{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UGKJ46NYDEWIBXH7NONLWMBZR7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ba66bd0e7decba7b866560fd4a1d45066f3582328d9a66eb0bd0183365d8adb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-12-12T02:26:15Z","title_canon_sha256":"51f30e83ba992ccf9d8d1d146626806af67c86cd8ec1614d814e732838eec2cf"},"schema_version":"1.0","source":{"id":"1612.03519","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.03519","created_at":"2026-05-17T23:56:58Z"},{"alias_kind":"arxiv_version","alias_value":"1612.03519v1","created_at":"2026-05-17T23:56:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.03519","created_at":"2026-05-17T23:56:58Z"},{"alias_kind":"pith_short_12","alias_value":"UGKJ46NYDEWI","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UGKJ46NYDEWIBXH7","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UGKJ46NY","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:c23897267f9c44a1aaae6a6d3460298830fe1fdd2a16d30e88d3716f864f8558","target":"graph","created_at":"2026-05-17T23:56:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study a generalization of the classical Riemannian Tonnetz to N-tone equally tempered scales (for all N) and arbitrary triads. We classify all the spaces that result. The torus turns out to be the most common possibility, especially as N grows. Other spaces include 2-simplices, tetrahedra boundaries, and the harmonic strip (in both its cylinder and Mobius band variants). The final and most exotic space we find is something we call a `circle of tetrahedra boundaries'. These are the Tonnetze for spaces of triads which contain a tritone. They are closely related to Peck's Klein bottle Tonnetz.","authors_text":"Michael J. Catanzaro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-12-12T02:26:15Z","title":"Generalized Tonnetze"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.03519","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:822fa1225b72f1f396f89507b8d54ca46cd3eae7c6ccf6034be53d7d716d202f","target":"record","created_at":"2026-05-17T23:56:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ba66bd0e7decba7b866560fd4a1d45066f3582328d9a66eb0bd0183365d8adb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-12-12T02:26:15Z","title_canon_sha256":"51f30e83ba992ccf9d8d1d146626806af67c86cd8ec1614d814e732838eec2cf"},"schema_version":"1.0","source":{"id":"1612.03519","kind":"arxiv","version":1}},"canonical_sha256":"a1949e79b8192c80dcff6b9abb30398fc4d88389feb219003bf1fd2609478ea2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a1949e79b8192c80dcff6b9abb30398fc4d88389feb219003bf1fd2609478ea2","first_computed_at":"2026-05-17T23:56:58.914485Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:58.914485Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4RwPvwQtFal4zaw+FM9eaKUz69NHrYbc2gG+cJ4+53s8vyddf8A0ShakKobpzZXZmIZ5jwadg7THgmaawnFZCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:58.915136Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.03519","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:822fa1225b72f1f396f89507b8d54ca46cd3eae7c6ccf6034be53d7d716d202f","sha256:c23897267f9c44a1aaae6a6d3460298830fe1fdd2a16d30e88d3716f864f8558"],"state_sha256":"3e691ba407fe836408d68aec1ebb7779db01c1b9a75ca41b4931aa759ba1a0b9"}