u751f\u7406\u7cfb\u7edf\u7684\u8303\u56f4\u3001\u5bf9\u6bd4\u4f8b\u7684\u7f16\u7801\u3001\u5bf9\u516b\u5ea6\u5173\u7cfb\u7684\u654f\u611f\uff0c\u4ee5\u53ca\u628a\u8de8\u516b\u5ea6\u97f3\u5f52\u5165\u540c\u4e00\u7c7b\u522b\u7684\u884c\u4e3a\uff0c\u5e76\u975e\u4e00\u4ef6\u4e8b\u3002<\/p>\n<p>\u540e\u7eed\u7814\u7a76\u4ecd\u5728\u68c0\u9a8c\u513f\u7ae5\u3001\u975e\u4eba\u52a8\u7269\u3001\u97f3\u4e50\u8bad\u7ec3\u548c\u5b9e\u9a8c\u4efb\u52a1\u4e4b\u95f4\u7684\u5dee\u5f02\u3002\u6700\u7a33\u59a5\u7684\u5224\u65ad\u662f\uff1a2:1 \u63d0\u4f9b\u975e\u5e38\u663e\u8457\u7684\u7269\u7406\u4e0e\u77e5\u89c9\u7ebf\u7d22\uff1b\u8bb8\u591a\u97f3\u4e50\u6587\u5316\u5229\u7528\u5b83\uff1b\u800c\u201c\u540c\u4e00\u4e2a\u97f3\u201d\u8fd9\u79cd\u660e\u786e\u5206\u7c7b\u8fd8\u4f1a\u88ab\u8bed\u8a00\u3001\u4e50\u5668\u3001\u8c03\u5f8b\u548c\u957f\u671f\u8046\u542c\u5851\u9020\u3002<\/p>\n<h2>\u79fb\u8c03\u63ed\u793a\u4e86\u76f8\u5bf9\u5173\u7cfb\u600e\u6837\u80dc\u8fc7\u7edd\u5bf9\u9891\u7387<\/h2>\n<p>\u5982\u679c\u628a\u4e00\u9996\u719f\u6089\u65cb\u5f8b\u6574\u4f53\u63d0\u9ad8\u4e00\u4e2a\u516b\u5ea6\uff0c\u5927\u591a\u6570\u4eba\u4ecd\u80fd\u8ba4\u51fa\u5b83\u3002\u751a\u81f3\u6574\u4f53\u63d0\u9ad8\u4e00\u4e2a\u4e0d\u662f\u516b\u5ea6\u7684\u97f3\u7a0b\uff0c\u65cb\u5f8b\u4e5f\u5e38\u4fdd\u6301\u8eab\u4efd\u3002\u8fd9\u8bf4\u660e\u97f3\u4e50\u8bc6\u522b\u5e76\u4e0d\u4e3b\u8981\u4f9d\u8d56\u6bcf\u4e2a\u97f3\u7684\u7edd\u5bf9\u9891\u7387\uff0c\u800c\u4f9d\u8d56\u97f3\u4e0e\u97f3\u4e4b\u95f4\u7684\u76f8\u5bf9\u65b9\u5411\u3001\u6bd4\u4f8b\u3001\u8282\u594f\u548c\u7ed3\u6784\u3002<\/p>\n<p>\u516b\u5ea6\u79fb\u8c03\u5c24\u5176\u5bb9\u6613\uff0c\u56e0\u4e3a\u5b83\u540c\u65f6\u4fdd\u7559\u97f3\u7a0b\u7ed3\u6784\u4e0e\u97f3\u7ea7\u540d\u79f0\u3002\u53ef\u662f\u6f14\u594f\u6548\u679c\u4e0d\u4f1a\u4fdd\u6301\u4e0d\u53d8\u3002\u628a\u5927\u63d0\u7434\u65cb\u5f8b\u63d0\u9ad8\u4e00\u4e2a\u516b\u5ea6\uff0c\u8eab\u4f53\u5171\u632f\u3001\u5f13\u5f26\u5173\u7cfb\u3001\u97f3\u8272\u548c\u53ef\u7528\u529b\u5ea6\u90fd\u4f1a\u6539\u53d8\uff1b\u628a\u94a2\u7434\u4f4e\u97f3\u79fb\u5230\u9ad8\u97f3\u533a\uff0c\u548c\u58f0\u57fa\u7840\u53ef\u80fd\u53d8\u5f97\u7a00\u8584\u3002\u6570\u5b66\u4fdd\u5b58\u4e86\u9891\u7387\u6bd4\uff0c\u97f3\u4e50\u5374\u91cd\u65b0\u5206\u914d\u4e86\u91cd\u91cf\u3001\u7a7a\u95f4\u548c\u89d2\u8272\u3002<\/p>\n<p>\u8fd9\u4e5f\u662f\u4f5c\u66f2\u5bb6\u4e0d\u4f1a\u628a\u516b\u5ea6\u5f53\u4f5c\u65e0\u610f\u4e49\u91cd\u590d\u7684\u539f\u56e0\u3002\u7ba1\u5f26\u4e50\u4e2d\u7684\u516b\u5ea6\u52a0\u500d\u53ef\u4ee5\u589e\u5f3a\u8f6e\u5ed3\uff0c\u5374\u4e0d\u7b49\u4e8e\u53ea\u628a\u58f0\u97f3\u201c\u653e\u5927\u201d\uff1b\u9ad8\u4f4e\u58f0\u90e8\u7684\u9891\u8c31\u3001\u65b9\u5411\u611f\u548c\u542c\u89c9\u5206\u7ec4\u4f1a\u5171\u540c\u6539\u53d8\u97f3\u54cd\u3002\u516b\u5ea6\u65e2\u662f\u7b49\u4ef7\u5173\u7cfb\uff0c\u4e5f\u662f\u914d\u5668\u8d44\u6e90\u3002<\/p>\n<h2>\u94a2\u7434\u4e0a\u7684\u516b\u5ea6\u8fd8\u4e0d\u662f\u4e25\u683c\u7684 2:1<\/h2>\n<p>\u5982\u679c\u516b\u5ea6\u7684\u5b9a\u4e49\u662f 2:1\uff0c\u94a2\u7434\u8c03\u5f8b\u4f3c\u4e4e\u53ea\u9700\u8ba9\u6240\u6709\u540c\u540d\u97f3\u51c6\u786e\u500d\u9891\u3002\u771f\u5b9e\u94a2\u7434\u5374\u5e38\u628a\u9ad8\u97f3\u533a\u516b\u5ea6\u8c03\u5f97\u7565\u5bbd\uff0c\u628a\u4f4e\u97f3\u533a\u4e5f\u5411\u5916\u4f38\u5c55\u3002\u539f\u56e0\u5c06\u5728\u7b2c\u4e03\u7bc7\u8be6\u7ec6\u8ba8\u8bba\uff1a\u6709\u521a\u5ea6\u7684\u7434\u5f26\u4ea7\u751f\u7684\u5206\u97f3\u4f1a\u6bd4\u7406\u60f3\u6574\u6570\u500d\u7565\u9ad8\uff0c\u8c03\u5f8b\u5e08\u5fc5\u987b\u8ba9\u4e0d\u540c\u97f3\u7684\u5b9e\u9645\u5206\u97f3\u5c3d\u91cf\u534f\u8c03\u3002\u5bf9\u94a2\u7434\u800c\u8a00\uff0c\u542c\u8d77\u6765\u5408\u9002\u7684\u516b\u5ea6\u53ef\u80fd\u5728\u57fa\u9891\u4e0a\u7565\u5927\u4e8e 2:1\u3002<\/p>\n<p>\u8fd9\u4e0d\u662f\u516b\u5ea6\u6982\u5ff5\u88ab\u63a8\u7ffb\uff0c\u800c\u662f\u8bf4\u660e\u542c\u89c9\u5224\u65ad\u9488\u5bf9\u590d\u5408\u58f0\u97f3\uff0c\u4e0d\u53ea\u9488\u5bf9\u9891\u7387\u8868\u4e0a\u7684\u57fa\u9891\u3002\u4e00\u4e2a\u7406\u8bba\u5173\u7cfb\u8fdb\u5165\u6750\u6599\u4e16\u754c\u4ee5\u540e\uff0c\u9700\u8981\u5728\u591a\u4e2a\u5206\u97f3\u3001\u591a\u4e2a\u97f3\u533a\u548c\u6574\u4ef6\u4e50\u5668\u4e4b\u95f4\u5b9e\u73b0\u3002\u6570\u5b66\u7ed9\u51fa\u4e2d\u5fc3\u5173\u7cfb\uff0c\u4e50\u5668\u8feb\u4f7f\u5b9e\u8df5\u8005\u89e3\u91ca\u4f55\u8c13\u201c\u5b9e\u73b0\u201d\u3002<\/p>\n<h2>\u516b\u5ea6\u540c\u4e00\u6027\u662f\u4e00\u79cd\u5728\u5dee\u5f02\u4e2d\u4fdd\u6301\u5173\u7cfb\u7684\u80fd\u529b<\/h2>\n<p>\u9891\u7387\u7ffb\u500d\u540e\uff0c\u6211\u4eec\u542c\u89c1\u7684\u4e0d\u662f\u539f\u97f3\u539f\u6837\u56de\u6765\u3002\u8033\u8717\u54cd\u5e94\u4f4d\u7f6e\u6539\u53d8\uff0c\u97f3\u8272\u4e0e\u529b\u5ea6\u6761\u4ef6\u6539\u53d8\uff0c\u97f3\u4e50\u89d2\u8272\u4e5f\u53ef\u80fd\u6539\u53d8\u3002\u4ecd\u7136\u628a\u4e24\u4e2a\u97f3\u5f52\u4e3a\u540c\u4e00\u97f3\u7ea7\uff0c\u662f\u56e0\u4e3a\u542c\u89c9\u7cfb\u7edf\u80fd\u5728\u5de8\u5927\u9ad8\u5ea6\u5dee\u5f02\u4e2d\u6293\u4f4f\u7a33\u5b9a\u5173\u7cfb\uff0c\u800c\u97f3\u4e50\u6587\u5316\u53c8\u628a\u8fd9\u79cd\u5173\u7cfb\u4fdd\u5b58\u4e3a\u540d\u79f0\u3001\u952e\u76d8\u5e03\u5c40\u3001\u8bb0\u8c31\u4e0e\u548c\u58f0\u529f\u80fd\u3002<\/p>\n<p>\u56e0\u6b64\uff0c2:1 \u65e2\u4e0d\u662f\u65e0\u5173\u7d27\u8981\u7684\u5de7\u5408\uff0c\u4e5f\u4e0d\u662f\u5168\u90e8\u7b54\u6848\u3002\u5b83\u63d0\u4f9b\u5171\u540c\u5468\u671f\u3001\u6cdb\u97f3\u5d4c\u5957\u548c\u5bb9\u6613\u5229\u7528\u7684\u7ed3\u6784\uff1b\u8eab\u4f53\u51b3\u5b9a\u8fd9\u4e9b\u7ebf\u7d22\u600e\u6837\u88ab\u7f16\u7801\uff1b\u5b66\u4e60\u51b3\u5b9a\u54ea\u4e9b\u76f8\u4f3c\u88ab\u8d4b\u4e88\u5206\u7c7b\u6743\uff1b\u4f5c\u54c1\u5219\u5229\u7528\u76f8\u540c\u4e0e\u4e0d\u540c\u4e4b\u95f4\u7684\u5f20\u529b\u521b\u9020\u65b9\u5411\u3002<\/p>\n<p>\u542c\u5df4\u8d6b\u5927\u63d0\u7434\u7ec4\u66f2\u65f6\uff0c\u53ef\u4ee5\u8bd5\u7740\u4e0d\u53ea\u8ba4\u51fa\u201c\u8fd9\u53c8\u662f G\u201d\u3002\u8fd8\u8981\u542c\u540c\u540d\u97f3\u6bcf\u6b21\u56de\u6765\u65f6\u5904\u5728\u4ec0\u4e48\u9ad8\u5ea6\u3001\u7531\u54ea\u6839\u5f26\u53d1\u51fa\u3001\u627f\u62c5\u4f4e\u97f3\u57fa\u7840\u8fd8\u662f\u4e0a\u65b9\u5f20\u529b\u3002\u516b\u5ea6\u771f\u6b63\u6709\u97f3\u4e50\u610f\u4e49\u7684\u5730\u65b9\uff0c\u4e0d\u662f\u5b83\u8ba9\u5dee\u5f02\u6d88\u5931\uff0c\u800c\u662f\u5b83\u8ba9\u4e00\u4e2a\u5173\u7cfb\u7a7f\u8fc7\u5dee\u5f02\u4ecd\u7136\u53ef\u4ee5\u88ab\u8ba4\u51fa\u3002<\/p>\n<h3>\u4e3b\u8981\u8d44\u6599\u4e0e\u8fdb\u4e00\u6b65\u8046\u542c<\/h3>\n<ul>\n<li><a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC9907018\/\" target=\"_blank\" rel=\"noopener\">Jacoby \u7b49\uff1a\u8de8\u6587\u5316\u97f3\u9ad8\u77e5\u89c9\u4e2d\u7684\u666e\u904d\u4e0e\u975e\u666e\u904d\u7279\u5f81<\/a><\/li>\n<li><a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC3481156\/\" target=\"_blank\" rel=\"noopener\">Oxenham\uff1a\u300aPitch Perception\u300b<\/a><\/li>\n<li><a href=\"https:\/\/imslp.org\/wiki\/Cello_Suite_No.1_in_G_major%2C_BWV_1007_%28Bach%2C_Johann_Sebastian%29\" target=\"_blank\" rel=\"noopener\">\u5df4\u8d6b\u300a\u7b2c\u4e00\u65e0\u4f34\u594f\u5927\u63d0\u7434\u7ec4\u66f2\u300b\u4e50\u8c31\u4e0e\u5f55\u97f3\u8d44\u6599<\/a><\/li>\n<\/ul>\n<p><strong>\u7ee7\u7eed\u9605\u8bfb\uff1a<\/strong><a href=\"https:\/\/geoffreychen.com\/zh\/category\/piano\/hearing-mathematics\/\">\u6d4f\u89c8\u300a\u542c\u89c1\u6570\u5b66\u300b\u7cfb\u5217\u6587\u7ae0<\/a><\/p>\n<p>\u5982\u679c\u4f60\u4e5f\u60f3\u628a\u8fd9\u4e9b\u5173\u4e8e\u8046\u542c\u3001\u7406\u89e3\u4e0e\u7ec3\u4e60\u7684\u60f3\u6cd5\u5e26\u5230\u7434\u952e\u4e0a\uff0c\u53ef\u4ee5\u8bd5\u8bd5\u6211\u5f00\u53d1\u7684 <a href=\"https:\/\/apps.apple.com\/us\/app\/scoreflow\/id6796205656\" rel=\"noopener\" target=\"_blank\">ScoreFlow<\/a>\uff0c\u8ba9\u4e50\u8c31\u9605\u8bfb\u4e0e\u65e5\u5e38\u7ec3\u4e60\u8854\u63a5\u5f97\u66f4\u81ea\u7136\u3002<\/p>\n<p>","protected":false},"excerpt":{"rendered":"","protected":false},"author":17897162,"featured_media":0,"comment_status":"closed","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":true,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_geoff_meta_description_en":"Explains why octave-doubled tones can feel identical despite different frequencies, focusing on logarithmic pitch, harmonic overlap, and musical categorisation shaped by culture and instruments.","_geoff_meta_description_zh":"\u9610\u660e\u4e3a\u4f55\u516b\u5ea6\u500d\u9891\u7684\u97f3\u867d\u9891\u7387\u4e0d\u540c\u4ecd\u50cf\u540c\u4e00\u4e2a\u97f3\uff1a\u4ece\u5bf9\u6570\u5316\u97f3\u9ad8\u3001\u6cdb\u97f3\u8c31\u7ebf\u91cd\u53e0\u5230\u7531\u6587\u5316\u4e0e\u4e50\u5668\u5851\u9020\u7684\u5206\u7c7b\u65b9\u5f0f\u3002","_wpcom_ai_launchpad_first_post":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[787296608],"tags":[787296626,787296646,787296617,787296619,787296615,787296620,787296614,787296623],"class_list":["post-16913","post","type-post","status-publish","format-standard","hentry","category-hearing-mathematics","tag-hearing-mathematics","tag-how-sound-enters-the-body-and-mind","tag-logarithmic-pitch","tag-octave-equivalence","tag-octave-equivalence-zh","tag-hearing-mathematics-zh","tag-how-sound-enters-the-body-and-mind-zh","tag-logarithmic-pitch-zh",""],"translatepress_titles":{"en_AU":"Why 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