Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1508.07739

Transposition of Notations in Just Intonation

David Ryan
View original source ↗Read source document ↗Export metadata ↓

Abstract

A notation system was previously presented which can notate any rational frequency in free Just Intonation. Transposition of music is carried out by multiplying each member of a set of frequencies by a single frequency. Transposition of JI notations up by a fixed amount requires multiplication to be defined for any two notations. Transposition down requires inversion to be defined for any notation, which allows division to also be defined for any two notations. Each notation splits into four components which in decreasing size order are octave, diatonic scale note, sharps or flats, rational comma adjustment. Multiplication can be defined for each of the four notation components. Since rational number multiplication is commutative, this leads to a definition of multiplication for frequencies and thus notations. Examples of notation inversion and multiplication are given. Examples of transposing melodies are given. These are checked for accuracy using the rational numbers which each notation represents. Calculation shortcuts are considered which make notation operations quicker to carry out by hand. A question regarding whether rational commas should be extended from 5-rough rational numbers to all rational numbers is considered which would greatly simplify notation multiplication. This approach is rejected since it leads to confusion about octave number. The four component notation system is recommended instead. Extensions to computer notation systems and stave representations are briefly mentioned.

Explore related subjects

cs.SD

Keep this discovery

BibTeXRIS

David Ryan. 2016-12-14. Transposition of Notations in Just Intonation. https://arxiv.org/abs/1508.07739

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Xiaomi-CocktailASR-1 Technical Report

Recently, large language model (LLM) based ASR models have achieved significant progress, yet they generally lack support for multi-speaker scenarios, where the cocktail party problem remains a critical bottleneck for further advancing ASR. Existing TS-ASR methods, including end-to-end architectures with speaker embeddings and latest LLM-based explorations suffer from degraded single-speaker performance and the inability to reject when the target speaker is absent. In this paper, we propose Xiaomi-CocktailASR-1, an LLM-based end-to-end TS-ASR architecture. By utilizing reference speech as voiceprint prompts, it directly transcribes the target speaker's speech without requiring speech separation. Xiaomi-CocktailASR-1 maintains competitive performance in single-speaker scenarios, comparable to mainstream ASR models. It also features a negative sample rejection capability, outputting empty text when the target speaker is absent from the mixed speech. Additionally, Xiaomi-CocktailASR-1 supports a Chain-of-Thought (CoT) reasoning mode to provide explicit reasoning steps. Extensive experiments on various synthetic and real-world multispeaker benchmarks demonstrate that Xiaomi-CocktailASR-1 achieves state-of-the-art performance, effectively addressing the cocktail party problem through a unified architecture that balances multispeaker and single-speaker recognition accuracy, along with rejection capability.

cs.SD↗
arXiv2026

EConv-TasNet: Efficient Conv-TasNet for Effective Speech Separation

Conv-TasNet has served as a strong baseline for time-domain speech separation, and many studies have extended it with advanced architectures such as dual-path networks, U-Nets, and attention mechanisms. However, these methods often introduce high computational cost and complexity, limiting their deployment in resource-constrained scenarios. To address this issue, we propose eConv-TasNet, an efficient variant of Conv-TasNet that improves both effectiveness and efficiency without relying on resource-intensive modules. The proposed model consists of a group-wise early-splitting (GES) module and a multi-group feature aggregation (MGFA) module. GES generates discriminative speaker embeddings at intermediate stages, while MGFA progressively aggregates these group-level representations for refined mask estimation. Experimental results show that eConv-TasNet reduces model size by 22.4%, accelerates inference by 18.9%, and improves SI-SNRi by 14.0%-28.0% across three public benchmarks. Moreover, it achieves competitive performance compared with state-of-the-art methods while requiring significantly fewer parameters and lower inference cost. These results demonstrate a favorable efficiency-effectiveness trade-off for edge deployment.

cs.SD↗